{"id":15180,"date":"2024-01-01T08:58:20","date_gmt":"2024-01-01T03:28:20","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=15180"},"modified":"2024-01-02T14:49:48","modified_gmt":"2024-01-02T09:19:48","slug":"envision-math-common-core-grade-7-answer-key-topic-1","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-7-answer-key-topic-1\/","title":{"rendered":"enVision Math Common Core Grade 7 Answer Key Topic 1 Rational Number Operations"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 7 Answer Key<\/a>\u00a0Topic 1 Rational Number Operations<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 7th Grade Answers Key Topic 1 Rational Number Operations<\/h2>\n

Topic 1 Essential Question<\/h3>\n

How can the properties of operations be used to solve problems involving integers and rational numbers?<\/p>\n

3-ACT MATH<\/strong><\/p>\n

\"Envision
\nWin Some, Lose Some Are you the kind of person who has a lot of knowledge about history, literature, or science? What about pop culture, music, sports, and current events? Some schools have an academic bowl team that competes in tournaments against other schools. The teams are made up of members with strengths in different subject areas.
\nIn any quiz competition, it’s important to understand the rules and scoring. Think about this during the 3-Act Mathematical Modeling lesson.
\n\"Envision<\/p>\n

Topic 1 enVision STEM Project<\/h3>\n

Did You Know?
\nThe lowest recorded temperature in the world, -136\u00b0F (-93.2\u00b0C), occurred in Antarctica.
\n\"Envision
\nThe highest recorded temperature in the world, 134\u00b0F (56.7\u00b0C), occurred in Death Valley, California.
\n\"Envision<\/p>\n

\"Envision The Celsius scale (\u00b0C) is commonly used for temperature measurement in most of the world.
\n\"Envision Only a small number of nations, including the United States, regularly use the Fahrenheit scale (\u00b0F).<\/p>\n

Windchill, based on the rate of heat loss from exposed skin, can make it feel colder outside than the actual air temperature indicates. Wind chills in some places of the world can dip into the – 100\u00b0F range.
\n\"Envision<\/p>\n

Your Task: How Cold is Too Cold?
\nThere are many regions of the world with cold temperatures and extreme conditions. How do the inhabitants of these regions adapt and thrive? Do conditions exist that make regions too cold for human living? You and your classmates will explore and describe the habitability of regions with low temperatures.
\n\"Envision<\/p>\n

Topic 1 Get Ready!<\/h3>\n

Review What You Know!<\/strong><\/p>\n

Vocabulary<\/strong>
\nChoose the best term from the box. Write it on the blank.<\/p>\n

    \n
  • absolute value<\/li>\n
  • Associative Property<\/li>\n
  • Commutative Property<\/li>\n
  • Distributive Property<\/li>\n
  • integers<\/li>\n
  • rational number<\/li>\n<\/ul>\n

    \"Rational<\/p>\n

    Question 1.
    \nThe __________ explains why a \u00d7 b = b \u00d7 a and a + b = b + a.<\/p>\n

    Answer:
    \nThe commutative property explains why a x b = b x a and a + b = b + a.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe commutative property explains why a x b = b x a and a + b = b + a.
    \nthe commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.
    \nthe property holds for addition and multiplication, but not for subtraction and division.
    \nfor example:
    \n4 + 3 = 7.
    \n3 + 4 = 7.
    \n2 x 6 = 12.
    \n6 x 2 = 12.<\/p>\n

    Question 2.
    \nThe __________ of -6 is 6, because it is 6 units from zero on the number line.<\/p>\n

    Answer:
    \nThe absolute value of -6 is 6, because it is 6 units from zero on the number line.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe absolute value of -6 is 6, because it is 6 units from zero on the number line.
    \nfor example:
    \n6 and -6 are at the same distance from zero on the number line.<\/p>\n

    Question 3.
    \nThe number \\(\\frac{5}{3}\\) is a _________ because 5 and 3 are integers and 3 \u2260 0.<\/p>\n

    Answer:
    \nThe number 5\/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe number 5\/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
    \nfor example:
    \n3, 5, 6, and 7 are rational numbers.
    \nrational numbers are also integers.<\/p>\n

    Question 4.
    \nThe set of _________ consists of the counting numbers, their opposites, and zero.<\/p>\n

    Answer:
    \nThe set of integers consists of the counting numbers, their opposites, and zero.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe set of integers consists of the counting numbers, their opposites, and zero.
    \nIntegers are counting numbers, their opposites, and zero.
    \nfor example:
    \nintegers can either be negative, positive, or zero.
    \n1, -1, 2, -2, and 3, -3.<\/p>\n

    Question 5.
    \nThe sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the _________<\/p>\n

    Answer:
    \nThe sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.
    \nthe distributive property is sometimes called the distributive law of multiplication and division.
    \nfor example:
    \n4(x – 3) = 20.
    \n4x – 12 = 20.
    \n4x – 12 + 12 = 20 + 12.
    \n4x = 32.
    \nx = 32\/4.
    \nx = 8.<\/p>\n

    \"Rational<\/p>\n

    Question 6.
    \nIf you evaluate n \u00d7 (v + z) by writing it as (n \u00d7 y) + (n \u00d7 z), you have used the __________.<\/p>\n

    Answer:
    \nIf you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIf you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.
    \nfor example:
    \n4(x – 3) = 20.
    \n4x – 12 = 20.
    \n4x – 12 + 12 = 20 + 12.
    \n4x = 32.
    \nx = 32\/4.
    \nx = 8.<\/p>\n

    Add and Subtract Fractions and Decimals<\/strong><\/p>\n

    Add or subtract.
    \nQuestion 7.
    \n\\(2 \\frac{1}{3}+6 \\frac{3}{5}\\)<\/p>\n

    Answer:
    \n2(1\/3) + 6(3\/5) = 8.9.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n2(1\/3) + 6(3\/5).
    \n7\/3 + 33\/5.
    \n2.3 + 6.6.
    \n8.9.
    \n2(1\/3) + 6(3\/5) = 8.9.<\/p>\n

    Question 8.
    \n\\(9 \\frac{1}{10}-4 \\frac{3}{4}\\)<\/p>\n

    Answer:
    \n9(1\/10) – 4(3\/4) = 4.35.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n9(1\/10) – 4(3\/4).
    \n91\/10 – 19\/4.
    \n9.1 – 4.75.
    \n4.35.
    \n9(1\/10) – 4(3\/4) = 4.35.<\/p>\n

    Question 9.
    \n19.86 + 7.091<\/p>\n

    Answer:
    \n19.86 + 7.091 = 26.951.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nadd the numbers.
    \n19.86 + 7.091.
    \n19.86 + 7.091 = 26.951.<\/p>\n

    Question 10.
    \n57 – 10.62<\/p>\n

    Answer:
    \n57 – 10.62 = 46.38.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nsubtract the numbers.
    \n57 – 10.62.
    \n46.38.
    \n57 – 10.62 = 46.38.<\/p>\n

    Multiply and Divide Fractions and Decimals<\/strong><\/p>\n

    Multiply or divide.
    \nQuestion 11.
    \n4.08 \u00d7 29.7<\/p>\n

    Answer:
    \n4.08 x 29.7 = 121.176.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nmultiply the numbers.
    \n4.08 x 29.7 = 121.176.<\/p>\n

    Question 12.
    \n15,183.3 \u00f7 473<\/p>\n

    Answer:
    \n15,183.3 \/ 473 = 32.1.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \ndivide the numbers.
    \n15,183.3 \/ 473 = 32.1.<\/p>\n

    Question 13.
    \n\\(\\frac{15}{16} \\times 9 \\frac{1}{5}\\)<\/p>\n

    Answer:
    \n15\/16 x 41\/5 = 7.6875.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nmultiply the numbers.
    \n15\/16 x 41\/5.
    \n0.9375 x 8.2.
    \n7.6875.
    \n15\/16 x 41\/5 = 7.6875.<\/p>\n

    \"Rational<\/p>\n

    Question 14.
    \n\\(4 \\frac{7}{9} \\div 1 \\frac{7}{12}\\)<\/p>\n

    Answer:
    \n4(7\/9) \/ 1(7\/12) = 2.974.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \ndivide the numbers.
    \n4(7\/9) \/ 1(7\/12).
    \n43\/9 \/ 19\/12.
    \n4.7 \/ 1.58.
    \n2.974.
    \n4(7\/9) \/ 1(7\/12) = 2.974.<\/p>\n

    Question 15.
    \nByron has \\(1 \\frac{7}{10}\\) kilograms of black pepper. He uses of the pepper and splits it between 7 pepper shakers. How much pepper will be in each shaker?
    \nA. \\(\\frac{119}{80} \\mathrm{~kg}\\)
    \nB. \\(\\frac{1}{8} \\mathrm{~kg}\\)
    \nC. 1.4125 kg
    \nD. \\(\\frac{17}{80} \\mathrm{~kg}\\)<\/p>\n

    Answer:
    \nThe quantity of pepper will be in each shaker = 1\/8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nByron has \\(1 \\frac{7}{10}\\) kilograms of black pepper.
    \nHe uses the pepper and splits it between 7 pepper shakers.
    \n1(7\/10).
    \n17\/10 = 1.7.<\/p>\n

    Language Development<\/strong>
    \nFill in the word map with new terms, definitions, and supporting examples or illustrations.
    \n\"Envision<\/p>\n

    Answer:
    \nDistributive property.
    \ncommutative property.
    \nInteger.
    \nRational number.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe commutative property explains why a x b = b x a and a + b = b + a.
    \nthe commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer.
    \nthe property holds for addition and multiplication, but not for subtraction and division.
    \nfor example:
    \n4 + 3 = 7.
    \n3 + 4 = 7.
    \n2 x 6 = 12.
    \n6 x 2 = 12.
    \nThe number 5\/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
    \nfor example:
    \n3, 5, 6, and 7 are rational numbers.
    \nrational numbers are also integers.
    \nThe set of integers consists of the counting numbers, their opposites, and zero.
    \nIntegers are counting numbers, their opposites, and zero.
    \nfor example:
    \nintegers can either be negative, positive, or zero.
    \n1, -1, 2, -2, and 3, -3.<\/p>\n

    Pick A Project<\/strong><\/p>\n

    PROJECT 1A
    \nWhat is something you can make?
    \nPROJECT: DESIGNATIANDMADE IM TO SIIT
    \n\"Envision<\/p>\n

    PROJECT 1B
    \nHow old were you when petroglyphs were being painted?
    \nPROJECT: MAKE A TIMELINE
    \n\"Envision<\/p>\n

    PROJECT 1C
    \nWhat makes an obstacle course fun?
    \nPROJECT: BUILD A MODEL OF AN OBSTACLE COURSE
    \n\"Envision<\/p>\n

    PROJECT 1D
    \nWhat are your favorite ways to exercise ?
    \nPROJECT: FILM AN EXERCISE VIDEO
    \n\"Envision<\/p>\n

    Lesson 1.1 Relate Integers and Their Opposites<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nWhen preparing for a rocket launch, the mission control center uses the phrase “T minus” before liftoff.
    \n…T minus 3, T minus 2, T minus 1, …
    \nAfter the rocket has launched, “T plus” is used while the rocket is in flight.
    \n…T plus 1, T plus 2, T plus 3, …
    \nWhen does the rocket launch? What could “T” represent?<\/p>\n

    I can… relate integers, their opposites, and their absolute values.
    \n\"Envision<\/p>\n

    Reasoning
    \nWhat integers can you use to represent this situation?<\/p>\n

    \"Rational<\/p>\n

    Focus on math practices
    \nReasoning How are “T minus 4” and “T plus 4” related?<\/p>\n

    Answer:
    \nThe situation is T plus.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe rocket is launching.
    \nso we have to use the “T plus”.
    \n“T plus 4”.<\/p>\n

    Essential Question<\/strong>
    \nHow are integers and their opposites related?<\/p>\n

    Try It!<\/p>\n

    Xavier climbs 9 feet up into an apple tree. What integer represents the direction and how far he will climb to get back down to the ground? What does the integer 0 represent in this situation?
    \nThe integer ________ represents Xavier’s climb down.
    \nThe integer 0 represents __________.<\/p>\n

    Answer:
    \nThe integer -9 represents Xavier’s climb down.
    \nthe integer 0 represents that he did not move.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nXavier climbs 9 feet up into an apple tree.
    \nXavier climbs feet down into an apple tree is -9.
    \nthe integer 0 represents that he did not move.
    \nso the integer 0 is related to +0ft or -0ft.<\/p>\n

    Convince Me! How are the absolute values of opposite integers related?<\/p>\n

    Try It!<\/strong><\/p>\n

    The temperature was 75\u00b0. At noon, the temperature increased 7\u00b0. By evening, the temperature decreased by 7\u00b0. How did the temperature change?<\/p>\n

    Answer:
    \nThe temperature change by 75\u00b0 + 7 and 75\u00b0 – 7.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature was 75\u00b0. At noon, the temperature increased 7\u00b0
    \n75\u00b0 + 7.
    \nincreased means plus.
    \ndecreased means minus.
    \nthe temperature change by 75\u00b0 + 7 and 75\u00b0 – 7.<\/p>\n

    Try It!<\/strong><\/p>\n

    Shaniqua has $45 in her wallet. She spends $4 on snacks and $8 on a movie ticket. What integer represents the change in the amount of money in Shaniqua’s wallet? How much money does she have left?<\/p>\n

    Answer:
    \nThe money does she have left = $33.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nShaniqua has $45 in her wallet.
    \nShe spends $4 on snacks and $8 on a movie ticket.
    \n$45 – $4 = $41.
    \n$41 – $8 = $33.
    \nwe are using the minus operator.
    \nso the money does she have left = $33.<\/p>\n

    KEY CONCEPT<\/strong>
    \nAn integer, n, and its opposite, -n, combine to make 0.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question How are integers and their opposites related?<\/p>\n

    Answer:
    \nThe integers and their opposites are related because an integer’s opposite has to be the same number away from zero as the integer in question.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe integers and their opposites are related because an integer’s opposite has to be the same number away from zero as the integer in question.
    \nfor example:
    \n67 is the integer and its opposite side is -67.
    \nits opposite has to be on the other side of zero and the same amount of numbers away from zero.<\/p>\n

    \"Rational<\/p>\n

    Question 2.
    \nReasoning In order for an atom to have a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of -1. A helium atom has 2 protons and 2 electrons. Explain why a helium atom has a zero charge.
    \n\"Envision<\/p>\n

    Answer:
    \nHelium atom has integers are +2 and -2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \natom has a zero charge, every proton, which has a charge of +1, must be matched with an electron, which has a charge of -1.
    \nA helium atom has 2 protons and 2 electrons.
    \n+2, and -2.
    \nso helium atom has integers that are +2 and -2.<\/p>\n

    Question 3.
    \nModel with Math Explain how to use a number line to show that opposite quantities combine to make 0.<\/p>\n

    Answer:
    \nThe number line shows the + 3 and – 3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthere are some numbers on the number line.
    \n-3, 0, and 3.
    \nso the number line shows the integers +3 and -3.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nMarcus dives from the surface of the ocean to a reef 18 meters below sea level. What integer represents Marcus’s location relative to the surface? How far does Marcus have to go to return to the surface?<\/p>\n

    Answer:
    \nThe Marcus have to go to return to the surface = -18 meters.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMarcus dives from the surface of the ocean to a reef 18 meters below sea level.
    \n-18 represents Marcus’s location relative to the surface.
    \n-18, 0, and 18.
    \nso the Marcus has to go to return to the surface = -18 meters.<\/p>\n

    Question 5.
    \nThe temperature of the water in Emily’s fish tank was 78\u00b0F on Sunday. The water temperature changed by -3\u00b0 on Monday, and then by 3o on Tuesday. What integer represents the temperature change of the water from Sunday to Tuesday? What was the water temperature on Tuesday?<\/p>\n

    Answer:
    \nThe temperature change of the water from Sunday to Tuesday = plus.
    \nThe water temperature on Tuesday = 105\u00b0F.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature of the water in Emily’s fish tank was 78\u00b0F on Sunday.
    \nThe water temperature changed by -3\u00b0 on Monday.
    \n75\u00b0F.
    \nand then by 3o on Tuesday.
    \n105\u00b0F.
    \nthe temperature change of the water from Sunday to Tuesday = plus.
    \nthe water temperature on Tuesday = 105\u00b0F.<\/p>\n

    Question 6.
    \nThe scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?
    \n\"Envision<\/p>\n

    Answer:
    \nTravis’s score = -4.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe scores of players on a golf team are shown in the table.
    \nThe team’s combined score was 0.
    \nTravis’s score is -4.<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice In 7-9, write the integer that represents the situation.
    \nQuestion 7.
    \nMax spent $53 and now has no money left. He had $ _______ before his purchase.<\/p>\n

    Answer:
    \nHe had $53 before his purchase.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMax spent $53 and now has no money left.
    \n$53 – $53 = 0.
    \nso he had $53 before his purchase.<\/p>\n

    Question 8.
    \nThe temperature was 8\u00b0F. It dropped so that the temperature was 0\u00b0F. ________ \u00b0F represents the change in temperature.<\/p>\n

    Answer:
    \nThe change in temperature represents = -8\u00b0F.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature was 8\u00b0F.
    \nIt dropped so that the temperature was 0\u00b0F.
    \n8 – 0 = 8.
    \nso the change in the temperature represents = -8\u00b0F.<\/p>\n

    Question 9.
    \nAn airplane descended 4,000 feet before landing. The integer that represents how many feet the airplane was above the ground before its descent is _________.<\/p>\n

    Answer:
    \nThe number of feet the airplane was above the ground before its descent = -4000.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn airplane descended 4,000 feet before landing.
    \nbefore landing = 4000 meters.
    \nafter landing = -4000 m.
    \nso the number of feet the airplane was above the ground before its descent = -4000 m.<\/p>\n

    Question 10.
    \nCarolyn says that point A and point B represent opposite integers.
    \na. What is the opposite of the integer represented by point A? By point B?
    \n\"Envision<\/p>\n

    Answer:
    \nThe opposite of the integer represented by point A is -7 and by point, B is 8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nCarolyn says that point A and point B represent opposite integers.
    \nin the number line, the numbers are -10, -5, 0, 5, and 10.
    \nso the missing numbers are points A and B.
    \nThe opposite of the integer represented by point A is -7 and by point, B is 8.<\/p>\n

    b. Construct Arguments Do you agree with Carolyn? Explain.<\/p>\n

    Answer:
    \nYes, Arguments agree with Carolyn.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nCarolyn says that point A and point B represent opposite integers.
    \nin the number line, the numbers are -10, -5, 0, 5, and 10.
    \nso the missing numbers are points A and B.
    \nThe opposite of the integer represented by point A is -7 and by point, B is 8.<\/p>\n

    Question 11.
    \nA football team lost 9 yards during a play. The team had a combined gain or loss of 0 yards after the next play. What integer represents the yards gained or lost on the next play? Show this on the number line.
    \n\"Envision<\/p>\n

    Answer:
    \nThe integer represents the yards gained = 9 yards.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA football team lost 9 yards during a play.
    \nThe team had a combined gain or loss of 0 yards after the next play.
    \nthe integer represents the yards gained\u00a0 = 9 yards.
    \n-9, 0, and 9.
    \nso the integer represents the yards gained = 9 yards.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Question 12.
    \nA roller coaster car goes above and below ground. Use the number line to show its changes in height. What is the height of the car at the end of the ride?
    \n\"Envision<\/p>\n

    Answer:
    \nThe height of the car at the end of the ride = 5 m.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA roller coaster car goes above and below ground.
    \nThe roller coaster starts at 1 meter.
    \ndrops 4 meters down.
    \nrises 13 meters up.
    \ndrops 6 meters.
    \n13 – 6 = 7.
    \nso the height of the car at the end of the ride = 5 m.<\/p>\n

    Question 13.
    \nDimitri is buying a car. He chooses Option 1 to add a new sound system to his car. What integer represents the change from the base price of the car to its final price?
    \n\"Envision<\/p>\n

    Answer:
    \nThe integer represents the change from the base price of the car to its final price = $1400.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nDimitri is buying a car.
    \nHe chooses Option 1 to add a new sound system to his car.
    \nthe base price of the car = -$700.
    \noption 1 = $1400.
    \nso the integer represents the change from the base price of the car to its final price = $1400.<\/p>\n

    Question 14.
    \nMake Sense and Persevere What values do x and y have if |x| = 16, |y| = 16, and when x and y are combined they equal 0? Explain your reasoning.<\/p>\n

    Answer:
    \nx and y are equal to 0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nif |x| = 16.
    \n|y| = -16.
    \n16 – 16 = 0.
    \nso x and y are equal to 0.<\/p>\n

    Question 15.
    \nWrite a situation that can be represented by the opposite of -42.<\/p>\n

    Answer:
    \nThe situation that can be represented by the opposite of -42 is 42.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe situation that can be represented by the opposite of -42 is 42.
    \n42 is the positive integer.
    \n-42 is the negative integer.
    \nso the situation that can be represented by the opposite 0f -42 is 42.<\/p>\n

    Question 16.
    \nHigher Order Thinking<\/strong> Three friends all live on the same street that runs west to east. Beth lives 5 blocks from Ann. Carl lives 2 blocks from Beth. If the street is represented by a number line and Ann’s house is located at 0, what are the possible locations for Carl’s house? Assume that each unit on the number line represents 1 block.<\/p>\n

    Answer:
    \nThe possible locations for Carl’s house = 3 and 5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThree friends all live on the same street that runs west to east.
    \nBeth lives 5 blocks from Ann.
    \nCarl lives 2 blocks from Beth.
    \n5 – 2 = 3.
    \nso the location of Ann is 0.
    \nthe location of Carl is 3.
    \nthe location of Beth is 5.
    \nso the possible locations for Carl’s house = 3 and 5.<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 17.
    \nWhich of these situations can be represented with an integer that when combined with -9 makes 0? Select all that apply.
    \n\u2610 You walk down 9 flights of stairs.
    \n\u2610 You climb up 9 flights of stairs.
    \n\u2610 The temperature drops 9\u00b0F.
    \n\u2610 You spend $9 on a book.
    \n\u2610 You earn $9 from your job.<\/p>\n

    Answer:
    \nYou climb up 9 flights of stairs, you spend $9 on a book, and you earn $9 from your job.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nan integer that when combined with -9 makes 0.
    \nYou climb up 9 flights of stairs.
    \nyou spend $9 on a book.
    \nyou earn $9 from your job.<\/p>\n

    Question 18.
    \nWhich of these situations can be represented by the opposite of 80? Select all that apply.
    \n\u2610 An airplane descends 80 m.
    \n\u2610 An elevator ascends 80 m.
    \n\u2610 The cost of a train ticket drops by $80.
    \n\u2610 You remove 80 songs from an MP3 player.
    \n\u2610 Suzy’s grandmother is 80 years old.<\/p>\n

    Answer:
    \nAn airplane descends 80m, the cost of a train ticket drops by $80, and you remove 80 songs from an MP3 player.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe situations can be represented by the opposite of 80.
    \nAn airplane descends 80m = -80m.
    \nthe cost of a train ticket drops by $80 = -$80.
    \nYou remove 80 songs from an MP3 player = -80.<\/p>\n

    Lesson 1.2 Understand Rational Numbers<\/h2>\n

    Solve & Discuss It!<\/strong>
    \nCalvin wants to customize his surfboard so that it is wider than the 82 model but narrower than the 92 model. What measurement could be the width of his surfboard? Explain.<\/p>\n

    I can… recognize rational numbers and write them in decimal form.
    \n\"Envision<\/p>\n

    Answer:
    \nThe width of his surfboard = 1 wide.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nCalvin wants to customize his surfboard so that it is wider than the 82 models but narrower than the 92 models.
    \n82 model has a wide of 221\/2 and 3(1\/4) of thick.
    \n92 model has a wide of 23(1\/4) and 3(1\/2) thick.
    \n23 – 22 = 1.
    \nso the width of his surfboard = 1 wide.<\/p>\n

    Focus on math practices
    \nUse Structure Lindy’s surfboard is 23 inches wide. Between which two surfboard models is her custom surfboard’s width? How do you know?<\/p>\n

    Answer:
    \nThe two surfboard models are her custom surfboard’s width = surfboards 82 and 92.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nLindy’s surfboard is 23 inches wide.
    \nBetween which, two surfboard models are her custom surfboard’s width.
    \nso the two surfboard models is her custom surfboard’s width = surfboards 82 and 92.<\/p>\n

    Essential Question<\/strong>
    \nHow are rational numbers written as decimals?<\/p>\n

    Try It!<\/strong><\/p>\n

    In the next several games, the pitcher threw a total of 384 pitches and used a fastball 240 times. What decimal should Juanita use to update her report?
    \n\"Envision
    \nJuanita should use the decimal ________ to update her report.<\/p>\n

    Answer:
    \nJuanita should use the decimal 0 to update her report.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe pitcher threw a total of 384 pitches and used a fastball 240 times.
    \n380 \/ 240.
    \n380 x 6 = 2304.
    \n380 x 2 = 768.
    \n380 x 5 = 1920.
    \n1920 – 1920 = 0.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Convince Me! How do you know that the answer is a terminating decimal?<\/p>\n

    Try It!<\/strong><\/p>\n

    What is the decimal form of \\(\\frac{100}{3}, \\frac{100}{5}\\) and \\(\\frac{100}{6}\\)? Determine whether each decimal repeats or terminates.<\/p>\n

    Answer:
    \nThe decimal form of 100\/3 = 33.3.
    \nthe decimal form of 100\/5 = 20.
    \nthe decimal form of 100\/6 = 16.6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe decimal form of 100\/3, 100\/5, and 100\/6.
    \n100\/3 = 33.3.
    \n100\/5 = 20.
    \n100\/6 = 16.6.
    \nthe decimal 100\/3 and 100\/6 repeats.
    \nthe decimal 100\/5 terminates.<\/p>\n

    Try It!<\/strong><\/p>\n

    Is -0.3 a rational number? Is 3.14144144414444… a rational number? Explain your reasoning.<\/p>\n

    Answer:
    \nYes, both are rational numbers.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -0.3 and 3.14144144414444…
    \n-0.3 is a rational number.
    \n3.14144144414444 is a rational number.<\/p>\n

    KEY CONCEPT<\/strong>
    \nTo convert from the fraction form of a rational number to its decimal form, divide the numerator by the denominator. The decimal form of a rational number either terminates in 0s or eventually repeats.
    \n\"Envision<\/p>\n

    Do You Understand?
    \nQuestion 1.
    \nEssential Question How are rational numbers written as decimals?<\/p>\n

    Answer:
    \nRational numbers can be written as decimals in two ways.
    \nthey are terminating decimal and repeating decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe rational numbers can be written as a decimal in two ways.
    \nfor example:
    \n100\/5 = 20.
    \n20 is a terminating decimal.
    \n100\/3 = 33.3.
    \n33.3 is a repeating decimal.<\/p>\n

    Question 2.
    \nReasoning How can you use division to find the decimal equivalent of a rational number?<\/p>\n

    Answer:
    \n100\/3 = 33.3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n100\/3 = 33.3.
    \n33.3 is a terminating decimal.
    \n100\/5 = 20.
    \n20 is a repeating decimal.<\/p>\n

    Question 3.
    \nBe Precise What is the difference between a terminating decimal and a repeating decimal?<\/p>\n

    Answer:
    \nThe difference is one decimal will stop and one decimal will continue.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two terms are terminating and repeating decimals.
    \n100\/3 = 33.3.
    \nit is a repeating decimal.
    \n100\/5 = 20.
    \n20 is a terminating decimal.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nWhat is the decimal equivalent of each rational number?
    \na. \\(\\frac{7}{20}\\)<\/p>\n

    Answer:
    \n7\/20 = 0.35.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 7\/20.
    \n7\/20 = 0.35.
    \n0.35 is a terminating decimal.<\/p>\n

    b. –\\(\\frac{23}{20}\\)<\/p>\n

    Answer:
    \n-23\/20 = -1.15.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is -23\/20.
    \n-23\/20 = -1.15.
    \nit is a terminating decimal.<\/p>\n

    c. \\(\\frac{1}{18}\\)<\/p>\n

    Answer:
    \n1\/18 = 0.055.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 1\/18.
    \n1\/18 = 0.055.
    \n0.055 is a repeating decimal.<\/p>\n

    d. –\\(\\frac{60}{22}\\)<\/p>\n

    Answer:
    \n-60\/22 = 2.727 is a repeating decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is -60\/22.
    \n-60\/22 = 2.727.
    \n2.727 is a repeating decimal.<\/p>\n

    Question 5.
    \nThere are 5,280 feet in a mile. What part of a mile, in decimal form, will you drive until you reach the exit?
    \n\"Envision<\/p>\n

    Answer:
    \n5280\/1000 = 5.28 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThere are 5,280 feet in a mile.
    \n5280\/1000 = 5.28.
    \nso the part of the mile in decimal form = 5.28 miles.<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice In 6-8, write the decimal equivalent for each rational number. Use a bar over any repeating digits.
    \nQuestion 6.
    \n\\(\\frac{2}{3}\\)<\/p>\n

    Answer:
    \n2\/3 = 0.666.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 2\/3.
    \n2\/3 = 0.666.
    \n0.666 is a repeating decimal.<\/p>\n

    Question 7.
    \n\\(\\frac{3}{11}\\)<\/p>\n

    Answer:
    \n3\/11 = 0.272727.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 3\/11.
    \n3\/11 = 0.272727.
    \n0.272727 is a repeating decimal.<\/p>\n

    Question 8.
    \n\\(8\\frac{4}{9}\\)<\/p>\n

    Answer:
    \n8(4\/9) = repeating decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 8(4\/9).
    \n9 x 8 = 72.
    \n72 + 4 = 76.
    \n76\/9 = 8.4444.
    \n8.444 is a repeating decimal.<\/p>\n

    Question 9.
    \nIs 1.0227 a rational number? Explain.<\/p>\n

    Answer:
    \nYes, 1.0227 is a rational number.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 1.0227.
    \n1.0227 is a rational number.<\/p>\n

    Question 10.
    \nWhich should Aaron use to convert a fraction to a decimal?
    \n\"Envision<\/p>\n

    Answer:
    \nOption A is correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nnumerator\/denominator is used to convert a fraction to a decimal.
    \nso option A is correct.<\/p>\n

    Question 11.
    \nIs the fraction \\(\\frac{1}{3}\\) equivalent to a terminating decimal or a decimal that does not terminate?<\/p>\n

    Answer:
    \n1\/3 is a repeating decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 1\/3.
    \n1\/3 = 0.333.
    \nso 1\/3 is a repeating decimal.<\/p>\n

    Question 12.
    \nDetermine whether the given number belongs to each set.
    \n\"Envision<\/p>\n

    Answer:
    \n-34 is an Integer.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthey have given the whole numbers, integers, and rational numbers.
    \nso -34 is an integer.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Question 13.
    \nAriel incorrectly says that 2\\(\\frac{5}{8}\\) is the same as 2.58.
    \na. Convert 2\\(\\frac{5}{8}\\) to a decimal.<\/p>\n

    Answer:
    \n2(5\/8) = 2.625.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAriel incorrectly says that 2\\(\\frac{5}{8}\\) is the same as 2.58.
    \nthe number is 2(5\/8).
    \n8 x 2 = 16.
    \n16 + 5 = 21.
    \n21\/8 = 2.625.<\/p>\n

    b. What was Ariel’s likely error?<\/p>\n

    Answer:
    \nAriel’s error was the decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAriel incorrectly says that 2\\(\\frac{5}{8}\\) is the same as 2.58.
    \nthe number is 2(5\/8).
    \n8 x 2 = 16.
    \n16 + 5 = 21.
    \n21\/8 = 2.625.<\/p>\n

    Question 14.
    \nUse Structure Consider the rational number \\(\\frac{3}{11}\\)
    \na. What are the values of a and b in \\(a\\sqrt {b}\\) when you use division to find the decimal form?<\/p>\n

    Answer:
    \n3\/11 = 0.272727.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \na = 3, and b = 11.
    \n3\/11 = 0.272727.
    \nso 3\/11 is a repeating decimal.
    \n3\/11 = 0.272727.<\/p>\n

    b. What is the decimal form for \\(\\frac{3}{11}\\)?<\/p>\n

    Answer:
    \nThe decimal form is 0.272727.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \na = 3, and b = 11.
    \n3\/11 = 0.272727.
    \nso 3\/11 is a repeating decimal.
    \n3\/11 = 0.272727.<\/p>\n

    Question 15.
    \nAt a grocery store, Daniel wants to buy 3\\(\\frac{1}{5}\\) lb of ham. What decimal should the digital scale show?
    \nWrite 3\\(\\frac{1}{3}\\) as a fraction and then divide.
    \n\"Envision
    \nThe scale should read ________ lb.<\/p>\n

    Answer:
    \nThe scale should read 3.3 lb.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAt a grocery store, Daniel wants to buy 3\\(\\frac{1}{5}\\) lb of ham.
    \n3(1\/3).
    \n3 x 3 = 9.
    \n9 + 1 = 10.
    \n10\/3 = 3.333.
    \n3.333 is a repeating decimal.
    \nso the scale should read 3.3 lb.<\/p>\n

    Question 16.
    \nReasoning<\/strong> At a butcher shop, Hilda bought beef and pork. She left with 18\\(\\frac{8}{25}\\) pounds of meat. Express the number of pounds of pork she bought using a decimal.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of pounds of pork she bought using a decimal = 9.47 pounds.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAt a butcher shop, Hilda bought beef and pork.
    \nShe left with 18\\(\\frac{8}{25}\\) pounds of meat.
    \n18(8\/25).
    \n18 x 25 = 450.
    \n458\/25 = 18.32.
    \nthe cost of the beef is 8(17\/20).
    \n20 x 8 = 160.
    \n177\/20 = 8.85.
    \n18.32 – 8.85 = 9.47.<\/p>\n

    Question 17.
    \nBe Precise Is 9.373 a repeating decimal? Is it rational? Explain your reasoning,<\/p>\n

    Answer:
    \nYes, it is a repeating decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number is 9.373 is a rational number.
    \nit is also a repeating decimal.<\/p>\n

    Question 18.
    \nReasoning Aiden has one box that is 3\\(\\frac{3}{11}\\), feet tall and a second box that is 3.27 feet tall. If he stacks the boxes, about how tall will the stack be?<\/p>\n

    Answer:
    \nThe tall will the stack be = 6.54 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAiden has one box that is 3\\(\\frac{3}{11}\\), feet tall and a second box that is 3.27 feet tall.
    \n3(3\/11).
    \n33 + 3 = 36.
    \n36\/11 = 3.272727.
    \n3.27 + 3.27 = 6.54,
    \nso the heght of the stack = 6.54 feets.<\/p>\n

    Question 19.
    \nYou are adding air to a tire. The air pressure in the tire should be \\(32\\frac{27}{200}\\) pounds per square inch. What decimal should you watch for on the digital pressure gauge?<\/p>\n

    Answer:
    \nThe decimal should I watch for on the digital pressure gauge = terminating decimal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nYou are adding air to a tire.
    \nThe air pressure in the tire should be \\(32\\frac{27}{200}\\) pounds per square inch.
    \n32(27\/200).
    \n32 x 200 = 6400.
    \n6400 + 27 = 6427.
    \n6427\/200 = 32.135.
    \nso the decimal should I watch for on the digital pressure gauge = terminating decimal.<\/p>\n

    Question 20.
    \nHigher Order Thinking<\/strong> Dion has a pizza with a diameter of 10\\(\\frac{1}{3}\\) in. Is the square box shown big enough to fit the pizza inside? Justify your answer.
    \n\"Envision<\/p>\n

    Answer:
    \nThe square box shown is big enough to fit the pizza inside.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nDion has a pizza with a diameter of 10\\(\\frac{1}{3}\\) in.
    \n10(1\/3).
    \n30 + 1\/3.
    \n31\/3 = 10.33.
    \nso the square box shown is big enough to fit the pizza inside.<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 21.
    \nWhich of the following mixed numbers has the same decimal value as \\(110 \\frac{147}{168}\\)?
    \nA. \\(110 \\frac{49}{56}\\)
    \nB. \\(110 \\frac{170}{180}\\)
    \nC. \\(110 \\frac{56}{72}\\)
    \nD. \\(110 \\frac{247}{268}\\)<\/p>\n

    Answer:
    \nOption A is the correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n110\/ 147\/168.
    \n110\/ 0.875.
    \nso option A is correct.<\/p>\n

    Question 22.
    \nSelect all the true statements about the negative fractions –\\(\\frac{4}{5}\\) and –\\(\\frac{5}{6}\\)
    \n–\\(\\frac{4}{5}\\) can be expressed as a repeating decimal.
    \no –\\(\\frac{5}{6}\\) can be expressed as a repeating decimal.
    \nBoth fractions can be expressed as repeating decimals.
    \nThe digit that repeats is 3.
    \nThe digit that repeats is 8.<\/p>\n

    Answer:
    \n-5\/6 can be expressed as a repeating decimal.
    \nthe digit that repeats is 3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -4\/5 and -5\/6.
    \n-4\/5 and -5\/6.
    \n-4\/5 = -0.8.
    \n-5\/6 = -0.8.
    \nso -5\/6 can be expressed as a repeating decimal.<\/p>\n

    Lesson 1.3 Add Integers<\/h3>\n

    Explore It!<\/strong>
    \nRain increases the height of water in a kiddie pool, while evaporation decreases the height. The pool water level is currently 2 inches above the fill line.<\/p>\n

    I can… add integers.
    \n\"Envision<\/p>\n

    A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice.<\/p>\n

    Answer:
    \nThe missing numbers are 2, -1, -2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe top of the pool is 4.
    \nthe bottom of the pool is -4.
    \nwe can add integers.
    \nso the missing numbers are 2, -1, and -2.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    B. Will the sum of 2 and (-6) be a positive or negative number? Explain.<\/p>\n

    Answer:
    \nThe sum of 2 and -6 be a negative number.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n2 + (-6).
    \n2 – 6 = -4.
    \nthe sum of two +ve numbers will always be a positive number.
    \nthe sum of two -ve numbers will also be positive number.
    \nthe sum of one +ve and one -ve is a negative number.
    \nso the sum of 2 and -6 be a negative number.<\/p>\n

    Focus on math practices
    \nLook for Relationships Suppose the water level of the pool started at 2 inches below the fill line. Make a table to show the starting height of the water, the change in inches, and the new final height of the water.<\/p>\n

    Essential Question<\/strong>
    \nHow do you use what you know about the absolute value to add integers?<\/p>\n

    Try It!<\/strong><\/p>\n

    Dana recorded a temperature drop of 2o and a second temperature drop of 3\u00b0. What is the total change in temperature?
    \n_______ + ________ = _________
    \nThe sign of the sum is _______. The total change in temperature is ________\u00b0.<\/p>\n

    Answer:
    \nThe sign of the sum is negative.
    \nthe total change in temperature is 5\u00b0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nDana recorded a temperature drop of 2o and a second temperature drop of 3\u00b0.
    \n-2 – 3 = -5.
    \nso the sign of the sum is negative.
    \nthe total change in temperature is 5\u00b0.<\/p>\n

    Convince Me! Would the sum of two positive integers be positive or negative? Explain.<\/p>\n

    Answer:
    \nThe sum of two positive integers is positive.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe sum of two positive integers is positive.
    \n2 + 3 = 5.
    \nso the sum is also a positive number.<\/p>\n

    Try It!<\/strong><\/p>\n

    Find the sum for each expression.
    \na. -66 + 42<\/p>\n

    Answer:
    \n-66 + 42 = -24.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -66 and 42.
    \nadd the numbers.
    \nthe sum of one positive and one negative is also a negative integer.
    \n-66 + 42 = -24.<\/p>\n

    b. -57 + 57<\/p>\n

    Answer:
    \n-57 + 57 = 0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -57 and 57.
    \nadd the numbers.
    \nthe sum of one positive and one negative is also a negative integer.
    \n-57 + 57 = 0.<\/p>\n

    c. 29 + (-28)<\/p>\n

    Answer:
    \n29 + (-28) = 1.
    \nExplanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -28 and 29.
    \nadd the numbers.
    \nthe sum of one positive and one negative is also a negative integer.
    \n29 + (-28) = 1.<\/p>\n

    KEY CONCEPT<\/strong>
    \nWhen adding integers with the same sign, find the sum of the absolute values.
    \n(-36) + (-12)
    \n|-36| = 36 and |-12| = 12
    \n36 + 12 = 48
    \nSo, (-36) + (-12) = -48
    \nUse the same sign as the addends.<\/p>\n

    When adding integers with different signs, find the difference of the absolute values.
    \n18+ (-14)
    \n|18| = 18 and |-14| = 14
    \n18 – 14 = 4
    \nSo, 18 + (-14) = 4
    \nUse the sign of the greater absolute value.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question How do you use what you know about absolute value to add integers?<\/p>\n

    Answer:
    \nWhen adding the integers with the same sign, we will find the sum of absolute values.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nWhen adding the integers with same sign, we will find the sum of absolute values.
    \nfor example:
    \n(-36) + (-12)
    \n|-36| = 36 and |-12| = 12
    \n36 + 12 = 48
    \nSo, (-36) + (-12) = -48.<\/p>\n

    Question 2.
    \nReasoning How can you tell the sign of the sum of a positive and negative integer without doing any calculations?<\/p>\n

    Answer:
    \nThe sum of positive and negative integers is always negative.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe sum of positive and negative integers is always negative.
    \nfor example:
    \n-33 + 22.
    \n-11.
    \nso the sum of positive and negative integers is always negative.<\/p>\n

    Question 3.
    \nModel with Math How would you use a number line to determine the sum of two negative integers?<\/p>\n

    Answer:
    \nThe sum of two negative integers is also a positive integer.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe sum of two negative integers is also a positive integer.
    \nfor example:
    \n-33 + -22.
    \n-33 – 22.
    \n55.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nSarah bought a bike that cost $260. She had a coupon that was worth $55 off the cost of any bike. Use the expression 260 + (-55) to find how much Sarah paid for her bike.<\/p>\n

    Answer:
    \nThe amount Sarah paid for her bike = $205.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nSarah bought a bike that cost $260.
    \nShe had a coupon that was worth $55 off the cost of any bike.
    \n260 + (-55).
    \n260 – 55.
    \n205.
    \nso the amount Sarah paid for her bike = $205.<\/p>\n

    Question 5.
    \nA shark is swimming 60 feet below the surface of the ocean. There is a fish that is 25 feet deeper in the water. Use the expression (-60) + (-25) to describe the fish’s location relative to the surface of the ocean.
    \n\"Envision<\/p>\n

    Answer:
    \nThe fish’s location is relative to the surface of the ocean = -85 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA shark is swimming 60 feet below the surface of the ocean.
    \nThere is a fish that is 25 feet deeper in the water.
    \n(-60) + (-25).
    \n-60 – 25 = -85.
    \nso the fish’s location relative to the surface of the ocean = -85 feet.<\/p>\n

    Question 6.
    \nThe high temperature one day was 30\u00b0F. Then the temperature dropped 23 degrees during the night. Does the expression 30 + (-23) represent the temperature at night? Explain.<\/p>\n

    Answer:
    \nThe expression 30 + (-23) represent the temperature at night = 7 degrees.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe high temperature one day was 30\u00b0F.
    \nThen the temperature dropped 23 degrees during the night.
    \n30 + (-23).
    \n30 – 23 = 7.
    \nso the expression represents the temperature at night = 7 degrees.<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice For 7-9, use the number lines to help find each sum.<\/p>\n

    Question 7.
    \n5 + (-3) is ________ units from 5, in the ________ direction.
    \n\"Envision<\/p>\n

    Answer:
    \n5 + (-3) is ____2____ units from 5, in the _positive____ direction.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n5 + (-3).
    \n5 – 3 = 2.
    \n2 units from 5, in the positive direction.<\/p>\n

    Question 8.
    \n-1 + (-3) is _______ units from -1, in the ________ direction.
    \n\"Envision<\/p>\n

    Answer:
    \n-1 + (-3) is __-4_____ units from -1, in the ___negative_____ direction<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n-1 + (-3).
    \n-1 – 3 = -4.
    \n-4 units from -1.
    \nin the negative direction.<\/p>\n

    Question 9.
    \nIn City A, the temperature rises 9\u00b0 from 8 A.M. to 9 A.M. Then the temperature drops 8\u00b0 from 9 A.m. to 10 A.M. In City B, the temperature drops 5\u00b0 from 8 A.M. to 9 A.M. Then the temperature drops 4\u00b0 from 9 A.M. to 10 A.M.
    \n\"Envision
    \na. What expression represents the change in temperature for City A?<\/p>\n

    Answer:
    \nThe expression represents the change in temperature for City A is 9\u00b0 – 8\u00b0 = 1\u00b0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn City A, the temperature rises 9\u00b0 from 8 A.M. to 9 A.M.
    \nThen the temperature drops 8\u00b0 from 9 A.m. to 10 A.M.
    \n9\u00b0 – 8\u00b0 = 1\u00b0.
    \nso the expression represents the change in temperature for City A is 9\u00b0 – 8\u00b0 = 1\u00b0.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    b. What integer represents the change in temperature for City A?<\/p>\n

    Answer:
    \nThe positive integer represents the change in temperature for City A.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn City A, the temperature rises 9\u00b0 from 8 A.M. to 9 A.M.
    \nThen the temperature drops 8\u00b0 from 9 A.m. to 10 A.M.
    \n9\u00b0 – 8\u00b0 = 1\u00b0.
    \nso the positive integer represents the change in temperature for City A.<\/p>\n

    c. What expression represents the change in temperature for City B?<\/p>\n

    Answer:
    \nThe expression represents the change in temperature for City B is 5\u00b0 – 4\u00b0 = 1\u00b0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn City B, the temperature rises 5\u00b0 from 8 A.M. to 9 A.M.
    \nThen the temperature drops 4\u00b0 from 9 A.m. to 10 A.M.
    \n5\u00b0 – 4\u00b0 = 1\u00b0.
    \nso the expression represents the change in temperature for City B is 5\u00b0 – 4\u00b0 = 1\u00b0.<\/p>\n

    d. What integer represents the change in temperature for City B?<\/p>\n

    Answer:
    \nA positive integer represents the change in temperature for City B.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn City B, the temperature rises 5\u00b0 from 8 A.M. to 9 A.M.
    \nThen the temperature drops 4\u00b0 from 9 A.m. to 10 A.M.
    \n5\u00b0 – 4\u00b0 = 1\u00b0.
    \nso the positive integer represents the change in temperature for City B.<\/p>\n

    e. Which city has the greater change in temperature from 8 A.M. to 10 A.M.?<\/p>\n

    Answer:
    \nBoth the cities have an equal change in temperature.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn City A, the temperature rises 9\u00b0 from 8 A.M. to 9 A.M.
    \nThen the temperature drops 8\u00b0 from 9 A.m. to 10 A.M.
    \n9\u00b0 – 8\u00b0 = 1\u00b0.
    \n5\u00b0 – 4\u00b0 = 1\u00b0.
    \nso both the cities have an equal change in temperature.<\/p>\n

    Question 10.
    \nAn airplane flying at an altitude of 30,000 feet flies up to avoid a storm. Immediately after passing the storm, the airplane returns to its original altitude.
    \n\"Envision
    \na. What integer represents the airplane’s change in altitude to avoid the storm?<\/p>\n

    Answer:
    \nThe positive integer represents the airplane’s change in altitude to avoid the storm.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn airplane flying at an altitude of 30,000 feet flies up to avoid a storm.
    \nthe airplane flies up to 38,000 feet to avoid a storm.
    \n38000 – 30000.
    \n8000.
    \nso the positive integer represents the airplane’s change in altitude to avoid the storm.<\/p>\n

    b. What integer represents the airplane’s change in altitude immediately after passing the storm?<\/p>\n

    Answer:
    \nThe positive integer represents the airplane’s change in altitude immediately after passing the storm.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn airplane flying at an altitude of 30,000 feet flies up to avoid a storm.
    \nthe airplane flies up to 38,000 feet to avoid a storm.
    \n38000 – 30000.
    \n8000.
    \nso the positive integer represents the airplane’s change in altitude immediately after passing the storm.<\/p>\n

    c. Use Appropriate Tools Draw a number line to represent the airplane’s change in altitude.
    \n\"Envision<\/p>\n

    Answer:
    \nThe change in altitude is 38000.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn airplane flying at an altitude of 30,000 feet flies up to avoid a storm.
    \nthe airplane flies up to 38,000 feet to avoid a storm.
    \n30000 + 8000 = 38000.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Question 11.
    \nA deep-sea diver dives 81 feet from the surface. He then dives 14 more feet. The diver’s depth can be represented by -81 + (-14). What is the diver’s present location?<\/p>\n

    Answer:
    \nThe diver’s present location is -95\u00a0 feet below the sea.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA deep-sea diver dives 81 feet from the surface.
    \nHe then dives 14 more feet.
    \nThe diver’s depth can be represented by -81 + (-14).
    \n-81 + (-14).
    \n-81 – 14 = -95.
    \nso the diver’s present location is -95 feet below the sea.<\/p>\n

    Question 12.
    \nRena’s rowboat drifts 23 feet from shore, followed by 9 more feet. The rowboat’s current position can be represented by –23 + (-9). What integer represents the rowboat’s position?<\/p>\n

    Answer:
    \nThe integer represents the rowboat’s position as negative.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nRena’s rowboat drifts 23 feet from shore, followed by 9 more feet.
    \nThe rowboat’s current position can be represented by –23 + (-9).
    \n-23 – 9 = -32.
    \nso the integer represents the rowboat’s position is negative.<\/p>\n

    Question 13.
    \nCritique Reasoning<\/strong> A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters. Adam says that the new location of the submarine is -155 meters. Describe an error Adam could have made that would result in the answer he gave.<\/p>\n

    Answer:
    \nYes, Adam was incorrect.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters.
    \n200 + 45 = 245.
    \nbut Adam says that the new location of the submarine is -155 meters.
    \nso Adam was incorrect.<\/p>\n

    Question 14.
    \nKim has $45 to spend for a day at the zoo. She pays $17 for admission, $8 for lunch, and $4 for a snack.
    \na. Model with Math Use integers to write an addition expression that represents the amount of money Kim has left.<\/p>\n

    Answer:
    \nThe amount of money Kim has left = $16.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nKim has $45 to spend for a day at the zoo.
    \nShe pays $17 for admission, $8 for lunch, and $4 for a snack.
    \n$45 + (-$17) + (-$8) + (-$4).
    \n$45 – $17 – $8 – $4.
    \n$45 – $29.
    \n$16.
    \nso the amount of money Kim has left = $16.<\/p>\n

    b. Kim goes to the gift shop and finds a T-shirt she likes for $19. Does she have enough money to buy the T-shirt? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nNo, she does not have enough money to buy the T-shirt.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nKim has $45 to spend for a day at the zoo.
    \nShe pays $17 for admission, $8 for lunch, and $4 for a snack.
    \nshe has left $16.
    \nso she does not have enough money to buy the T-shirt.<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong> Samantha has $300 for guitar lessons to learn her favorite song. Mrs. Jones charges $80 per lesson and requires three lessons to teach Samantha the song. Mr. Beliz charges $62 per lesson and will require four lessons to teach Samantha the song. Use integers to represent what each teacher charges. Which is the better deal for Samantha?<\/p>\n

    Answer:
    \nThe better deal for Samantha is Mr. Beliz charges $62 per lesson.
    \n$62 + $62 + $62 + $62 = $248.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nSamantha has $300 for guitar lessons to learn her favorite song.
    \nMrs. Jones charges $80 per lesson and requires three lessons to teach Samantha the song.
    \nMr. Beliz charges $62 per lesson and will require four lessons to teach Samantha the song.
    \n$80 + $80 + $80 = $240.
    \n$62 + $62 + $62 + $62 = $248.
    \nso the better deal for Samantha is Mr. Beliz charges $62 per lesson.<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 16.
    \nA fish swims at 10 ft below sea level, and then swims another 10 ft deeper to avoid a shark. Write an addition expression that represents this situation.<\/p>\n

    Answer:
    \nThe addition expression that represents this situation is (-10) + (-10) = -20.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA fish swims at 10 ft below sea level and then swims another 10 ft deeper to avoid a shark.
    \n(-10) + (-10) = -20.
    \nbelow sea level = -10.
    \nso the addition expression that represents this situation is (-10) + (-10) = -20.<\/p>\n

    Question 17.
    \nThe temperature drops 10 degrees and then rises 10 degrees. Write an addition expression that represents this situation.<\/p>\n

    Answer:
    \nThe addition expression that represents this situation is 10 + (-10) = 0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature drops 10 degrees and then rises 10 degrees.
    \n10 + (-10).
    \n10 – 10 = 0.
    \nso the addition expression that represents this situation is 10 + (-10) = 0.<\/p>\n

    Lesson 1.4 Subtract Integers<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nA library database shows the total number of books checked out at any given time as a negative number. What are the possible numbers of books that were checked out and checked in on Monday? Explain.
    \nI can.. subtract integers.
    \n\"Envision<\/p>\n

    Make Sense and Persevere
    \nHow can you use the data to understand what happened during the day?<\/p>\n

    Focus on math practices
    \nReasoning Suppose the library database showed 0 for Monday evening. What do you know about the number of books checked out and checked in that day?<\/p>\n

    Answer:
    \nThe number of books checked out and checked in that day = 45 books.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n0 + (-45).
    \n0 – 45 = -45.
    \nso the number of books checked out and checked in that day = -45.<\/p>\n

    Essential Question<\/strong>
    \nHow is subtracting integers related to adding integers?<\/p>\n

    Try It!<\/strong><\/p>\n

    On the next play, the team gained 5 yards and then lost 6 yards. What is the total change in yards?
    \n\"Envision
    \n5 – ______
    \n= 5 + ______
    \n= _______
    \nThe total change in yards is ________, so they had a total loss of ________ yard.<\/p>\n

    Answer:
    \nThe total change in yards is 1.
    \nso they had a total loss of 1 yard.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nOn the next play, the team gained 5 yards and then lost 6 yards.
    \n5 – 6 = -1.
    \n5 + (-6) = -1.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Convince Me! Is the additive inverse of an integer always negative? Explain.<\/p>\n

    Try It!<\/strong><\/p>\n

    Subtract. Use a number line to help you find the answer.
    \n\"Envision
    \na. -4 – 6<\/p>\n

    Answer:
    \n-4 -6 = 10.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -4 and -6.
    \n-4 + (-6).
    \n-4 – 6 = -10.<\/p>\n

    b. -6 – (-4).<\/p>\n

    Answer:
    \n-6 – (-4) = -2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -4 and -6.
    \n-6 – (-4).
    \n-6 + 4.
    \n-2.<\/p>\n

    c. 4 – (-6)<\/p>\n

    Answer:
    \n4 – (-6) = 10.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 4 and -6.
    \n4 – (-6).
    \n4 + 6.
    \n10.<\/p>\n

    d. 6 – 4<\/p>\n

    Answer:
    \n6 – 4 = 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 6 and -4.
    \n6 – 4 = 2.<\/p>\n

    e. 4 – 6<\/p>\n

    Answer:
    \n4 – 6 = -2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 4 and -6.
    \n4 – 6 = -2.<\/p>\n

    f. -4 – (-6)<\/p>\n

    Answer:
    \n-4 + 6 = 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -4 and -6.
    \n-4 – (-6).
    \n-4 + 6 = 2.<\/p>\n

    KEY CONCEPT<\/strong>
    \nWhen subtracting integers, such as a – b, you can use the additive inverse to write subtraction as an equivalent addition expression.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How is subtracting integers related to adding integers?<\/p>\n

    Answer:
    \nSubtracting integers related to additive inverse as an equivalent addition.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nSubtracting integers related to additive inverse as an equivalent addition.
    \nfor example:
    \na + (-b) = a -b.<\/p>\n

    Question 2.
    \nReasoning<\/strong> Explain how to simplify the expression -98 – 31 using the additive inverse.<\/p>\n

    Answer:
    \n-98 + (-31) = – 129.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -98 and -31.
    \n-98 + (-31).
    \n-129.<\/p>\n

    Question 3.
    \nModel with Math How can you use a number line to represent the subtraction between two integers?<\/p>\n

    Answer:
    \n-5 – 2 = -7.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers on the number line = -5 and -2.
    \n-5 + (-2).
    \n-5 – 2 = -7.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nIt was 12\u00b0C when Preston got home from school. The weather report shows a storm front moving in that will drop the temperature by 17\u00b0C. What is the expected temperature?<\/p>\n

    Answer:
    \nThe expected temperature is -5 degrees c.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIt was 12\u00b0C when Preston got home from school.
    \nThe weather report shows a storm front moving in that will drop the temperature by 17\u00b0C.
    \n12 – 17 = -5.
    \nso the expected temperature is -5\u00b0C.<\/p>\n

    Question 5.
    \nComplete the equation.
    \n-67 – _____ = 0<\/p>\n

    Answer:
    \n-67 – (-67) = 0.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n-67 – (-67).
    \n-67 + 67 = 0.<\/p>\n

    Question 6.
    \nFind the difference.
    \na. 41 – 275<\/p>\n

    Answer:
    \n41 – 275 = -234.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are 41 and 275.
    \nsubtract the numbers.
    \n41 – 275.
    \n– 234.<\/p>\n

    b. -15 – 47<\/p>\n

    Answer:
    \n-15 – 47 = -32.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -15 and -47.
    \nsubtract the numbers
    \n-15 – 47.
    \n47 – 15 = -32.<\/p>\n

    c. -72 – (-151)<\/p>\n

    Answer:
    \n-72 – (-151) = 79.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -72 and -151.
    \nsubtract the numbers.
    \n-72 – (-151).
    \n-72 + 151.
    \n79.<\/p>\n

    d. 612 – (-144)<\/p>\n

    Answer:
    \n612 – (-144) = 756.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are 612 and -144.
    \nsubtract the numbers.
    \n612 – (-144).
    \n612 + 144 = 756.<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice In 7-8, fill in the boxes to solve.
    \nQuestion 7.
    \nWhat subtraction expression does the number line model show?
    \n\"Envision<\/p>\n

    Answer:
    \n-2 + (-9) = 7.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe lines are moving from -2 and -9.
    \n-2 + (-9).
    \n-2 – 9 = -11.<\/p>\n

    Question 8.
    \nWhat is the value of the expression -9 – (-5)?
    \n-9 – (-5)
    \n= -9 ______ 5
    \n= _______<\/p>\n

    Answer:
    \nThe value of the expression -9 – (-5) = -4.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -9 and -5.
    \nsubtract the numbers.
    \n-9 – (-5) = -4.<\/p>\n

    Question 9.+e
    \nThe temperature at the beginning of the day was 6\u00b0F. The temperature dropped 9\u00b0F by the end of the day. Use the number line to find the temperature at the end of the day.
    \n\"Envision<\/p>\n

    Answer:
    \nThe temperature at the end of the day = -3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature at the beginning of the day was 6\u00b0F.
    \nThe temperature dropped 9\u00b0F by the end of the day.
    \n6 + (- 9) = -3.
    \nso the temperature at the end of the day = -3.<\/p>\n

    Question 10.
    \nMurphy and Naryam do their math homework together. When they find 9- (-8), they get different answers. Murphy claims the difference is 17. Naryam claims the difference is \u2013 1.
    \na. Who is correct?<\/p>\n

    Answer:
    \nMurphy was correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMurphy and Naryam do their math homework together.
    \nWhen they find 9- (-8), they get different answers.
    \n9 – (-8).
    \n9 + 8 = 17.
    \nso Murphy was correct.<\/p>\n

    b. What error likely led to the incorrect answer?<\/p>\n

    Answer:
    \nThe sign likely led to the incorrect answer.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMurphy and Naryam do their math homework together.
    \nWhen they find 9- (-8), they get different answers.
    \n9 – (-8).
    \n9 + 8 = 17.<\/p>\n

    Question 11.
    \nThe news reports that today’s high temperature is 16\u00b0F colder than yesterday’s high temperature. Yesterday’s high temperature was -2\u00b0F.
    \na. Write an expression to represent today’s high temperature.<\/p>\n

    Answer:
    \nThe expression is -2\u00b0F + 16\u00b0F = 14\u00b0F.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe news reports that today’s high temperature is 16\u00b0F colder than yesterday’s high temperature.
    \nYesterday’s high temperature was -2\u00b0F.
    \nso the expression is -2\u00b0F + 16\u00b0F = 14\u00b0F.<\/p>\n

    b. Reasoning is today’s high temperature positive or negative? Why?<\/p>\n

    Answer:
    \nToday’s high temperature is positive.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe news reports that today’s high temperature is 16\u00b0F colder than yesterday’s high temperature.
    \nYesterday’s high temperature was -2\u00b0F.
    \nso the expression is -2\u00b0F + 16\u00b0F = 14\u00b0F.<\/p>\n

    Question 12.
    \nMax sprints forward 10 feet and then stops and sprints back 15 feet. Use subtraction to explain where Max is relative to where he started.
    \n\"Envision<\/p>\n

    Answer:
    \nHe started at -5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMax sprints forward 10 feet and then stops and sprints back 15 feet.
    \n10 + (-15).
    \n10 – 15.
    \n-5.
    \nso he started at -5.<\/p>\n

    Question 13.
    \nHigher Order Thinking<\/strong> Use the number line at the right.
    \n\"Envision
    \na. What subtraction equation does the number line represent?<\/p>\n

    Answer:
    \nThe subtraction equation is -2 + (-6) = -8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe starting point is -2.
    \nhe then increases by -6.
    \n-2 + (-6).
    \n-2 -6 = -8.<\/p>\n

    b. Use the number line to represent a different subtraction equation that has the same difference shown in the number line. Write the subtraction equation.
    \n\"Envision<\/p>\n

    Answer:
    \n0 – 2 = -2.
    \n-4 – (- 2) = -2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe subtraction equations are:
    \n0 – 2 = -2.
    \n-4 – (-2).
    \n-4 + 2 = -2.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Question 14.
    \nA crane lifts a pallet of concrete blocks 8 feet from the back of a truck. The truck drives away and the crane lowers the pallet 13 feet: What is the final position of the pallet relative to where it started in the back of the truck?
    \n\"Envision<\/p>\n

    Answer:
    \nThe final position of the pallet relative to where it started in the back of the truck = 5 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA crane lifts a pallet of concrete blocks 8 feet from the back of a truck.
    \nThe truck drives away and the crane lowers the pallet 13 feet:
    \n13 – 8 = 5.
    \nso the final position of the pallet relative to where it started in the back of the truck = 5 feet.<\/p>\n

    Question 15.
    \nMake Sense and Persevere At its highest point, the elevation of a county is 5,762 feet above sea level. At its lowest point, the elevation of the county is 9 feet below sea level.
    \na. Write an expression using integers to represent the difference between the elevations.<\/p>\n

    Answer:
    \nThe expression using integers to represent the difference between the elevations is 5753.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAt its highest point, the elevation of a county is 5,762 feet above sea level.
    \nAt its lowest point, the elevation of the county is 9 feet below sea level.
    \n5762 + (-9) = 5753.
    \nso the expression using integers to represent the difference between the elevations is 5753.<\/p>\n

    b. Will the answer be written as a positive or negative integer?<\/p>\n

    Answer:
    \nThe answer will be written as a positive number.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAt its highest point, the elevation of a county is 5,762 feet above sea level.
    \nAt its lowest point, the elevation of the county is 9 feet below sea level.
    \n5762 + (-9) = 5753.
    \nso the answer will be written as a positive number.<\/p>\n

    c. What is the difference between the highest and lowest points of the county?<\/p>\n

    Answer:
    \nThe difference between the highest and lowest points of the county is 5753.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAt its highest point, the elevation of a county is 5,762 feet above sea level.
    \nAt its lowest point, the elevation of the county is 9 feet below sea level.
    \n5762 + (-9) = 5753.
    \nso the difference between the highest and lowest points of the county is 5753.<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 16.
    \nWhich number line model shows the subtraction 2 – 4?
    \n\"Envision<\/p>\n

    Answer:
    \nOption D is the correct answer.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number line from 0 to -2 and -2 to -6.
    \n-2 + (-6).
    \n-2 -6 = -8.
    \nso option D is the correct answer.<\/p>\n

    Lesson 1.5 Add and Subtract Rational Numbers<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nMalik hikes Castle Trail from point A to point B. The elevation at point A is below sea level. What are possible beginning and ending elevations of Malik’s hike?<\/p>\n

    I can… add and subtract rational numbers.
    \n\"Envision<\/p>\n

    Look for Relationships
    \nHow are elevation values of point A and point B related?<\/p>\n

    Focus on math practices
    \nReasoning What would be different about the hike from point B to point A?<\/p>\n

    Answer:
    \nThe difference between the hike from point B to point A is 120.5 meters.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMalik hikes Castle Trail from point A to point B.
    \nThe elevation at point A is below sea level.
    \n120(1\/2).
    \n120 x 2 = 240.
    \n240 + 1 = 241.
    \n241\/2 = 120.5.
    \nso the difference between the hike from point B to point A is 120.5 meters.<\/p>\n

    Essential Question<\/strong>
    \nHow are adding and subtracting integers related to adding and subtracting other rational numbers?<\/p>\n

    Try It!<\/strong><\/p>\n

    A dolphin is at the surface of the water and then descends to a depth of 4\\(\\frac{1}{2}\\) feet. Then the dolphin swims down another 2\\(\\frac{3}{4}\\) feet. What is the location of the dolphin relative to the surface of the water?
    \n-4\\(\\frac{1}{2}\\) – ________
    \n-4\\(\\frac{1}{2}\\) + _______ = __________
    \n\"Envision
    \nThe location of the dolphin relative to the surface of the water is ________ feet.<\/p>\n

    Answer:
    \nThe location of the dolphin relative to the surface of the water is -7.25 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA dolphin is at the surface of the water and then descends to a depth of 4\\(\\frac{1}{2}\\) feet.
    \nThen the dolphin swims down another 2\\(\\frac{3}{4}\\) feet.
    \n-4(1\/2) – 2(3\/4).
    \n-9\/2 – 11\/4.
    \n-4.5 – 2.75.
    \n-7.25.
    \nso the location of the dolphin relative to the surface of the water is -7.25 feet.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Convince Me! How are adding and subtracting two rational numbers with different signs related to adding and subtracting two integers with different signs?<\/p>\n

    Try It!<\/strong><\/p>\n

    Find the sum or difference of the rational numbers.
    \na. \\(-2.5+\\left(-5 \\frac{6}{10}\\right)\\)<\/p>\n

    Answer:
    \n-2.5 + (-5 x 6\/10) = -8.1.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe sum of two numbers.
    \n-2.5 – 5 x 6\/10.
    \n-2.5 – 56\/10.
    \n-2.5 – 5.6.
    \n-8.1.<\/p>\n

    b. \\(-4.4-\\left(-1 \\frac{1}{2}\\right)\\)<\/p>\n

    Answer:
    \n-5.9.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe difference between two numbers.
    \n-4.4 – 3\/2.
    \n-4 . 4 – 1.5.
    \n-5.9.<\/p>\n

    c. \\(-135.4+78 \\frac{1}{2}\\)<\/p>\n

    Answer:
    \n-56.9.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n-135.4 + 78(1\/2).
    \n-135.4 + 157\/2.
    \n-135.4 + 78.5.
    \n-56.9.<\/p>\n

    Try It!<\/strong><\/p>\n

    Two divers are swimming at different depths below sea level. One diver is at -25.5 feet. The other diver is at –40.75 feet. How much farther below sea level is the diver who is farthest below sea level?<\/p>\n

    Answer:
    \nThe much farther below sea level is the diver who is farthest below sea level = -15.25 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nTwo divers are swimming at different depths below sea level.
    \nOne diver is at -25.5 feet.
    \nThe other diver is at –40.75 feet.
    \n-25.5 – 40.75.
    \n– 15.25.
    \nso the much farther below sea level is the diver who is farthest below sea level = -15.25 feet.<\/p>\n

    KEY CONCEPT<\/strong>
    \nThe rules for adding and subtracting all rational numbers are the same as those for adding and subtracting integers.
    \nThe distance between any two rational numbers p and q on a number line is the absolute value of their difference.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How are adding and subtracting integers related to adding and subtracting other rational numbers?<\/p>\n

    Answer:
    \nThe integers are related to adding and subtracting other rational numbers is the absolute value of their difference.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
    \nfor example:
    \n-5 – (-2).
    \n-2 – (-5).
    \n-2 + 5 = 3.<\/p>\n

    Question 2.
    \nReasoning When finding the distance between two rational numbers on a number line, does the order of the numbers you subtract matter? Explain.<\/p>\n

    Answer:
    \nThe distance between two rational numbers on a number line does the order of the numbers you subtract matter.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
    \nfor example:
    \n-5 – (-2).
    \n-2 – (-5).
    \n-2 + 5 = 3.<\/p>\n

    Question 3.
    \nCritique Reasoning<\/strong> Gwen says that the sum of – 1\\(\\frac{3}{4}\\) and 2\\(\\frac{1}{2}\\) is the same as the difference between 2\\(\\frac{1}{2}\\) and 1\\(\\frac{3}{4}\\). Is Gwen correct? Explain why or why not.<\/p>\n

    Answer:
    \nGwen was correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nGwen says that the sum of – 1\\(\\frac{3}{4}\\) and 2\\(\\frac{1}{2}\\).
    \n-1(3\/4) and 2(1\/2).
    \n-4 + 3 = -7\/4.
    \n4 + 1 = 5\/2.
    \n-7\/4 = -1.75.
    \n5\/2 = 2.5.
    \n1(3\/4) = 7\/4.
    \n7\/4 = 1.75.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nWhat is the distance between the top of the fishing pole and the fish?
    \n\"Envision<\/p>\n

    Answer:
    \nThe distance between the top of the fishing pole and the fish = 4.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe top of the pole = 8(1\/2).
    \nthe distance of the fish = 4(1\/2).
    \n8(1\/2) – 4(1\/2).
    \n8 x 2 = 16.
    \n16 + 1 = 17\/2.
    \n4 x 2 = 8.
    \n8 + 1 = 9.
    \n9\/2 = 4.5.
    \n17\/2 = 8.5.
    \n8.5 – 4.5 = 4.<\/p>\n

    Question 5.
    \nA shark began at 172.5 meters below sea level and then swam up 137.1 meters. Where is the shark’s location now in relation to sea level?<\/p>\n

    Answer:
    \nThe shark’s location now in relation to sea level = 35.4 meters.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA shark began at 172.5 meters below sea level and then swam up 137.1 meters.
    \n172.5 – 137.1.
    \n35.4.
    \nso the shark’s location now in relation to sea level = 35.4 meters.<\/p>\n

    Question 6.
    \nFind the sum or difference.
    \na. -12\\(\\frac{1}{2}\\) + 4\\(\\frac{1}{2}\\)<\/p>\n

    Answer:
    \n-8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers.
    \nadd the numbers.
    \n-12(1\/2) + 4(1\/2).
    \n-25\/2 + 9\/2.
    \n-12.5 + 4.5.
    \n-8.<\/p>\n

    b. -0.35 – (-0.25)<\/p>\n

    Answer:
    \n-0.10.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers.
    \nsubtract the numbers.
    \n-0.35 – (-0.25).
    \n-0.35 + 0.25.
    \n-0.10.<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    Leveled Practice In 7-8, complete the expressions to find the sum or difference.
    \nQuestion 7.
    \n3.2 – (-5.7)
    \n= 3.2 + _____
    \n= ______<\/p>\n

    Answer:
    \n3.2 – (-5.7) = 8.9.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers.
    \nsubtract the numbers.
    \n3.2 – (-5.7).
    \n3.2 + 5.7.
    \n8.9.<\/p>\n

    Question 8.
    \n\\(\\frac{12}{13}+\\left(\\frac{-1}{13}\\right)\\)
    \n= _____ – ______
    \n= ______<\/p>\n

    Answer:
    \n12\/13 + (-1\/13) = 0.85.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers.
    \nadd the numbers.
    \n12\/13 = 0.92.
    \n1\/13 = 0.07.
    \n0.92 – 0.07 = 0.85.<\/p>\n

    Question 9.
    \nReasoning<\/strong> When Tom simplified the expression -2.6 + (-5.4), he got 2.8. What mistake did Tom likely make?<\/p>\n

    Answer:
    \nThe mistake did Tom likely make = -8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n-2.6 + (-5.4).
    \n-2.6 – 5.4.
    \n– 8.<\/p>\n

    Question 10.
    \nThe temperature in a town is 36.6\u00b0F during the day and -12.6\u00b0F at night. What is the temperature change from day to night?<\/p>\n

    Answer:
    \nThe temperature change from day to night = 24\u00b0F.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature in a town is 36.6\u00b0F during the day and -12.6\u00b0F at night.
    \n36.6 + (-12.6).
    \n36.6 – 12.6.
    \n24.
    \nso the temperature change from day to night = 24\u00b0F.<\/p>\n

    Question 11.
    \nSimplify each expression.
    \na. 50\\(\\frac{1}{2}\\) + (-12.3)<\/p>\n

    Answer:
    \n50(1\/2) + (-12.3) = 38.2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are 50(1\/2) and -12.3.
    \nadd the numbers.
    \n101\/2 and -12.3.
    \n50.5 + (-12.3).
    \n50.5 – 12.3 = 38.2.<\/p>\n

    b. -50\\(\\frac{1}{2}\\) + (-12.3)<\/p>\n

    Answer:
    \n-50(1\/2) + (-12.3) = -62.8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -50(1\/2) and -12.3.
    \nadd the numbers.
    \n-101\/2 and -12.3.
    \n-50.5 – 12.3.
    \n– 62.8.<\/p>\n

    C. \u201350\\(\\frac{1}{2}\\) + 12.3<\/p>\n

    Answer:
    \n-50(1\/2) + 12.3 = -32.8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -50(1\/2) and 12.3.
    \nadd the numbers.
    \n-101\/2 + 12.3.
    \n-50.5 + 12.3.
    \n-38.2.<\/p>\n

    Question 12.
    \nAt the beginning of the day, the stock market goes up 30\\(\\frac{1}{2}\\) points. At the end of the day, the stock market goes down 120\\(\\frac{1}{4}\\) points. What is the total change in the stock market from the beginning of the day to the end of the day?<\/p>\n

    Answer:
    \nThe total change in the stock market from the beginning of the day to the end of the day = 89.75 points.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAt the beginning of the day, the stock market goes up 30\\(\\frac{1}{2}\\) points.
    \nAt the end of the day, the stock market goes down 120\\(\\frac{1}{4}\\) points.
    \n30(1\/2) = 61\/2.
    \n61\/2 = 30.5.
    \n120(1\/4) = 481\/4.
    \n481\/4 = 120.25.
    \n120.25 – 30.5 = 89.75.
    \nso the total change in the stock market from the beginning of the day to the end of the day = 89.75 points.<\/p>\n

    Question 13.
    \nA dolphin is swimming 18 feet below the surface of the ocean. There is a coast guard helicopter 75.5 feet above the surface of the water that is directly above the dolphin. What is the distance between the dolphin and the helicopter?<\/p>\n

    Answer:
    \nThe distance between the dolphin and the helicopter = 57.5 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA dolphin is swimming 18 feet below the surface of the ocean.
    \nThere is a coast guard helicopter 75.5 feet above the surface of the water that is directly above the dolphin.
    \n75.5 – 18 = 57.5 feet.
    \nso the distance between the dolphin and the helicopter = 57.5 feet.<\/p>\n

    Question 14.
    \nA bird flies from its nest to the bottom of the canyon. How far did the bird fly?
    \n\"Envision<\/p>\n

    Answer:
    \nThe far did the birds fly = 438.6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA bird flies from its nest to the bottom of the canyon.
    \nthe top of the nest is 528(1\/5).
    \n528 x 1\/5.
    \n528 x 5 = 2640.
    \n2641\/5 = 528.2.
    \nthe height of canyon floor = -89(3\/5).
    \n-89 x 5 = 445.
    \n-448 \/ 5 = -89.6.
    \n528.2 – 89.6 = 438.6.<\/p>\n

    Question 15.
    \nA scuba diving instructor takes a group of students to a depth of 54.96 feet. Then they ascend 22.38 feet to see some fish. Where are the fish in relation to the surface?<\/p>\n

    Answer:
    \nThe fish in relation to the surface = 32.58 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA scuba diving instructor takes a group of students to a depth of 54.96 feet.
    \nThen they ascend 22.38 feet to see some fish.
    \n54.96 – 22.38 = 32.58.
    \nso the fish in relation to the surface = 32.58 feet.<\/p>\n

    Question 16.
    \nModel with Math Write an addition expression that is represented by the number line.
    \n\"Envision<\/p>\n

    Answer:
    \nThe addition expression that is represented by the number line = 0.5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n0 + 1 = 1.
    \n0 + 0.5 = 0.5.
    \n1 – 0.5 = 0.5.
    \nso the addition expression that is represented by the number line is 1 + (-0.5).<\/p>\n

    Question 17.
    \nThe roots of a plant reach down 3\\(\\frac{3}{4}\\) inches below ground. How many inches is the plant above the ground?
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of inches is the plant above the ground = 8.75 inches.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe roots of a plant reach down 3\\(\\frac{3}{4}\\) inches below ground.
    \n3(3\/4) = 15\/4.
    \n15\/4 = 3.75.
    \n12(1\/2) = 25\/2.
    \n25\/2 = 12.5
    \n12.5 – 3.75 = 8.75.
    \nso the number of inches is the plant above the ground = 8.75 inches.<\/p>\n

    Question 18.
    \nHigher Order Thinking<\/strong>
    \na. Simplify the expression (-13.2) + 8.1.<\/p>\n

    Answer:
    \n-13.2 + 8.1 = -5.1.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -13.2 and 8.1.
    \nadd the numbers.
    \n-13.2 + 8.1.
    \n– 5.1.
    \n-13.2 + 8.1 = -5.1.<\/p>\n

    b. How are (-13.2) + 8.1 and 13.2 + (-8.1) related? Explain without computing.<\/p>\n

    Answer:
    \n-13.2 + 8.1 = -5.1.
    \n13.2 – 8.1 = 5.1.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are 13.2 and 8.1.
    \nadd the numbers.
    \nthey have an equal value but the sign is different.<\/p>\n

    c. Using a property of operations, what can you say about the sum of the two expressions?<\/p>\n

    Answer:
    \nThe sum of the two expressions is also positive.<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 19.
    \nThe temperatures at sunrise and sunset are shown in the table.
    \n\"Envision
    \nPART A
    \nWrite an expression that represents the change in temperature for Day 1. Show how you can use properties of operations to find the value of the expression.<\/p>\n

    Answer:
    \nThe properties of operations to find the value of the expression is -11.31 + 13.49.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe temperatures at sunrise and sunset are shown.
    \nsunrise on day 1 is -11.31 and day 2 is -7.69.
    \nsunset on day 1 is 13.49 and day 2 is 25.25.
    \n-11.31 + (13.49).
    \n-11.31 + 13.49.
    \n2.18.
    \nso the properties of operations to find the value of the expression is -11.31 + 13.49 is 2.18.<\/p>\n

    PART B
    \nOn which day did the temperature change more? Explain your reasoning.<\/p>\n

    Answer:
    \nOn day 2 the temperature change more.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe temperatures at sunrise and sunset are shown.
    \nsunrise on day 1 is -11.31 and day 2 is -7.69.
    \nsunset on day 1 is 13.49 and day 2 is 25.25.
    \n-11.31 + (13.49).
    \n-11.31 + 13.49.
    \n2.18.
    \n-7.69 + 25.25.
    \n17.56.
    \nso on day 2 the temperature change more.<\/p>\n

    Question 20.
    \nMischa dives from a platform that is 5 meters above water. Her dive takes her 2.1 meters below the surface of the water. Which expression could represent the distance, in meters, that Mischa dives? Select all that apply.
    \n\u2610 |5 – (-2.1)|
    \n\u2610 |-(2.1) \u2013 (-5)|
    \n\u2610 |2.1 – 5|
    \n\u2610 |-(2.1) \u2013 5|
    \n\u2610 |5 + (-2.1)|<\/p>\n

    Answer:
    \nThe expression could represent the distance in meters is 5 – (-2.1).<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMischa dives from a platform that is 5 meters above the water.
    \nHer dive takes her 2.1 meters below the surface of the water.
    \n5 – (-2.1).
    \n5 + 2.1 = 7.1.
    \nso the expression could represent the distance in meters is 7.1.<\/p>\n

    Topic 1 Mid-Topic Checkpoint<\/p>\n

    Question 1.
    \nVocabulary How do you find the additive inverse of a number? Give an example of a number and its additive inverse. Lesson 1-3<\/p>\n

    Answer:
    \nThe additive inverse of a number is always negative.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -4 and -6.
    \n-4 + (-6).
    \n-4 – 6 = -10.<\/p>\n

    Question 2.
    \nA plastic toy submarine is held 15 centimeters below the water surface in a bath tub. The submarine is let go and rises 15 centimeters. What integer represents the toy submarine’s position with respect to the surface of the water? Lesson 1-1<\/p>\n

    Answer:
    \nThe positive integer represents the toy submarine’s position with respect to the surface of the water.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA plastic toy submarine is held 15 centimeters below the water surface in a bathtub.
    \nThe submarine is let go and rises 15 centimeters.
    \n-15 + (15).
    \n-15 + 15 = 0.
    \nso the integer is zero.<\/p>\n

    Question 3.
    \nThe temperature in the late afternoon was -7.5\u00b0C. It dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight. What was the temperature at midnight? Lessons 1-3, 1-4, and 1-5<\/p>\n

    Answer:
    \nThe temperature at the midnight = -21\u00b0C.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe temperature in the late afternoon was -7.5\u00b0C.
    \nIt dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight.
    \n-7.5 – 5 – 8.5.
    \n-7.5 – 13.5.
    \n-21.
    \nso the temperature at the midnight is -21\u00b0C.<\/p>\n

    Question 4.
    \nThe floor of an elevator in a building is 30 feet above ground level. It travels down to the lower level of the building, where the floor is 10 feet below ground level. What distance has the elevator’s floor traveled? Lessons 1-4 and 1-5<\/p>\n

    Answer:
    \nThe distance has the elevator’s floor traveled = 20 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe floor of an elevator in a building is 30 feet above ground level.
    \nIt travels down to the lower level of the building, where the floor is 10 feet below ground level.
    \n30 + (-10).
    \n30 – 10.
    \n20.
    \nso the distance has the elevator’s floor traveled = 20 feet.<\/p>\n

    Question 5.
    \nGreg says that 3.3. is a rational number. Kari says 3.3 is not a terminating decimal. Who is correct and why? Lesson 1-2<\/p>\n

    Answer:
    \nYes. both of them are correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nGreg says that 3.3. is a rational number.
    \n3.3 is a rational number.
    \nKari says 3.3 is not a terminating decimal.
    \n3.3 is also a terminating decimal.
    \nso both of them are correct.<\/p>\n

    Question 6.
    \nCece is hiking on a mountain and stops at 15\\(\\frac{5}{8}\\) feet above sea level. The base of the mountain is 10.2 feet below sea level. What is the vertical distance between Cece and the base of the mountain? Lesson 1-5
    \nA. 5.425 feet
    \nB. 25.825 feet
    \nC. 25\\(\\frac{3}{8}\\) feet
    \nD. 5\\(\\frac{1}{4}\\) feet<\/p>\n

    Answer:
    \nOption A is correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nCece is hiking on a mountain and stops at 15\\(\\frac{5}{8}\\) feet above sea level.
    \nThe base of the mountain is 10.2 feet below sea level.
    \n15(5\/8) – 10.2.
    \n125\/8 – 10.2.
    \n15.625 – 10.2.
    \n5.425.
    \nso option A is correct.<\/p>\n

    Topic 1 Mid-Topic Performance Task<\/p>\n

    An oceanographer, Dr. Price, is studying the types of sea life at various depths.
    \n\"Envision<\/p>\n

    PART A<\/p>\n

    Dr. Price uses a table to organize the types of sea life and the positions relative to sea level of each location.<\/p>\n

    Complete each sentence.
    \nThe difference between Location A and Location B is ________ meters.
    \nThe difference between Location B and Location C is ________ meters.
    \nThe difference between Location C and Location D is ________ meters.<\/p>\n

    Answer:
    \nThe difference between Location A and Location B is 202.25 meters.
    \nThe difference between Location B and Location C is 1686.6 meters.
    \nThe difference between Location C and Location D is474.875 meters.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nlocation A = -895.9.
    \nlocation B = – 1098(3\/20).
    \nlocation C = -2784.75.
    \nlocation D = – 3259(5\/8).
    \n-895.9 – (-1098(3\/20).
    \n-895.9 + 21963\/20.
    \n-895.9 + 1098.15.
    \n202.25.
    \n-1098.15 – (-2784.75).
    \n-1098.15 + 2784.75.
    \n1686.6.
    \n-2784.75 + 3259.625.
    \n474.875.<\/p>\n

    PART B
    \nAfter observing Location B, Dr. Price returns to Location A before descending to Location C. What is the total distance she travels?<\/p>\n

    Answer:
    \nThe total distance she travels = -2987 meters.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nlocation B = -1098.15.
    \nlocation A = – 895.9.
    \nlocation C = -2784.75.
    \n-1098.15 + (-895.9) + (- 2784.75).
    \n-1098.15 – 895.9 – 2784.75.
    \n-1098.15 – 1888.85.
    \n-2987.
    \nso the total distance she travels = -2987 meters.<\/p>\n

    PART C
    \nDr. Price descends to Location:D to observe shrimp. She then ascends and stops to observe sea life that is halfway between Location B and Location C. What is the total distance between Location D and where Dr. Price stopped to observe?<\/p>\n

    Answer:
    \nThe distance between Location D and where Dr. Price stopped to observe = -272.625 meters.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nDr. Price descends to Location:D to observe shrimp.
    \nShe then ascends and stops to observe sea life that is halfway between Location B and Location C.
    \n-3259.625 – (-2987).
    \n-3259.625 + 2987.
    \n-272.625.
    \nso the distance between Location D and where Dr. Price stopped to observe = -272.625 meters.<\/p>\n

    Lesson 1.6 Multiply Integers<\/h3>\n

    Explore It!<\/strong>
    \nA popular beach erodes 4 inches per year on average.
    \nI can… multiply integers.
    \n\"Envision<\/p>\n

    A. How many years will it take for the coastline to erode one foot?<\/p>\n

    Answer:
    \nThe number of years will it take for the coastline to erode one foot = 12 years.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA popular beach erodes 4 inches per year on average.
    \n1 year = 4 inches.
    \none feet = 12 inches.
    \nso the number of years will it take for the coastline to erode one foot = 12 years.<\/p>\n

    B. The number line below shows the expected change in the coastline as years pass. How could you use the number line to show the erosion after 10 years?
    \n\"Envision<\/p>\n

    Answer:
    \nThe erosion after 10 years = 10.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe number line below shows the expected change in the coastline as the years pass.
    \ncoastline this year = 0.
    \n0 + 10 = 10.
    \nso the erosion after 10 years = 10.<\/p>\n

    Focus on math practices
    \nBe Precise What expression could you use to represent the change in the coastline in 5 years?<\/p>\n

    Essential Question<\/strong>
    \nHow do the signs of factors affect their product?<\/p>\n

    Try It!<\/strong><\/p>\n

    A race car game takes 6 points from a player each time the player hits a cone. What integer represents the change in total points if the player hits 10 cones?
    \n10 \u2022 ______ = ______
    \nThe change in total points is ______.<\/p>\n

    Answer:
    \nIf the player hits 10 cones = 60 points.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA race car game takes 6 points from a player each time the player hits a cone.
    \n10 x 6 = 60.
    \nso the change in total points is 60 points.<\/p>\n

    Convince Me! Could the product of a positive integer and a negative integer be positive? Explain.<\/p>\n

    Try It!<\/strong><\/p>\n

    Find each product.
    \na. -7 \u2022 (-2)<\/p>\n

    Answer:
    \nThe product is 14.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -7 and -2.
    \nmultiply the numbers.
    \n-7 x -2.
    \n14.
    \nso the product is 14.<\/p>\n

    b. 7 \u2022 (-13)<\/p>\n

    Answer:
    \nThe product is -91.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are 7 and -13.
    \nmultiply the numbers.
    \n7 x -13.
    \n-91.
    \nso the product is -91.<\/p>\n

    c.-6 \u2022 8<\/p>\n

    Answer:
    \nThe product is -48.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -6 and 8.
    \nmultiply the numbers.
    \n-6 x 8.
    \n-48.
    \nso the product is -48.<\/p>\n

    d.(-1) \u2022 (-1)<\/p>\n

    Answer:
    \nThe product is 1.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two numbers are -1 and -1.
    \nmultiply the numbers.
    \n-1 x -1.
    \n1.
    \nso the product is 1.<\/p>\n

    KEY CONCEPT<\/strong>
    \nWhen multiplying two integers, the sign of the product depends on the sign of the factors.
    \nIf the signs of the factors are the same, the product is positive.
    \n7 \u2022 3 = 21
    \n– 7 \u2022 (-3) = 21<\/p>\n

    If the signs of the factors are different, the product is negative.
    \n-4 \u2022 5 = -20
    \n4 \u2022 (-5) = -20<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question<\/strong> How do the signs of factors affect their product?<\/p>\n

    Answer:
    \nThe signs of factors affect their product according to the sign.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that.
    \nIf the signs of the factors are the same, the product is positive.
    \n7 x 3 = 21
    \n– 7 x (-3) = 21
    \nIf the signs of the factors are different, the product is negative.
    \n-4 x 5 = -20
    \n4 x (-5) = -20<\/p>\n

    Question 2.
    \nConstruct Arguments What is the sign of the product if you multiplied three negative integers? Explain your answer.<\/p>\n

    Answer:
    \nThe product is also a negative integer.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that.
    \nthe numbers are -7, -8, and -3.
    \n-7 x -8 x -3.
    \n-7 -24.
    \n-168.
    \nso the product is also a negative integer.<\/p>\n

    Question 3.
    \nReasoning<\/strong> Explain why the product of two negative integers is not negative. Use (-1)(-1) as an example.<\/p>\n

    Answer:
    \nyes, the product of two negative integers is positive.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe product of two negative integers is positive.
    \nfor example:
    \n-1 x -1 = 1.<\/p>\n

    Question 4.
    \nUse Structure is the product the same when multiplying 22 \u00d7 (-5) and multiplying (-5) \u00d7 22? Explain.<\/p>\n

    Answer:
    \nYes, the product is the same.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 22 and -5.
    \n22 x -5 = 110.
    \nthe numbers are -5 and 22.
    \n-5 x 22 = 110.
    \nso both the product is the same.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 5.
    \nRepresent 2 \u2022 (-3) on the number line.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number is -6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 2 and -3.
    \n2 x -3 = -6.
    \nso the number is -6.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

    Question 6.
    \nWhich of these products is negative? Select all that apply.
    \n\u2610 -8 \u2022 (-3)
    \n\u2610 -2 \u2022 8
    \n\u2610 0 \u2022 (-2)
    \n\u2610 15 \u2022 (-5)
    \n\u2610 -8 \u2022 (-9)<\/p>\n

    Answer:
    \n-2 x 8 = -16.
    \n15 x -5 = -75.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe product is negative.
    \nthe numbers are – 2 and 8.
    \n-2 x 8 = -16.
    \n15 x -5 = -75.<\/p>\n

    Question 7.
    \nFind each product.
    \na. -9 \u2022 (-4)
    \nb.-7 \u2022 12
    \nc. 8 \u2022 (-8)
    \nd. 9 \u2022 15<\/p>\n

    Answer:
    \nThe products are 36, -84, -64, and 135.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n-9 x -4 = 36.
    \n12 x -7 = -84.
    \n8 x -8 = 64.
    \n15 x 9 = 135.
    \nso the products are 36, -84, -64, and 135.<\/p>\n

    Question 8.
    \nA game show contestant starts a game by answering two questions incorrectly. Each incorrect answer costs the contestant $600. Use a product of two integers to show the point total that would appear for the contestant.
    \n\"Envision<\/p>\n

    Answer:
    \nThe numbers are $20 x $30.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA game show contestant starts a game by answering two questions incorrectly.
    \nEach incorrect answer costs the contestant $600.
    \nthe product of two integers is $600.
    \n$20 x $30.
    \nso the numbers are $20 and $30.<\/p>\n

    Practice & Problem Solving<\/strong><\/p>\n

    In 9-14, multiply.
    \nQuestion 9.
    \n(-6) \u2022 (-2)<\/p>\n

    Answer:
    \nThe product is 12.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -6 and -2.
    \n-6 x -2.
    \n12.
    \nso the product is 12.<\/p>\n

    Question 10.
    \n4 \u2022 (-8)<\/p>\n

    Answer:
    \nThe product is -32.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -8 and 4.
    \n-8 x 4.
    \n-32.
    \nso the product is -32.<\/p>\n

    Question 11.
    \n7 \u2022 (-5)<\/p>\n

    Answer:
    \nThe product is -35.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -5 and -7.
    \n-5 x 7.
    \n-35.
    \nso the product is -35.<\/p>\n

    Question 12.
    \n-5 \u2022 2<\/p>\n

    Answer:
    \nThe product is -10.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -5 and 2.
    \n-5 x 2.
    \n-10.
    \nso the product is -10.<\/p>\n

    Question 13.
    \n-1 \u2022 (-24)<\/p>\n

    Answer:
    \nThe product is 24.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -1 and -24.
    \n-1 x -24.
    \n24.
    \nso the product is 24.<\/p>\n

    Question 14.
    \n(5) \u2022 (-9) \u2022 (-2)<\/p>\n

    Answer:
    \nThe product is 90.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 5, -9, and -2.
    \n-9 x -2.
    \n18.
    \n18 x 5 = 90.
    \nso the product is 90.<\/p>\n

    Question 15.
    \nA football team lost the same number of yards on each of 3 consecutive plays. What is the total change in yards from where the team started?
    \n\"Envision<\/p>\n

    Answer:
    \nThe total change in yards from where the team started = 18 yards.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA football team lost the same number of yards on each of 3 consecutive plays.
    \n6 x 3 = 18.
    \nso the total change in yards from where the team started = 18 yards.<\/p>\n

    Question 16.
    \na. Find the product.
    \n-41 \u2022 (-1)<\/p>\n

    Answer:
    \nThe product is 41.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -41 and -1.
    \nmultiply the numbers.
    \n-41 x -1 = 41.
    \nso the product is 41.<\/p>\n

    b. Construct Arguments Describe how you use the properties of multiplication to find the product.<\/p>\n

    Answer:
    \nThe properties of multiplication depend on the sign.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIf the signs of the factors are the same, the product is positive.
    \n7 x 3 = 21
    \n– 7 x (-3) = 21
    \nIf the signs of the factors are different, the product is negative.
    \n-4 x 5 = -20
    \n4 x (-5) = -20<\/p>\n

    Question 17.
    \nAlex is working to simplify 5 \u2022 (-8) \u2022 2.
    \na. What is the product?<\/p>\n

    Answer:
    \nThe product is -80.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that.
    \nthe numbers are 5, 2, and -8.
    \nmultiply the numbers.
    \n5 x -8 x 2 = -80.
    \nso the product is -80.<\/p>\n

    b. Suppose Alex found the opposite of the correct product. Describe an error he could have made that resulted in that product.<\/p>\n

    Answer:
    \nThe opposite of the correct product is 80.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 5, 2, and -8.
    \nmultiply the numbers.
    \n5 x -8 x 2 = -80.
    \nso the product is -80.<\/p>\n

    Question 18.
    \nWhich product is greater, (-4) \u2022 (-6) or (-7) \u2022 (-8)? Explain.<\/p>\n

    Answer:
    \n-7 x -8 = 56.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe product of two numbers.
    \n-4 x -6 = 24.
    \n-7 x -8 = 56.
    \nso 56 is the greatest product.<\/p>\n

    Question 19.
    \nMake Sense and Persevere While playing a board game, Cecilia had to move back 6 spaces 9 times. What integer represents Cecilia’s movement on the board for those 9 turns?<\/p>\n

    Answer:
    \nThe integer represents Cecilia’s movement on the board for those 9 turns = 54 times.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nWhile playing a board game, Cecilia had to move back 6 spaces 9 times.
    \n6 x 9 = 54.
    \nso the integer represents Cecilia’s movement on the board for those 9 turns = 54 times.<\/p>\n

    Question 20.
    \nAnya makes withdrawals from and deposits into her bank account.
    \na. What integer represents the change in the amount in her account if Anya withdraws $12 once each day for four days?<\/p>\n

    Answer:
    \nThe integer represents the change in the amount in her account = $48.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAnya makes withdrawals and deposits into her bank account.
    \n$12 x 4 = $48.
    \nso the integer represents the change in the amount in her account = $48.<\/p>\n

    b. What integer represents the change in the amount in her account if Anya deposits $12 once each day for four days?<\/p>\n

    Answer:
    \nThe integer represents the change in the amount in her account = $48.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAnya makes withdrawals and deposits into her bank account.
    \n$12 x 4 = $48.
    \nso the integer represents the change in the amount in her account = $48.<\/p>\n

    c. Look for Relationships Explain the difference between the integer for the withdrawals and the integer for the deposits.<\/p>\n

    Answer:
    \nThe difference is zero.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAnya makes withdrawals and deposits into her bank account.
    \n$12 x 4 = $48.
    \nso the integer represents the change in the amount in her account = $48.
    \n$48 – $48 = 0.<\/p>\n

    Question 21.
    \nHigher Order Thinking<\/strong> A gold mine has two elevators, one for equipment and one for miners One day, the equipment elevator begins to descend. After 28 seconds, the elevator for the miners begins to descend. What is the position of each elevator relative to the surface after another 14 seconds? At that time, how much deeper is the elevator for the miners?
    \n\"Envision<\/p>\n

    Answer:
    \nAt 14 seconds the elevator of the miners is 210 feet.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA gold mine has two elevators, one for equipment and one for miners One day, the equipment elevator begins to descend.
    \nAfter 28 seconds, the elevator for the miners begins to descend.
    \n14 x 4 = 56.
    \n14 x 15 = 210.
    \nso the elevator of the miners is 210 feet.<\/p>\n

    Assessment Practice<\/strong><\/p>\n

    Question 22.
    \nA number line is shown.
    \n\"Envision
    \nWrite a multiplication equation that is represented by the number line.<\/p>\n

    Answer:
    \nThe equation is -35 x 2 = -70.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number line is moving from 0 to -35.
    \n-35 to -70.
    \n-35 x 2 = -70.
    \nso the equation is -35 x 2 = -70.<\/p>\n

    Question 23.
    \nWhich of these expressions have the same product as (-6) \u2022 7? Select all that apply.
    \n\u2610 (-3) \u2022 14
    \n\u2610 16 \u2022 (-3)
    \n\u2610 -6 \u2022 (-7)
    \n\u2610 7 \u2022 (-6)
    \n\u2610 14 \u2022 (-3)<\/p>\n

    Answer:
    \n-3 x 14, 7 x -6 and 14 x -3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe product is -6 x 7 = -42.
    \n-3 x 14 = -42.
    \n7 x -6 = -42.
    \n14 x -3 = -42.<\/p>\n

    Lesson 1.7 Multiply Rational Numbers<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nStella is making the United States flag. She has blue fabric, red fabric, and white fabric. Choose a length for the flag. What length of blue fabric would Stella need to make this flag? Explain your thinking.<\/p>\n

    I can… multiply rational numbers.
    \n\"Envision<\/p>\n

    Focus on math practices
    \nBe Precise The blue region of the flag is \\(\\frac{7}{13}\\) the width and \\(\\frac{2}{5}\\) the length of the flag. What part of the total area is the blue region of the flag?<\/p>\n

    Answer:
    \nThe total area is the blue region of the flag = 0.2152.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe blue region of the flag is \\(\\frac{7}{13}\\) the width.
    \n\\(\\frac{2}{5}\\) the length of the flag.
    \n7\/13 x 2\/5.
    \n0.538 x 0.4.
    \n0.2152.
    \nso the total area is the blue region of the flag = 0.2152.<\/p>\n

    Essential Question<\/strong>
    \nHow is multiplying rational numbers like multiplying integers?<\/p>\n

    Try It!<\/strong><\/p>\n

    Meghan’s bank account is charged $9.95 per month for an online newspaper subscription. How could you represent the change in her account balance after three months of charges?
    \n\"Envision
    \n_____ \u2022 -9.95 = ______
    \nAfter three months, the change in her account balance is $ ________.<\/p>\n

    Answer:
    \nThe change in her account balance after three months of charges = -20.05.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMeghan’s bank account is charged $9.95 per month for an online newspaper subscription.
    \n$9.95 x -30 = -20.05.
    \nso the change in her account balance after three months of charges = -20.05.<\/p>\n

    Convince Me! Meghan’s bank account is charged 3 times. Without calculating, how can you determine whether this is a negative or positive change to her account? Explain.<\/p>\n

    Try It!<\/strong><\/p>\n

    Find each product.
    \na. -5.3 \u2022 (-2.6)<\/p>\n

    Answer:
    \nThe product is 13.78.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -5.3 and -2.6.
    \nmultiply the numbers.
    \n-5.3 x – 2.6.
    \n13.78.<\/p>\n

    b. \\(-\\frac{3}{5} \\cdot 4 \\frac{1}{6}\\)<\/p>\n

    Answer:
    \n-3\/5 x 4(1\/6) = -2.46.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are -3\/5 and 25\/6.
    \n-0.6 x 4.1.
    \n-2.46.
    \nthe product is -2.46.<\/p>\n

    c. 0.2 \u2022 (-1.78)<\/p>\n

    Answer:
    \n0.2 x – 1.78 = -0.356.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 0.2 x -1.78.
    \nmultiply the numbers.
    \n0.2 x -1.78 = -0.356.<\/p>\n

    d. -2.5 \u2022 (70)<\/p>\n

    Answer:
    \nThe product is -175.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers are 70 x -2.5.
    \nmultiply the numbers.
    \n-2.5 x 70 = -175.<\/p>\n

    KEY CONCEPT<\/strong>
    \nThe same rules for multiplying integers apply to multiplying all rational numbers. When multiplying two rational numbers:<\/p>\n

      \n
    • If the signs of the factors are the same, the product is positive.<\/li>\n
    • If the signs of the factors are different, the product is negative.<\/li>\n<\/ul>\n

      Do You Understand?<\/strong>
      \nQuestion 1.
      \nEssential Question How is multiplying rational numbers like multiplying integers?<\/p>\n

      Answer:
      \nThe same rules for multiplying integers apply to multiplying all rational numbers.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nIf the signs of the factors are the same, the product is positive.
      \nIf the signs of the factors are different, the product is negative.<\/p>\n

      Question 2.
      \nHow do you multiply a decimal greater than 0 and a fraction less than 0?<\/p>\n

      Answer:
      \nThe product is negative.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe decimal greater than 0 is 1.5.
      \na fraction less than 0 is – 2.2.
      \n1.5 x – 2.2 = -3.3.
      \nso the product is negative.<\/p>\n

      Question 3.
      \nModel with Math How does this number line represent multiplication of a negative number by a positive number? Explain.
      \n\"Envision<\/p>\n

      Answer:
      \n-2\/3 x -1\/3 = 0.108.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are -2\/3 and -1\/3.
      \nmultiply the numbers.
      \n-2\/3 x -1\/3.
      \n-0.6 x – 0.33.
      \n0.108.
      \n-2\/3 x -1\/3 = 0.108.<\/p>\n

      Do You Know How?<\/strong>
      \nQuestion 4.
      \nUse the number line to find the product
      \n\"Envision<\/p>\n

      Answer:
      \nThe product is -4.5.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are 3 and -1(1\/2).
      \nmultiply the numbers.
      \n3 x -3\/2.
      \n3 x – 1.5.
      \n-4.5.
      \nso the product is -4.5.
      \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

      Question 5.
      \nWhich of these products is positive? Select all that apply.
      \n\u2610 -0.2 \u2022 (12.5)
      \n\u2610 \\(\\frac{1}{12}\\) \u2022 \\(\\left(-6 \\frac{1}{2}\\right)\\)
      \n\u2610 3.2 \u2022 \\(\\left(-\\frac{1}{900}\\right)\\)
      \n\u2610 -3\\(\\frac{1}{2}\\) \u2022 0
      \n\u2610 -4.7 \u2022 (-1)<\/p>\n

      Answer:
      \n-4.7 x -1 = 4.7.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -4.7 and -1.
      \nmultiply the product.
      \n-4.7 x -1 = 4.7.
      \nso the product is 4.7.<\/p>\n

      Question 6.
      \nFind the product.
      \na. -3.1 \u2022 (-2.9)<\/p>\n

      Answer:
      \nThe product is 8.99.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -3.1 and -2.9.
      \nmultiply the product.
      \n-3.1 x -2.9 = 8.99.
      \nso the product is 8.99.<\/p>\n

      b. 1\\(\\frac{1}{2}\\) \u2022 \\(\\left(-\\frac{5}{3}\\right)\\)<\/p>\n

      Answer:
      \n1(1\/2) x -5\/3 = -2.4.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are 3\/2 and -5\/3.
      \nmultiply the product.
      \n3\/2 x – 5\/3.
      \n1.5 x – 1.6.
      \n-2.4.
      \nso the product is -2.4.<\/p>\n

      c. -3\\(\\frac{1}{2}\\) \u2022 0.5<\/p>\n

      Answer:
      \n-3(1\/2) x 0.5 = – 1.75.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -7\/2 and 0.5.
      \nmultiply the product.
      \n-7\/2 x 0.5.
      \n-3.5 x 0.5 = -1.75.<\/p>\n

      d. –\\(\\frac{4}{5}\\) \u2022 –\\(\\frac{1}{8}\\)<\/p>\n

      Answer:
      \n-4\/5 x – 1\/8 = 0.1.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -4\/5 and -1\/8.
      \nmultiply the product.
      \n-4\/5 x -1\/8.
      \n– 0.8 x – 0.125 = 0.1.<\/p>\n

      Practice & Problem Solving<\/strong><\/p>\n

      In 7-14, multiply.
      \nQuestion 7.
      \n(-2.655) \u2022 (18.44)<\/p>\n

      Answer:
      \n-2.655 x 18.44 = -48.9582.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -2.655 and 18.44.
      \nmultiply the product.
      \n-2.655 x 18.44 = -48.9582.<\/p>\n

      Question 8.
      \n-1\\(\\frac{5}{6}\\) \u2022 6\\(\\frac{1}{2}\\)<\/p>\n

      Answer:
      \n-1(5\/6) x 6(1\/2) = -11.895.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -1(5\/6) and 6(1\/2).
      \nmultiply the product.
      \n-11\/6 x 13\/2.
      \n-1.83 x 6.5.
      \n-11.895.<\/p>\n

      Question 9.
      \n-2\\(\\frac{1}{2}\\) \u2022 (-1\\(\\frac{2}{3}\\))<\/p>\n

      Answer:
      \n-2(1\/2) x -1(2\/3) = 4.1.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -(5\/2) and -(5\/3).
      \nmultiply the product.
      \n-5\/2 = -2.5.
      \n-5\/3 = – 1.6.
      \nso the product is 4.1.<\/p>\n

      Question 10.
      \n-3\\(\\frac{7}{8}\\) \u2022 (-5\\(\\frac{3}{4}\\))<\/p>\n

      Answer:
      \n-3(7\/8) x -5(3\/4) = 22.28125.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -31\/8 and -23\/4.
      \nmultiply the product.
      \n-31\/8 x -23\/4.
      \n3.875 x 5.75.
      \n22.28125.<\/p>\n

      Question 11.
      \n-7.5 \u2022 -2\\(\\frac{3}{4}\\)<\/p>\n

      Answer:
      \n-7.5 x -2(3\/4) = 20.625.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -7.5 and -11\/4.
      \nmultiply the product.
      \n-7.5 x – 2.75.
      \n20.625.<\/p>\n

      Question 12.
      \n-0.6 \u2022 (-0.62)<\/p>\n

      Answer:
      \n-0.6 x -0.62 = 0.372.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -0.6 and -0.62.
      \nmultiply the product.
      \n-0.6 x -0.62 = 0.372.<\/p>\n

      Question 13.
      \n-0.2 \u2022 –\\(\\frac{5}{6}\\)<\/p>\n

      Answer:
      \n-0.2 x – 5\/6 = 0.166.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -0.2 and -5\/6.
      \nmultiply the product.
      \n-0.2 x – 0.83.
      \n0.166.<\/p>\n

      Question 14.
      \n–\\(\\frac{5}{6}\\) \u2022 \\(\\frac{1}{8}\\)<\/p>\n

      Answer:
      \n-5\/6 x 1\/8 = 0.0375.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -5\/6 and -1\/8.
      \nmultiply the product.
      \n-5\/6 x -1\/8.
      \n-0.83 x – 0.125 = 0.0375.<\/p>\n

      Question 15.
      \nAt the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.
      \n\"Envision
      \na. Represent the total change in the cost of a ticket given their losses.<\/p>\n

      Answer:
      \nThe total change in the cost of a ticket given their losses = $20.3524.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nAt the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.
      \nticket prices decrease by $0.41 for every game the panthers lose this season.
      \n$0.41 x $49.64.
      \n$20.3524.
      \nso the total change in the cost of a ticket given their losses = $20.3524.<\/p>\n

      b. What is the cost of a ticket for the next game they play?<\/p>\n

      Answer:
      \nThe cost of a ticket for the next game they play = $20.3524.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nAt the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.
      \nticket prices decrease by $0.41 for every game the panthers lose this season.
      \n$0.41 x $49.64.
      \n$20.3524.
      \nso the total change in the cost of a ticket given their losses = $20.3524.<\/p>\n

      Question 16.
      \nThe price per share of ENVX stock is dropping at a rate of $1.45 each hour.
      \n\"Envision
      \na. Write the rate as a negative number.<\/p>\n

      Answer:
      \nThe rate as a negative number = $55.274.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe price per share of ENVX stock is dropping at a rate of $1.45 each hour.
      \n$1.45 x 38.12 = $55.274.
      \nso the rate as a negative number = $55.274.<\/p>\n

      b. What rational number represents the change in the price per share after 5 hours?<\/p>\n

      Answer:
      \nThe rational number represents the change in the price per share after 5 hours = 190.6.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe rational number is 38.12.
      \n38.12 x 5 = 190.6.
      \nso the rational number represents the change in the price per share after 5 hours = 190.6.<\/p>\n

      c. What is the price per share after 5 hours?<\/p>\n

      Answer:
      \nThe price per share after 5 hours = 953.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe price per share after 5 hours is:
      \n190.6 x 5 = 953.
      \nso the price per share after 5 hours = 953.<\/p>\n

      Question 17.
      \nMing incorrectly says that this product is \\(\\frac{4}{63}\\).
      \n\\(-\\left(-\\frac{4}{9}\\right)\\) \u2022 \\(\\left(-\\frac{1}{7}\\right)\\)
      \na. What is the correct product?<\/p>\n

      Answer:
      \nThe correct product is 0.0616.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nMing incorrectly says that this product is \\(\\frac{4}{63}\\).
      \n4\/63 = 0.063.
      \n-4\/9 x -1\/7.
      \n– 0.44 x – 0.14.
      \n0.0616.
      \nso the correct product is 0.0616.<\/p>\n

      b. What error could Ming have made?<\/p>\n

      Answer:
      \nThe error could ming have made = 0.0616.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nMing incorrectly says that this product is \\(\\frac{4}{63}\\).
      \n4\/63 = 0.063.
      \n-4\/9 x -1\/7.
      \n– 0.44 x – 0.14.
      \n0.0616.
      \nso the correct product is 0.0616.<\/p>\n

      Question 18.
      \nHigher Order Thinking<\/strong> place the products in order from least to greatest.
      \n\\(4 \\frac{4}{7}\\) \u2022 \\(4 \\frac{4}{7}\\)
      \n\\(5 \\frac{6}{7}\\) \u2022 \\(\\left(-6 \\frac{6}{7}\\right)\\)
      \n\\(-5 \\frac{1}{8}\\) \u2022 \\(\\left(-2 \\frac{1}{4}\\right)\\)<\/p>\n

      Answer:
      \n-39.44, 11.53, and 20.894.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n4(4\/7) x 4(4\/7).
      \n32\/7 x 32\/7.
      \n4.571 x 4.571.
      \n20.894.
      \n41\/7 x -48\/7.
      \n5.8 x – 6.8.
      \n-39.44.
      \n-41\/8 x -9\/4.
      \n-5.125 x -2.25.
      \n11.53.
      \nso the numbers from least to greatest are -39.44, 11.53, and 20.894.<\/p>\n

      Assessment Practice<\/strong><\/p>\n

      Question 19.
      \nSuppose there is a 1.3\u00b0F drop in temperature for every thousand feet that an airplane climbs into the sky. The temperature on the ground is -2.8\u00b0F.
      \nPART A
      \nWrite a multiplication equation to represent the change in temperature after the plane ascends 10,000 feet.<\/p>\n

      Answer:
      \nThe equation represents the change in temperature after the plane ascends 10,000 feet = 4.1\u00b0F.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nSuppose there is a 1.3\u00b0F drop in temperature for every thousand feet that an airplane climbs into the sky.
      \nThe temperature on the ground is -2.8\u00b0F.
      \n1.3\u00b0F – (-2.8\u00b0F).
      \n1.3\u00b0F + 2.8\u00b0F.
      \n4.1\u00b0F.<\/p>\n

      PART B
      \nWhat will the temperature be when the plane reaches an altitude of 10,000 feet?
      \nA. -15.8
      \nB. -10.2
      \nC. 10.2
      \nD. 15.8<\/p>\n

      Answer:
      \nThe temperature is when the plane reaches an altitude of 10,000 feet = -15.8.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe temperature on the ground is -2.8\u00b0F.
      \n1.3\u00b0F – (-2.8\u00b0F).
      \n1.3\u00b0F + 2.8\u00b0F.
      \n4.1\u00b0F.
      \nso option B is correct.<\/p>\n

      Lesson 1.8 Divide Integers<\/h3>\n

      Explain It!<\/strong>
      \nThe shapes below are used to show the relationship between each of the four equations in the same fact family.
      \nI can… divide integers.
      \n\"Envision<\/p>\n

      A. Suppose the star represents -24. What values could the other shapes represent?<\/p>\n

      Answer:
      \nSquare represents 8.
      \nthe circle represents 3.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n8 x 3 = 24.
      \n3 x 8 = 24.
      \n24 \/ 3 = 8.
      \n24 \/ 8 = 3.
      \nso the square represents 8.
      \nthe circle represents 3.<\/p>\n

      B. What do you know about the square and circle if the star represents a negative number?<\/p>\n

      Answer:
      \nThe square and circle if the star represents a negative number.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n8 x 3 = 24.
      \n3 x 8 = 24.
      \n24 \/ 3 = 8.
      \n24 \/ 8 = 3.
      \nso the square represents 8.
      \nthe circle represents 3.<\/p>\n

      C. What do you know about the star if the square and circle both represent a negative number?<\/p>\n

      Answer:
      \nThe square and circle both represent a positive numbers.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nif the square and circle both represent a negative number.
      \nso the square and circle both represent positive numbers.
      \nfor example:
      \n– 3 x -4 = 12.<\/p>\n

      Focus on math practices
      \nUse Structure Suppose the square represents -8 and the circle represents 3. Use what you know about integer multiplication and the relationship between multiplication and division to write the complete fact family.<\/p>\n

      Essential Question<\/strong>
      \nHow does dividing integers relate to multiplying integers?<\/p>\n

      Try It!<\/strong><\/p>\n

      Suppose the machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level. What was the change in the location of the bottom of the hole each day?
      \n\"Envision
      \nEach day, the location of the bottom of the hole changed by _______ feet, or decreased by ________ feet.<\/p>\n

      Answer:
      \n-84\/3 = -28.
      \n3 x 9 = 28.
      \n3 x -28 = -84.
      \n-84\/3 = -28.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level.
      \n84\/3 = 28.
      \nso the location of the bottom of the hole changed by 28 feet or decreased by 28 feet.
      \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

      Convince Me! Explain why the quotient of two integers with different signs is negative.<\/p>\n

      Try It!<\/strong><\/p>\n

      Simplify.
      \na. -40 \u00f7 (-5)<\/p>\n

      Answer:
      \n-40 \/ -5 = 8.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -40 and -5.
      \ndivide the numbers.
      \n-40 \/ -5 = 8.<\/p>\n

      b. 40 \u00f7 (-5)<\/p>\n

      Answer:
      \n40 \/ -5 = -8.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are 40 and -5.
      \ndivide the numbers.
      \n40 \/ -5 = -8.<\/p>\n

      c. 0 \u00f7 -40<\/p>\n

      Answer:
      \n0 \/ 40 = 0.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are 0 and 40.
      \ndivide the numbers.
      \n0 \/ 40 = 0.<\/p>\n

      Try It!<\/strong><\/p>\n

      Which of the following are equivalent to -5?
      \n\"Envision<\/p>\n

      Answer:
      \nThe following are equivalent to -5 is -(55\/11), -55\/11, 55\/-11, and -(-55\/-11).<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are 55 and -11.
      \ndivide the numbers.
      \n-55\/11 = -5.
      \n-(55\/11) = -5.
      \n55\/-11 = -5.
      \n-(-55\/-11) = -5.<\/p>\n

      KEY CONCEPT<\/strong>
      \nThe rules for dividing integers are related to the rules for multiplying integers.
      \nIf the signs of the dividend and the divisor are the same, the quotient is positive.
      \n24 \u00f7 4 = 6
      \n-24 \u00f7 (-4) = 6<\/p>\n

      If the signs of the dividend and the divisor are different, the quotient is negative.
      \n-15 \u00f7 3 = -5
      \n15 \u00f7 (-3) = -5<\/p>\n

      Do You Understand?<\/strong>
      \nQuestion 1.
      \nEssential Question<\/strong> How does dividing integers relate to multiplying integers?<\/p>\n

      Answer:
      \nThe integers are related to multiplying integers depending on their signs.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nIf the signs of the dividend and the divisor are the same, the quotient is positive.
      \n24 \u00f7 4 = 6
      \n-24 \u00f7 (-4) = 6
      \nIf the signs of the dividend and the divisor are different, the quotient is negative.
      \n-15 \u00f7 3 = -5
      \n15 \u00f7 (-3) = -5<\/p>\n

      Question 2.
      \nReasoning<\/strong> Why is the quotient of two negative integers positive?<\/p>\n

      Answer:
      \nYes, the quotient of two negative integers is positive.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -30 and -3.
      \ndivide the numbers.
      \n-30 \/ -3 = 10.
      \non both sides, the signs get canceled.<\/p>\n

      Question 3.
      \nHelen wrote the following facts to try to show that division by 0 results in 0. Explain her error.
      \n\"Envision<\/p>\n

      Answer:
      \nYes, the division by 0 results in 0.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n0 x -7 = 0.
      \n-7 \/ 0 = 0.
      \nso the division by 0 results in 0.<\/p>\n

      Do You Know How?<\/strong>
      \nQuestion 4.
      \nFind each quotient.
      \na. –\\(\\frac{18}{3}\\)<\/p>\n

      Answer:
      \n-18\/3 = -6.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe number is -18\/3.
      \ndivide the number.
      \n-18\/3 = -6.<\/p>\n

      b. \\(\\frac{-5}{-1}\\)<\/p>\n

      Answer:
      \n-5\/-1 = 5.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe number is -5\/-1.
      \ndivide the number.
      \n-5\/-1 = 5.<\/p>\n

      c. \\(\\frac{24}{-6}\\)<\/p>\n

      Answer:
      \n24\/-6 = -4.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe number is 24\/-6.
      \ndivide the number.
      \n24\/-6 = -4.<\/p>\n

      d. \\(\\frac{-10}{-1}\\)<\/p>\n

      Answer:
      \n-10\/-1 = 10.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe number is -10\/-1.
      \ndivide the number.
      \n-10\/-1 = 10.<\/p>\n

      e. \\(\\frac{-25}{5}\\)<\/p>\n

      Answer:
      \n-25\/5 = -5.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe number is -25\/5.
      \ndivide the number.
      \n-25\/5 = -5.<\/p>\n

      f. –\\(\\frac{8}{2}\\)<\/p>\n

      Answer:
      \n-8\/2 = -4.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe number is -8\/2.
      \ndivide the number.
      \n-8\/2 = -4.<\/p>\n

      Question 5.
      \nA scuba diver descends 63 feet in 18 seconds. What integer represents the change in the diver’s position in feet per second?<\/p>\n

      Answer:
      \nThe integer represents the change in the diver’s position in feet per second = 3.5 feet.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nA scuba diver descends 63 feet in 18 seconds.
      \n63\/18 = 3.5.
      \nso the integer represents the change in the diver’s position in feet per second = 3.5 feet.<\/p>\n

      Question 6.
      \nWhich of the following are equivalent to -7?
      \n\"Envision<\/p>\n

      Answer:
      \n49\/-7 = -7 and -21\/3 = -7.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe quotient is -7.
      \n49\/-7 = -7.
      \n-21\/3 = -7.<\/p>\n

      Practice & Problem Solving<\/strong><\/p>\n

      Leveled Practice In 7-8, fill in the boxes to find each quotient.
      \nQuestion 7.
      \n-16 \u00f7 4 = ?
      \n4 \u2022 ? = ______
      \n4 \u2022 _____ = _______
      \nSo, -16 \u00f7 4 = ________.<\/p>\n

      Answer:
      \n-16\/4 = -4.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -16 and 4.
      \ndivide the numbers.
      \n-16 \/ 4 = -4.
      \nthe quotient is -4.<\/p>\n

      Question 8.
      \n-56 \u00f7 – 7 = ?
      \n______ \u2022 ? = _______
      \n______ \u2022 ______ = ________
      \nSo, -56 \u00f7 -7 = ________<\/p>\n

      Answer:
      \n-56 \/ -7 = 8.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -56 and -7.
      \ndivide the numbers.
      \n-56\/ -7 = 8.<\/p>\n

      Question 9.
      \nClassify the quotient -50 \u00f7 5 as positive, negative, zero, or undefined.<\/p>\n

      Answer:
      \nThe quotient is -10.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are -50 and -5.
      \ndivide the numbers.
      \n-50\/5 = -10.<\/p>\n

      Question 10.
      \nIs the expression \\(\\frac{42}{-7}\\) undefined? If not, find the quotient.<\/p>\n

      Answer:
      \nThe quotient is -6.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe numbers are 42 and -7.
      \ndivide the numbers.
      \n42\/-7 = -6.
      \nso the quotient is -6.<\/p>\n

      Question 11.
      \nA company loses $780 as a result of a shipping delay. The 6 owners of the company must share the loss equally.
      \na. Write an expression to show the change in profit for each owner.<\/p>\n

      Answer:
      \nThe change in profit for each owner = $130.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nA company loses $780 as a result of a shipping delay.
      \nThe 6 owners of the company must share the loss equally.
      \n$780 \/ 6 = $130.
      \nso the change in profit for each owner = $130.<\/p>\n

      b. Evaluate the expression.<\/p>\n

      Answer:
      \nThe change in profit for each owner = $130.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nA company loses $780 as a result of a shipping delay.
      \nThe 6 owners of the company must share the loss equally.
      \n$780 \/ 6 = $130.
      \nso the change in profit for each owner = $130.<\/p>\n

      Question 12.
      \nWhich of the quotients are equivalent to 2.5? Select all that apply.
      \n\"Envision<\/p>\n

      Answer:
      \n-5\/12, 10\/4, -10\/ -4, and 5\/2 are equivalent to 2.5.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe quotient equal to 2.5 is
      \n10\/-4 = 5\/-2 = -2.5.
      \n-5\/-2 = 2.5.
      \n10\/4 = 2.5.
      \n-10\/-4 = 2.5.
      \n5\/2 = 2.5.<\/p>\n

      Question 13.
      \nUse Structure The price of a stock steadily decreased by a total of $127 over 15 months. Which expression shows the change in the stock’s value?
      \n\"Envision<\/p>\n

      Answer:
      \nOption A is correct.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe price of a stock steadily decreased by a total of $127 over 15 months.
      \n-$127\/ -15 months.
      \nso option A is correct.<\/p>\n

      Question 14.
      \nZak goes parachuting and descends at the rate shown. If he maintains a steady descent, what integer represents Zak’s change in elevation in feet per second?
      \n\"Envision<\/p>\n

      Answer:
      \nThe integer represents Zak’s change in elevation in feet per second = 12 feet.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nZak goes parachuting and descends at the rate shown.
      \n24 feet in 2 seconds.
      \n24\/2 = 12.
      \nso the integer represents Zak’s change in elevation in feet per second = 12 feet.<\/p>\n

      Question 15.
      \nModel with Math Find each quotient and plot it on the number line. Which of the expressions are undefined?
      \n\"Envision<\/p>\n

      Answer:
      \nThe expressions are undefined = 0\/-8.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n-8\/4 = 2.
      \n-21\/ -7 = 3.
      \n-4\/0 = 0.
      \n-25\/-5 = 5.
      \n36\/-9 = -4.
      \n9\/0 = 9.
      \n0\/-8 = 0.
      \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

      Question 16.
      \nUse Structure The temperature in a town increased 16\u00b0F in 5 hours. The temperature decreased 31\u00b0F in the next 8 hours. Which of the expressions shows the rate of the total change in temperature?
      \nA. \\(\\frac{-15^{\\circ} \\mathrm{F}}{13 \\text { hours }}\\)
      \nB. \\(\\frac{47^{\\circ} \\mathrm{F}}{13 \\text { hours }}\\)
      \nC. \\(\\frac{15^{\\circ} \\mathrm{F}}{13 \\text { hours }}\\)
      \nD. \\(\\frac{47{\\circ} \\mathrm{F}}{-13 \\text { hours }}\\)<\/p>\n

      Answer:
      \nOption C is correct.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe temperature in a town increased by 16\u00b0F in 5 hours.
      \nThe temperature decreased 31\u00b0F in the next 8 hours.
      \n31 – 16 = 15.
      \n5 + 8 = 13.
      \nso option C is correct.<\/p>\n

      Question 17.
      \nCamille takes a rock-climbing class. On her first outing, she rappels down the side of a boulder in three equal descents. What integer represents Camille’s change in altitude in feet each time she descends?
      \n\"Envision<\/p>\n

      Answer:
      \nThe integer represents Camille’s change in altitude in feet each time she descends = 165 feet.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nCamille takes a rock-climbing class.
      \nOn her first outing, she rappels down the side of a boulder in three equal descents.
      \n165 feet.
      \nso the integer represents Camille’s change in altitude in feet each time she descends = 165 feet.<\/p>\n

      Question 18.
      \nHigher Order Thinking<\/strong> If the fraction \\(\\frac{396}{x-10}\\) is equivalent to -22, find the value of x. Show your work.<\/p>\n

      Answer:
      \nx = 9.94.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n396 x x – 10 = -22.
      \n396x – 3960 = -22.
      \n396x = -22 + 3960.
      \n396x = 3938.
      \nx = 3938\/396.
      \nx = 9.94.<\/p>\n

      Assessment Practice<\/strong><\/p>\n

      Question 19.
      \nWhich of the quotients is equivalent to \\(-\\frac{5}{8}\\)? Select all that apply.
      \n\u2610 \\(\\frac{-5}{8}\\)
      \n\u2610 \\(\\frac{5}{8}\\)
      \n\u2610 \\(\\frac{5}{-8}\\)
      \n\u2610 \\(-\\left(\\frac{5}{-8}\\right)\\)
      \n\u2610 \\(\\frac{-5}{-8}\\)<\/p>\n

      Answer:
      \n-5\/8, and 5\/-8.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n-5\/8 = -0.625.
      \n5\/-8 = -0.625.<\/p>\n

      Question 20.
      \nWhich of the following pairs of quotients are equivalent?
      \nA. \\(\\frac{-4}{5}\\) and \\(-\\left(\\frac{20}{25}\\right)\\)
      \nB. \\(-\\left(\\frac{2}{-3}\\right)\\) and \\(\\frac{-4}{6}\\)
      \nC. \\(\\frac{-5}{7}\\) and \\(\\frac{35}{-40}\\)
      \nD. \\(\\frac{1}{5}\\) and \\(-\\left(\\frac{-2}{-10}\\right)\\)<\/p>\n

      Answer:
      \nOption A is correct.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe pairs of quotients are equivalent.
      \n-4\/5 = -20\/25.
      \n5 x 4 = 20.
      \n5 x 5 = 25.
      \nso option A is correct.<\/p>\n

      Question 21.
      \nAn elevator descends 36 feet in 3 seconds. What integer represents the elevator’s change in elevation in feet per a second?<\/p>\n

      Answer:
      \nThe integer represents the elevator’s change in elevation in feet per second = 12 seconds.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nAn elevator descends 36 feet in 3 seconds.
      \n36\/3 = 13.
      \nso the integer represents the elevator’s change in elevation in feet per second = 12 seconds.<\/p>\n

      Lesson 1.9 Divide Rational Numbers<\/h3>\n

      Explore It!<\/strong>
      \nThe number line shows the movement of a glacier that retreats 8 meters every year.
      \nI can… divide rational numbers.
      \n\"Envision<\/p>\n

      A. How could you use division to represent the yearly change in the glacier’s position over the next 4 years?<\/p>\n

      Answer:
      \nThe change in the glacier’s position over the next 4 years = -32.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe number line shows the movement of a glacier that retreats 8 meters every year.
      \n4 x -8 = -32.
      \n-32\/-8 = 4.
      \nso the change in the glacier’s position over the next 4 years = -32.<\/p>\n

      B. How could you use division to represent the yearly change in the glacier’s position over the past 4 years?<\/p>\n

      Answer:
      \nThe change in the glacier’s position over the past 4 years = -32.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe number line shows the movement of a glacier that retreats 8 meters every year.
      \n4 x -8 = -32.
      \n-32\/-8 = 4.
      \nso the change in the glacier’s position over the past 4 years = -32.<\/p>\n

      C. Suppose the glacier retreated 8.25 meters every year. Draw a number line to represent this movement.<\/p>\n

      Answer:<\/p>\n

      The change in the glacier’s position over the past 4 years = 33.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nThe number line shows the movement of a glacier that retreats 8 meters every year.
      \n4 x 8.25 = 33.
      \n33\/8.25 = 4.
      \nso the change in the glacier’s position over the past 4 years<\/p>\n

      Focus on math practices
      \nReasoning If the number of meters the glacier retreats each year changes, does it affect the signs of each part of the division statement in Part A? Explain.<\/p>\n

      Essential Question<\/strong>
      \nHow is dividing rational numbers like dividing integers?<\/p>\n

      Try It!<\/strong><\/p>\n

      Suppose that the volume of water in the rain barrel decreased by 4\\(\\frac{5}{8}\\) gallons in 4 minutes. What will be the change in the volume of water after 1 minute?
      \n\"Envision
      \nThe rain barrel will lose _________ gallons in 1 minute.<\/p>\n

      Answer:
      \nThe rain barrel will lose -37\/32 gallons in 1 minute.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nsuppose that the volume of water in the rain barrel decreased by 4\\(\\frac{5}{8}\\) gallons in 4 minutes.
      \n-4(5\/8) \/4.
      \n-37\/8 x 4.
      \n-37\/32.
      \nso the rain barrel will lose -37\/32 gallons in 1 minute.
      \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

      Convince Me! How are multiplicative inverses used in division with rational numbers?<\/p>\n

      Try It!<\/strong><\/p>\n

      Find each quotient.
      \na. \\(\\frac{1 \\frac{2}{5}}{-\\frac{1}{5}}\\)<\/p>\n

      Answer:
      \n1(2\/5) – 1\/5 = 1.2.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \n7\/5 – 1\/5.
      \n1.4 – 0.2.
      \n1.2.<\/p>\n

      b. -0.4 – 0.25<\/p>\n

      Answer:
      \n-0.4 – 0.25 = 0.65.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are -0.4 and -0.25.
      \nsubtract the numbers.
      \n-0.4 – 0.25.
      \n0.65.<\/p>\n

      c. \\(\\frac{7}{8} \\div-\\frac{3}{4}\\)<\/p>\n

      Answer:
      \n7\/8 – 3\/4 = 0.125.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are 7\/8 and -3\/4.
      \nsubtract the numbers.
      \n7\/8 = 0.875.
      \n-3\/4 = 0.75.
      \n0.875 – 0.75.
      \n0.125.<\/p>\n

      d. \\(0.7 \\div-1 \\frac{1}{6}\\)<\/p>\n

      Answer:
      \n0.7 \/ 1(1\/6) = 0.6.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are 0.7 and -7\/6.
      \ndivide the numbers.
      \n0.7\/ 1.16.
      \n0.6.<\/p>\n

      Try It!<\/strong><\/p>\n

      Find each quotient.
      \na. \\(-1 \\frac{1}{3} \\div(-1.6)\\)<\/p>\n

      Answer:
      \nThe quotient is 0.83.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are -1(1\/3) and -1.6.
      \ndivide the numbers.
      \n-4\/3 \/ -1.6.
      \n-1.33\/ -1.6.
      \n0.83.<\/p>\n

      b. \\(\\frac{-\\frac{2}{3}}{-\\frac{1}{4}}\\)<\/p>\n

      Answer:
      \nThe quotient is 2.4.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are -(2\/3) and -1\/4.
      \ndivide the numbers.
      \n-2\/3 \/ -1\/4.
      \n-0.6\/-0.25.
      \n2.4.<\/p>\n

      c. \\(-\\frac{9}{10} \\div\\left(-\\frac{3}{10}\\right)\\)<\/p>\n

      Answer:
      \nThe quotient is 3.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are -9\/10 and -3\/10.
      \ndivide the numbers.
      \n-9\/10\/-3\/10.
      \n-0.9\/-0.3.
      \n3.<\/p>\n

      d. \\(-0.5 \\div\\left(-\\frac{3}{13}\\right)\\)<\/p>\n

      Answer:
      \nThe quotient is 2.17.<\/p>\n

      Explanation:
      \nIn the above-given question,
      \ngiven that,
      \nthe two numbers are -0.5 and -3\/13.
      \ndivide the numbers.
      \n-0.5\/ -0.23.
      \n2.17.<\/p>\n

      KEY CONCEPT<\/strong>
      \nThe same rules for dividing integers apply to dividing rational numbers. When dividing two rational numbers:<\/p>\n

        \n
      • If the signs of the dividend and divisor are the same, the quotient is positive.<\/li>\n
      • If the signs of the dividend and divisor are different, the quotient is negative.<\/li>\n<\/ul>\n

        Do You Understand?<\/strong>
        \nQuestion 1.
        \nEssential Question<\/strong> How is dividing rational numbers like dividing integers?<\/p>\n

        Answer:
        \nThe division of rational numbers like dividing integers depends upon size.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nIf the signs of the dividend and divisor are the same, the quotient is positive.
        \nIf the signs of the dividend and divisor are different, the quotient is negative.
        \nfor example:
        \n12\/3 = 4.
        \n-12\/3 = -4.<\/p>\n

        Question 2.
        \nUse Structure How do you know the sign of the quotient \\(-\\frac{4}{5} \\div \\frac{1}{6}\\)?<\/p>\n

        Answer:
        \n-4\/5 \/ 1\/6 = -8.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -4\/5 and 1\/6.
        \ndivide the numbers.
        \n-4\/5 \/ 1\/6.
        \n-0.8 \/ 0.1.
        \n-8.<\/p>\n

        Question 3.
        \nReasoning<\/strong> When -4 is divided by a rational number between 0 and 1, where would the quotient be located on the number line? Why?<\/p>\n

        Answer:
        \nThe quotient will be located on the number line is -4.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nWhen -4 is divided by a rational number between 0 and 1.
        \n-4\/0 = -4.
        \n-4\/1 = -4.<\/p>\n

        Do You Know How?<\/strong>
        \nQuestion 4.
        \nFind each quotient.
        \na. \\(-\\frac{7}{12} \\div \\frac{1}{7}\\)<\/p>\n

        Answer:
        \n-7\/12 \/ 1\/7 = -4.14.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -7\/12 and 1\/7.
        \ndivide the numbers.
        \n-7\/12 \/ 1\/7.
        \n-0.58 \/ 0.14.
        \n-4.14.<\/p>\n

        b. \\(-0.05 \\div\\left(-\\frac{5}{8}\\right)\\)<\/p>\n

        Answer:
        \n-0.05 \/ -5\/8 =0.08.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -0.05 and -5\/8.
        \ndivide the numbers.
        \n-0.05 \/ -5\/8.
        \n-0.05 \/ -0.625.
        \n0.08.<\/p>\n

        c. \\(6 \\frac{1}{4} \\div\\left(-\\frac{5}{16}\\right)\\)<\/p>\n

        Answer:
        \n6(1\/4) \/ -5\/16 = -20.03.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 6(1\/4) and -5\/16.
        \ndivide the numbers.
        \n25\/4 \/ -5\/16.
        \n6.25 \/ -0.3125.
        \n-20.03.<\/p>\n

        d. \\(-1 \\div\\left(-\\frac{10}{13}\\right)\\)<\/p>\n

        Answer:
        \n-1 \/ -10\/3 = 0.300.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -1 and -10\/3.
        \ndivide the numbers.
        \n-1\/ -10\/3.
        \n-1 \/ -3.33.
        \n0.300.<\/p>\n

        Question 5.
        \nSimplify the complex fraction.
        \na. \\(\\frac{-\\frac{2}{7}}{1 \\frac{1}{3}}\\)<\/p>\n

        Answer:
        \n-2\/7 \/ 4\/3 = -0.21.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -2\/7 and 1(1\/3).
        \ndivide the numbers.
        \n-2\/7 \/ 4\/3.
        \n-0.28 \/ 1.3.
        \n-0.21.<\/p>\n

        b. \\(\\frac{-\\frac{3}{5}}{2 \\frac{1}{4}}\\)<\/p>\n

        Answer:
        \n-3\/5 \/ 9\/4 = -0.26.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -3\/5 and 2(1\/4).
        \ndivide the numbers.
        \n-3\/5 \/ 9\/4.
        \n– 0.6 \/ 2.25.
        \n-0.26.<\/p>\n

        c. \\(\\frac{-\\frac{9}{10}}{1 \\frac{3}{5}}\\)<\/p>\n

        Answer:
        \n-9\/10 \/ 8\/5 = – 0.5625.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -9\/10 and 8\/5.
        \ndivide the numbers.
        \n-9\/10 \/ 8\/5.
        \n-0.9 \/ 1.6.
        \n-0.5625.<\/p>\n

        Practice & Problem Solving<\/strong><\/p>\n

        Leveled Practice In 6-7, fill in the boxes to find the quotient.
        \nQuestion 6.
        \nFind the quotient \\(\\frac{5}{7} \\div\\left(-\\frac{11}{5}\\right)\\).
        \n\"Envision
        \nAnswer:
        \n5\/7 \/ -11\/5 = -0.32.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 5\/7 and -11\/5.
        \ndivide the numbers.
        \n5\/7\/ – 11\/5.
        \n0.71 \/ -2.2.
        \n– 0.32.
        \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

        Question 7.
        \nSimplify the complex fraction \\(\\frac{-\\frac{4}{5}}{\\frac{3}{10}}\\)
        \nRewrite the complex fraction ______ \u00f7 _______
        \nWrite the division as multiplication: ______ \u2022 _______
        \nThe product is _______ .<\/p>\n

        Answer:
        \nThe product is -0.24.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -4\/5 and 3\/10.
        \ndivide the numbers.
        \n-4\/5 x 3\/10.
        \n-0.8 x 0.3.
        \n-0.24.
        \nso the product is -0.24.<\/p>\n

        Question 8.
        \nWhich multiplication expression is equivalent to the division expression \\(-\\frac{7}{17} \\div \\frac{13}{34}\\)?
        \n\"Envision
        \nAnswer:
        \nOption A is correct.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -7\/17 and 13\/34.
        \ndivide the numbers.
        \n-7\/17 x 13\/34.
        \n-0.411 x 0.38.
        \n-0.156.
        \nso option A is correct.<\/p>\n

        Question 9.
        \nDerek says that the quotient
        \n\\(-\\frac{2}{7} \\div\\left(-\\frac{2}{21}\\right) \\text { is }-\\frac{1}{3} \\text { . }\\)
        \na. What is the correct quotient?<\/p>\n

        Answer:
        \nThe correct quotient is 3.11.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nDerek says that the quotient
        \n\\(-\\frac{2}{7} \\div\\left(-\\frac{2}{21}\\right) \\text { is }-\\frac{1}{3} \\text { . }\\).
        \n-2\/7 \/ -2\/21.
        \n-0.28\/ -0.09.
        \n3.11.<\/p>\n

        b. What mistake did Derek likely make?<\/p>\n

        Answer:
        \nThe two expressions are not equal.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nDerek says that the quotient
        \n\\(-\\frac{2}{7} \\div\\left(-\\frac{2}{21}\\right) \\text { is }-\\frac{1}{3} \\text { . }\\).
        \n-2\/7 \/ -2\/21.
        \n-0.28\/ -0.09.
        \n3.11.<\/p>\n

        Question 10.
        \nThe water level of a lake fell by \\(1 \\frac{1}{2}\\) inches during a \\(1 \\frac{2}{3}\\)-week-long dry spell. Simplify the complex fraction below to find the average rate at which the water level changed every week.
        \n\\(\\frac{-1 \\frac{1}{2}}{1 \\frac{2}{3}}\\) inches\/week
        \n\"Envision<\/p>\n

        Answer:
        \nThe average rate at which the water level changed every week = 0.9375 meters.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nThe water level of a lake fell by \\(1 \\frac{1}{2}\\) inches during a \\(1 \\frac{2}{3}\\)-week-long dry spell.
        \n1(1\/2) \/ 1(2\/3).
        \n3\/2 \/ 5\/3.
        \n1.5 \/ 1.6.
        \n0.9375.
        \nso the average rate at which the water level changed every week = 0.9375 meters.<\/p>\n

        Question 11.
        \nComplete the table. Simplify expressions.
        \n\"Envision<\/p>\n

        Answer:
        \nThe quotients are the same.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-3\/4 \/ 2\/5.
        \n-0.75 \/ 0.4 = -1.875.
        \n-0.75\/ 0.4.
        \n-1.875.
        \n3\/4 \/ -2\/5 = -1.875.
        \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

        Question 12.
        \na. Find the reciprocal of \\(-1 \\frac{1}{17}\\).<\/p>\n

        Answer:
        \nThe reciprocal of -1(1\/17) = 1.05.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-1(1\/17).
        \n-18\/17.
        \n-1.05.<\/p>\n

        b. Find the reciprocal of \\(-\\frac{17}{18}\\).<\/p>\n

        Answer:
        \nThe reciprocal of -17\/18 = -1.05.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe reciprocal of -17\/18 is -18\/17.
        \n-18\/17 = -1.05.<\/p>\n

        c. Reasoning Explain why the answer for part a is the multiplicative inverse of the answer for part b.<\/p>\n

        Answer:
        \nThe answer for part a is the multiplicative inverse of the answer for part b.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe reciprocal of -17\/18 is -18\/17.
        \n-18\/17 = -1.05.<\/p>\n

        Question 13.
        \nUse numbers \\(-\\frac{7}{13}, 1 \\frac{6}{7},-1 \\frac{6}{7}, \\frac{7}{13}\\)
        \na. Which is the reciprocal of \\(1 \\frac{6}{7}\\)?<\/p>\n

        Answer:
        \nThe reciprocal of 1(6\/7) = 7\/13.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n1(6\/7).
        \n13\/7.
        \n13\/7 = 1.85.<\/p>\n

        b. Which is the reciprocal of \\(\\frac{7}{13}\\)?<\/p>\n

        Answer:
        \nThe reciprocal of 7\/13 = 13\/7.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe numbers are 7 and 13.
        \ndivide the numbers.
        \n7\/13 = 0.538.
        \n13\/7 = 1.85.<\/p>\n

        c. Reasoning What do you notice about the reciprocals of \\(1 \\frac{6}{7}\\) and \\(\\frac{7}{13}\\)?<\/p>\n

        Answer:
        \nThe reciprocals of 1(6\/7) and 7\/13 is<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe number is 1(6\/7).
        \n13\/7 = 7\/13.
        \nthe reciprocals are the same.<\/p>\n

        Question 14.
        \nA water tank in Stewart’s home had a small, steady leak.
        \n\"Envision
        \na. Use a complex fraction to represent the change in the volume of water in 1 minute.
        \n\"Envision<\/p>\n

        Answer:
        \nThe change in the volume of water in 1 minute = 1.6 ml.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA water tank in Stewart’s home had a small, steady leak.
        \n1(3\/5) = 8\/5.
        \n8\/5 = 1.6.
        \nso the change in the volume of water in 1 minute = 1.6 ml.
        \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

        b. Simplify the complex fraction to find the change in the volume of water in the tank in 1 minute.<\/p>\n

        Answer:
        \nThe change in the volume of water in 1 minute = 1.6 ml.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA water tank in Stewart’s home had a small, steady leak.
        \n1(3\/5) = 8\/5.
        \n8\/5 = 1.6.
        \nso the change in the volume of water in 1 minute = 1.6 ml.<\/p>\n

        Question 15.
        \nFind the quotient. Express your answer as a simplified fraction.
        \n\\(\\frac{3}{10} \\div 3.8\\)<\/p>\n

        Answer:
        \nThe quotient is 1.14.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 3\/10 and 3.8.
        \ndivide the numbers.
        \n3\/10 = 0.3.
        \n0.3 x 3.8 = 1.14.<\/p>\n

        Question 16.
        \nHigher Order Thinking<\/strong> Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by \\(\\frac{13}{16}\\) inch.
        \nAssuming that the water level decreased at a constant rate, how much did it drop each hour?
        \nThe water level decreased by ______ inch each hour.<\/p>\n

        Answer:
        \nThe water level decreased by 0.8125 inches each hour.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nBetween 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by \\(\\frac{13}{16}\\) inch.
        \n13\/16.
        \n0.8125.
        \nso the water level decreased by 0.8125 inches each hour.<\/p>\n

        Question 17.
        \nCritique Reasoning<\/strong> Kayla wants to find \\(2 \\frac{2}{3} \\div\\left(-1 \\frac{3}{7}\\right)\\). She first rewrites the division as \\(\\left(2 \\frac{2}{3}\\right)\\left(-1 \\frac{7}{3}\\right)\\) What is wrong with Kayla’s reasoning?<\/p>\n

        Answer:
        \nThe wrong with Kayla’s reasoning = -1.8309.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nKayla wants to find \\(2 \\frac{2}{3} \\div\\left(-1 \\frac{3}{7}\\right)\\).
        \n2(2\/3) \/ -1(3\/7).
        \n8\/3 \/ -10\/7.
        \n2.6 \/ – 1.42.
        \n-1.8309.<\/p>\n

        Assessment Practice<\/strong><\/p>\n

        Question 18.
        \nWhich is an equivalent multiplication expression for \\(\\frac{-\\frac{3}{8}}{\\left(-\\frac{7}{54}\\right)}\\)?
        \n\"Envision<\/p>\n

        Answer:
        \nOption A is correct.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n\\(\\frac{-\\frac{3}{8}}{\\left(-\\frac{7}{54}\\right)}\\).
        \n-3\/8 \/ -7\/54.
        \n-3\/8 x 7\/54.
        \nso option A is correct.<\/p>\n

        Question 19.
        \nWhich is NOT a step you perform to divide \\(-2 \\frac{1}{8} \\div 6 \\frac{4}{5}\\). Select all that apply.
        \n\u2610 Rewrite the mixed numbers as fractions.
        \n\u2610 Divide 8 by 4.
        \n\u2610 Multiply by the multiplicative inverse of a \\(\\frac{34}{5}\\)
        \n\u2610 Multiply by the multiplicative inverse of \\(\\frac{17}{8}\\)
        \n\u2610 Multiply 8 and 34.<\/p>\n

        Answer:
        \nRewrite the mixed numbers as fractions.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n\\(-2 \\frac{1}{8} \\div 6 \\frac{4}{5}\\).
        \n-2(1\/8) \/ 6(4\/5).
        \n-17\/8 \/ 34\/5.
        \n-2.125 \/ 6.8.
        \n-0.3125.<\/p>\n

        Lesson 1-10 Solve Problems with Rational Numbers<\/h3>\n

        Solve & Discuss It!<\/strong>
        \nStefan estimates the income and expenses for renting a phone accessory store in the mall. He enters the amounts in the table below. Should Stefan rent a phone accessory store? Explain.<\/p>\n

        I can… solve problems with rational numbers.
        \n\"Envision<\/p>\n

        Focus on math practices
        \nReasoning How can you assess the reasonableness of your solution using mental math or estimation strategies?<\/p>\n

        Essential Question<\/strong>
        \nHow do you decide which rational number operations to use to solve problems?<\/p>\n

        Try It!<\/strong><\/p>\n

        A weather balloon ascended from an elevation of 18 feet below sea level to an elevation of 19\\(\\frac{1}{2}\\) feet above sea level. What distance did the weather balloon rise?
        \nThe distance between two points is the absolute value of their __________.
        \n\"Envision
        \nThe weather balloon rose a distance of ________ feet.<\/p>\n

        Answer:
        \nThe weather balloon rose a distance of 1.5 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA weather balloon ascended from an elevation of 18 feet below sea level to an elevation of 19\\(\\frac{1}{2}\\) feet above sea level.
        \n-18 + 19(1\/2).
        \n-18 + 39\/2.
        \n-18 + 19.5.
        \n1.5.
        \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

        Convince Me! How can you decide which operation to use to solve a problem?<\/p>\n

        Try It!<\/strong><\/p>\n

        Rashida had 18 correct answers and 12 incorrect answers. What was Rashida’s score?<\/p>\n

        Answer:
        \nRashida’s score is 6.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nRashida had 18 correct answers and 12 incorrect answers.
        \n18 – 12 = 6.
        \nso Rashida’s score is 6.<\/p>\n

        Try It!<\/strong><\/p>\n

        The temperature at 10:00 A.M. was -3\u00b0F and increased 2.25\u00b0F each hour for the next 5 hours. What was the temperature at 3:00 P.M.?<\/p>\n

        Answer:
        \nThe temperature at 3:00 P.M = -1.25\u00b0F.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nThe temperature at 10:00 A.M. was -3\u00b0F and increased 2.25\u00b0F each hour for the next 5 hours.
        \n-3 + 2.25.
        \n-1.25.
        \nso the temperature at 3:00 P.M = – 1.25\u00b0F.<\/p>\n

        KEY CONCEPT<\/strong>
        \nYou can solve a problem with rational numbers by making sense of the problem and deciding which operations to use.<\/p>\n

        Do You Understand?<\/strong>
        \nQuestion 1.
        \nEssential Question<\/strong> How do you decide which rational number operations to use to solve problems?<\/p>\n

        Question 2.
        \nReasoning<\/strong> A truck’s position relative to a car’s position is -60 feet. The car and the truck move in the same direction, but the car moves 5 feet per second faster for 8 seconds. What operations could be used to find the truck’s relative position after 8 seconds? Explain.<\/p>\n

        Answer:
        \nThe operations could be used to find the truck’s relative position after 8 seconds = -47 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA truck’s position relative to a car’s position is -60 feet.
        \nThe car and the truck move in the same direction, but the car moves 5 feet per second faster for 8 seconds.
        \n-60 + 5 + 8.
        \n-47.
        \nso the operations could be used to find the truck’s relative position after 8 seconds = -47 feet.<\/p>\n

        Question 3.
        \nConstruct Arguments Emilio used addition of two rational numbers to solve a problem. Jim used subtraction to solve the same problem. Is it possible that they both solved the problem correctly? Use a specific example to explain.<\/p>\n

        Answer:
        \nYes, they both have solved the problem correctly.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nEmilio used the addition of two rational numbers to solve a problem.
        \nfor example:
        \n4 + 3 = 7.
        \nJim used subtraction to solve the same problem.
        \n4 – 3 = 1.<\/p>\n

        Do You Know How?<\/strong>
        \nQuestion 4.
        \nKara had a savings account balance of $153 on Monday. On Tuesday, she had six withdrawals of $15.72 and a deposit of $235.15. What was her account balance after these transactions?<\/p>\n

        Answer:
        \nThe account balance after these transactions = $66.43.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nKara had a savings account balance of $153 on Monday.
        \nOn Tuesday, she had six withdrawals of $15.72 and a deposit of $235.15.
        \n$153 + $15.72 = $168.72.
        \n$235.15 – $168.72 = $66.43.
        \nso the account balance after these transactins = $66.43.<\/p>\n

        Question 5.
        \nA scuba diver is swimming at the depth shown, and then swims 0.5 foot toward the surface every 3 seconds. What is the location of the scuba diver, relative to the surface, after 15 seconds?
        \n\"Envision<\/p>\n

        Answer:
        \nThe location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA scuba diver is swimming at the depth shown and then swims 0.5 feet toward the surface every 3 seconds.
        \n0.5 x 3 = 1.5.
        \n1.5 x 15 = 22.5.
        \nso the location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.<\/p>\n

        Question 6.
        \nThe temperature of a cup of coffee changed by -54\u00b0F over 22\\(\\frac{1}{2}\\) minutes. What was the change in temperature each minute?<\/p>\n

        Answer:
        \nThe change in temperature each minute = 22.5 minutes.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nThe temperature of a cup of coffee changed by -54\u00b0F over 22\\(\\frac{1}{2}\\) minutes.
        \n22(1\/2).
        \n44 + 1 = 45.
        \n45\/2 = 22.5.
        \nso the change in temperature each minute = 22.5 minutes.<\/p>\n

        Practice & Problem Solving<\/strong><\/p>\n

        Question 7.
        \nSuppose there is a 1.1\u00b0F drop in temperature for every thousand feet that an airplane climbs into the sky. If the temperature on the ground is 59.7\u00b0F, what will be the temperature at an altitude of 11,000 ft?
        \n\"Envision<\/p>\n

        Answer:
        \nThe temperature at an altitude of 11,000 ft = 656700\u00b0F.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nSuppose there is a 1.1\u00b0F drop in temperature for every thousand feet that an airplane climbs into the sky.
        \nIf the temperature on the ground is 59.7\u00b0F.
        \n11000 x 59.7\u00b0F.
        \n656700\u00b0F
        \nso the temperature at an altitude of 11,000 ft = 656700\u00b0F.<\/p>\n

        Question 8.
        \nA farmer sells an average of 15\\(\\frac{3}{5}\\) bushels of corn each day. What integer represents the change in bushels of corn in his inventory after 6 days?<\/p>\n

        Answer:
        \nThe integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA farmer sells an average of 15\\(\\frac{3}{5}\\) bushels of corn each day.
        \n15(3\/5).
        \n75 + 3 = 78.
        \n78\/5 = 15.6.
        \n15.6 x 6 = 93.6.
        \nso the integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.<\/p>\n

        Question 9.
        \nA certain plant grows 1\\(\\frac{1}{6}\\) inches every week. How long will it take the plant to grow 6\\(\\frac{1}{6}\\) inches?<\/p>\n

        Answer:
        \nThe long will it take the plant to grow = 7.1456 inches.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA certain plant grows 1\\(\\frac{1}{6}\\) inches every week.
        \n1(1\/6) = 7\/6.
        \n7\/6 = 1.16.
        \n6(1\/6) = 37\/6.
        \n37\/6 = 6.16.
        \n6.16 x 1.16 = 7.1456.
        \nso the longer will it take the plant to grow = 7.1456 inches.<\/p>\n

        Question 10.
        \nAn object is traveling at a steady speed of 8\\(\\frac{2}{3}\\) miles per hour. How long will it take the object to travel 5\\(\\frac{1}{5}\\) miles?<\/p>\n

        Answer:
        \nThe long will it take for the object o travel is 5(1\/5) miles = 45.032.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nAn object is traveling at a steady speed of 8\\(\\frac{2}{3}\\) miles per hour.
        \n8(2\/3) = 26\/3.
        \n26\/3 = 8.66.
        \n8.66 x 26\/5.
        \n8.66 x 5.2.
        \n45.032.<\/p>\n

        Question 11.
        \nBrianna works as a customer service representative. She knows that the amount of her yearly bonus is $155, but $2.50 is taken away for each customer complaint about her during the year. What is her bonus if there are 12 complaints about her in the year?<\/p>\n

        Answer:
        \nThe bonus if there are 12 complaints about her in the year = $1890.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nBrianna works as a customer service representative.
        \nShe knows that the amount of her yearly bonus is $155, but $2.50 is taken away for each customer complaint about her during the year.
        \n$155 + $2.50 = 157.5.
        \n$157.5 x 12 = $1890.
        \nso the bonus if there are 12 complaints about her in the year = $1890.<\/p>\n

        Question 12.
        \nMake Sense and Persevere There are ten birdbaths in a park. On the first day of spring, the birdbaths are filled. Several weeks later, the overall change in the water level is found. The results are shown in the table. What is the range of the data?
        \n\"Envision<\/p>\n

        Answer:
        \nThe range of the data =<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nThere are ten birdbaths in a park.
        \nOn the first day of spring, the birdbaths are filled.
        \n2.4 – 0.9 = 1.5.
        \nso the range of the data = 1.5.<\/p>\n

        Question 13.
        \nModel with Math Marcelo played a carnival game at the Interstate Fair 6 times. He spent 3 tokens to play each game, and he won 7 tokens each game. Write two different expressions that can be used to find the total profit in tokens that Marcelo made.<\/p>\n

        Answer:
        \nThe different expressions that can be used to find the total profits in tokens that Marcelo made = 25.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nMarcelo played a carnival game at the Interstate Fair 6 times.
        \nHe spent 3 tokens to play each game.
        \nand he won 7 tokens each game.
        \n6 x 3 = 18
        \n18 + 7 = 25.
        \nso the different expressions that can be used to find the total profits in tokens that Marcelo made = 25.<\/p>\n

        Question 14.
        \nThe temperature of a pot of water is shown. The temperature of the water changed -2.5\u00b0F per minute.
        \n\"Envision
        \na. What was the temperature after 20 minutes?<\/p>\n

        Answer:
        \nThe temperature of the water after 20 minutes = -3.606\u00b0F.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nThe temperature of a pot of water is 180.3\u00b0F.
        \n-2.5\u00b0F x 20 = -50\u00b0F .
        \n180.3\u00b0F \/-50\u00b0F.
        \n-3.606\u00b0F.
        \nso the temperature of the water after 20 minutes = -3.606\u00b0F.<\/p>\n

        b. Make Sense and Persevere How many minutes did it take to cool to 100.3\u00b0F?<\/p>\n

        Answer:
        \nThe number of minutes did it take to cool to 100.3\u00b0F = 40 minutes.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-2.5 x 40 = 100.3\u00b0F.
        \nso the number of minutes did it take to cool to 100.3\u00b0F = 40 minutes.<\/p>\n

        Question 15.
        \nHigher Order Thinking<\/strong> the table shows the relationship between a hedgehog’s change in weight and the number of days of hibernation.
        \n\"Envision
        \na. What number represents the change in weight for each day of hibernation?<\/p>\n

        Answer:
        \nThe number represents the change in weight for each day of hibernation = -0.06.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe change in weight is -0.24 – 0.84 – 2.25 -2.79.
        \n-0.6 – 2.25 – 2.79.
        \n-2.85 – 2.79.
        \n-0.06.
        \nso the number represents the change in weight for each day of hibernation = -0.06.<\/p>\n

        b. What number represents the change in weight in ounces for the hedgehog in 115 days of hibernation?<\/p>\n

        Answer:<\/p>\n

        Assessment Practice<\/strong><\/p>\n

        Question 16.
        \nA basketball team played six games. In those games, the team won by 7 points, lost by 20, won by 8, won by 11, lost by 3, and won by 9. Which was the mean amount by which the team won or lost over the six games?
        \nA. -3 points
        \nB. 2 points
        \nC. 3 points
        \nD. 6 points<\/p>\n

        Answer:
        \nOption D is correct.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA basketball team played six games.
        \nIn those games, the team won by 7 points, lost by 20, won by 8, won by 11, lost by 3, and won by 9.
        \nwon points = 7 + 8 + 11 + 9.
        \n35.
        \nlost points = -20 -3.
        \n35\/6 = 5.8.
        \nso option D is correct.<\/p>\n

        Question 17.
        \nIn digging a hole, the construction crew records the location of the bottom of the hole relative to ground level. After 3 hours the hole is 8.25 feet deep.
        \nPART A
        \nWhat number represents the change in location in feet after 1 hour?<\/p>\n

        Answer:
        \nThe change in location in feet after 1 hour = 2.75 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nIn digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
        \nAfter 3 hours the hole is 8.25 feet deep.
        \n8.25\/3 = 2.75.
        \n2.75 + 2.75 + 2.75 = 8.25.
        \nso the change in locate in feet after 1 hour = 2.75 feet.<\/p>\n

        PART B
        \nIf the crew were to continue digging at the same rate, what number would they record for the location in feet after 8 hours?<\/p>\n

        Answer:
        \nThe number would they record for the location in feet after 8 hours = 1.03 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nIn digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
        \nAfter 8 hours the hole is 8.25 feet deep.
        \n8.25\/8 = 1.03.
        \nso the number would they record for the location in feet after 8 hours = 1.03 feet.<\/p>\n

        3-ACT MATH<\/strong><\/p>\n

        3-Act Mathematical Modeling: Win Some, Lose Some
        \n\"Envision
        \nACT 1<\/strong>
        \nQuestion 1.
        \nAfter watching the video, what is the first question that comes to mind?<\/p>\n

        Answer:
        \nIf the winning price is more than the loose price.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nwin some and lose some.
        \nfor example:
        \nif we win rs100\u00a0 and if we lose rs50.
        \nso the profit is rs50.<\/p>\n

        Question 2.
        \nWrite the Main Question you will answer.
        \nAnswer:<\/p>\n

        Question 3.
        \nMake a prediction to answer this Main Question.
        \nThe person who will win is _________.
        \nAnswer:<\/p>\n

        Question 4.
        \nConstruct Arguments Explain how you arrived at your prediction.
        \n\"Envision
        \nAnswer:<\/p>\n

        ACT 2<\/strong>
        \nQuestion 5.
        \nWhat information in this situation would be helpful to know? How would you use that information?
        \n\"Envision
        \nAnswer:<\/p>\n

        Question 6.
        \nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
        \nAnswer:<\/p>\n

        Question 7.
        \nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
        \nAnswer:<\/p>\n

        Question 8.
        \nWhat is your answer to the Main Question? Does it differ from your prediction? Explain.
        \n\"Envision
        \nAnswer:<\/p>\n

        ACT 3<\/strong>
        \nQuestion 9.
        \nWrite the answer you saw in the video.
        \nAnswer:<\/p>\n

        Question 10.
        \nReasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
        \n\"Envision
        \nAnswer:<\/p>\n

        Question 11.
        \nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
        \nAnswer:<\/p>\n

        Reflect<\/strong>
        \nQuestion 12.
        \nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
        \n\"Envision
        \nAnswer:<\/p>\n

        Question 13.
        \nReasoning How is each person’s starting score related to their final score?
        \nAnswer:<\/p>\n

        SEQUEL<\/strong>
        \nQuestion 14.
        \nConstruct Arguments If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning.
        \n\"Envision<\/p>\n

        Answer:
        \nI think that each person would wager at least $200.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nIf there were one final round where each contestant chooses how much to wager.
        \nso I am thinking that each person would wager at least $200.<\/p>\n

        Topic 2 Review<\/h3>\n

        Topic Essential Question<\/strong>
        \nHow can the properties of operations be used to solve problems involving integers and rational numbers?<\/p>\n

        Vocabulary Review
        \nComplete each definition and then provide an example of each vocabulary word.<\/p>\n

        \"Envision<\/p>\n

        Answer:
        \nA Rational number is a fraction a\/b where a and\/or b are fractions and b is not equal to 0.
        \nA terminating decimal that ends a(n).
        \nTwo numbers that have a sum of 0 are additive inverse.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA Rational number is a fraction a\/b where a and\/or b are fractions and b is not equal to 0.
        \nfor example:
        \nall fractions and decimals are rational numbers.
        \nA terminating decimal that ends a(n).
        \nfor example:
        \n2.30 is a terminating decimal.
        \nTwo numbers that have a sum of 0 are additive inverse.
        \nfor example.
        \n3 and -3.
        \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-1-Rational<\/p>\n

        Use Vocabulary in Writing
        \nExplain how you could determine whether \\(\\frac{\\frac{21}{3}}{\\frac{120}{12}}\\) and \\(\\frac{7}{9}\\) have the same decimal equivalent. Use vocabulary words in your explanation.<\/p>\n

        Answer:
        \nNo, both of them are not equal.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n\\(\\frac{\\frac{21}{3}}{\\frac{120}{12}}\\).
        \n21\/3 \/ 120\/12.
        \n7\/ 10.
        \n7\/10 and 7\/9 are not equal.
        \nso both of them are not equal.<\/p>\n

        Concepts and Skills Review<\/strong><\/p>\n

        Lesson 1.1 Relate Integers and Their Opposites<\/strong><\/p>\n

        Quick Review<\/strong>
        \nIntegers are the counting numbers, their opposites, and 0. Opposite integers are the same distance from 0 in opposite directions. Opposite quantities combine to make 0.<\/p>\n

        Example<\/strong>
        \nA climber descends 3 miles into a canyon. What integer represents the descent of her climb? How far does she have to climb to return to her starting point?
        \nThe descent of her climb is represented by -3. She has to climb 3 miles to return to her starting point.<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nOn a cold winter morning, the temperature was -4\u00b0F By noon, the temperature increased 4o. What was the temperature at noon?<\/p>\n

        Answer:
        \nThe temperature at noon = 4\u00b0F.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nOn a cold winter morning, the temperature was -4\u00b0F By noon, the temperature increased by 40.
        \nthe temperature at noon is 4\u00b0F.<\/p>\n

        Question 2.
        \nAudrey deposits $27 in her account. Then she makes two withdrawals, one for $15 and one for $12. What is the total change to the balance of Audrey’s account? Explain.<\/p>\n

        Answer:
        \nThe total change chance to the balance of Audrey’s account = $0.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nAudrey deposits $27 in her account.
        \nThen she makes two withdrawals, one for $15 and one for $12.
        \n$15 + $12 = $27.
        \n$27 – $27 = 0.
        \nso the total change chance to the balance of Audrey’s account = $0.<\/p>\n

        Lesson 1.2 Understand Rational Numbers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nAll rational numbers have an equivalent decimal form. The decimal equivalent will be either a terminating decimal or a repeating decimal. A terminating decimal ends in repeating zeros. A repeating decimal has a never-ending pattern of the same digits.<\/p>\n

        Example<\/strong>
        \nWrite the decimal equivalents for and i Are the decimals terminating or repeating?
        \nThe decimal equivalent for \\(\\frac{5}{8}\\) = 0.625, which is a terminating decimal.
        \nThe decimal equivalent for \\(\\frac{8}{11}\\) = 0.7272… = 0.72,
        \nwhich is a repeating decimal.<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nWhich fractions have a decimal equivalent that is a repeating decimal? Select all that apply.
        \n\u2610 \\(\\frac{13}{65}\\)
        \n\u2610 \\(\\frac{141}{47}\\)
        \n\u2610 \\(\\frac{11}{12}\\)
        \n\u2610 \\(\\frac{19}{3}\\)<\/p>\n

        Answer:
        \nOptions B and C are repeating decimals.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n13\/65 = 0.2.
        \n141\/47 = 3.
        \n11\/12 = 0.916.
        \n19\/3 = 6.33.
        \nso options B and C are repeating decimals.<\/p>\n

        Question 2.
        \nGreg bought 19\\(\\frac{11}{16}\\) gallons of gas. What decimal should the meter on the gas pump read?<\/p>\n

        Answer:
        \nThe decimal should the meter on the gas pump read = 19.6875.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nGreg bought 19\\(\\frac{11}{16}\\) gallons of gas.
        \n19(11\/16) = 315\/16.
        \n315\/16 = 19.6875.
        \nso the decimal should the meter on the gas pump read = 19.6875.<\/p>\n

        Question 3.
        \nWhat is the decimal equivalent of each rational number?
        \na. \\(\\frac{9}{11}\\)
        \nb. \\(\\frac{4}{5}\\)
        \nc. –\\(\\frac{17}{5}\\)
        \nd. \\(\\frac{5}{9}\\)<\/p>\n

        Answer:
        \n0.81, 0.8, 3.4, and 0.55.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n9\/11 = 0.81.
        \n4\/5 = 0.8.
        \n17\/5 = 3.4.
        \n5\/9 = 0.55.<\/p>\n

        Lesson 1.3 Add Integers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nTo add integers with the same sign, add the absolute value of each integer. The sign of the sum will be the same as the sign of the addends. To add integers with different signs, find the difference of the absolute value of each integer. The sign of the sum will be the same as the sign of the greater addend.<\/p>\n

        Example<\/strong>
        \nFind the sum of (-28) + (-19)
        \n|-28| + |-19| = 28 + 19 = 47
        \nThe sum of (-28) + (-19) = -47.
        \nFind the sum of (-28) + 19.
        \n|-28| – |19| = 28 \u2013 19 = 9
        \nThe sum of (-28) + 19 = -9, because
        \n|-28| > |19|<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nJonah’s cell phone came with 64 GB of memory. He has used 15 GB. He then uses 5 MB of memory to record photos and videos from a trip. Use the addition expression 64 + (-15) + (-5) to find how much memory is left on his phone.<\/p>\n

        Answer:
        \nThe memory is left on his phone = 44 MB.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nJonah’s cell phone came with 64 GB of memory.
        \nHe has used 15 GB.
        \nHe then uses 5 MB of memory to record photos and videos from a trip.
        \n64 + (-15) + (-5).
        \n64 -15 -5.
        \n49 – 5.
        \n44.
        \nso the memory is left on his phone = 44 MB.<\/p>\n

        Question 2.
        \nStella walks down a flight of stairs to the basement. Then she walks back up the stairs and up another flight of stairs to the second floor of her house. Each flight of stairs represents a change of 12 feet in height. How far is Stella above the ground?<\/p>\n

        Answer:
        \nThe far is Stella above the ground = 24 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nStella walks down a flight of stairs to the basement.
        \nThen she walks back up the stairs and up another flight of stairs to the second floor of her house.
        \nEach flight of stairs represents a change of 12 feet in height.
        \n12 +12 = 24.
        \nso the far is stella above the ground = 24 feet.<\/p>\n

        Question 3.
        \nFind the sum.
        \na. 64 + (-15)<\/p>\n

        Answer:
        \nThe sum is 49.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 64 and -15.
        \nadd the numbers.
        \n64 + (-15).
        \n64 – 15.
        \n49.<\/p>\n

        b. -121 + (-34)<\/p>\n

        Answer:
        \nThe sum is -155.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -121 and -34.
        \nadd the numbers.
        \n-121 – 34.
        \n-155.<\/p>\n

        c. -86 + 92<\/p>\n

        Answer:
        \nThe sum is 6.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -86 and 92.
        \nadd the numbers.
        \n-86 + 92 = 6.
        \nso the sum is 6.<\/p>\n

        d. 109 + (-162)<\/p>\n

        Answer:
        \nThe sum is -53.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 109 and -53.
        \nadd the numbers.
        \n109 + (-162).
        \n-53.
        \nso the sum is -53.<\/p>\n

        Lesson 1.4 Subtract Integers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nTo subtract integers, use the additive inverse to write an equivalent addition expression. Then follow the rules for addition. When the signs are the same, find the sum of the absolute values. When the signs are different, find the difference. Use the sign of the number with the greater absolute value.<\/p>\n

        Example<\/strong>
        \nFind -7 – (8).
        \n-7 + (-8) = -15
        \nThe signs are the same, so the sum has the same sign as the addends.<\/p>\n

        Find -7 – (-8).
        \n-7 + 8 = 1
        \nThe signs are different, so the sign of the difference is the same sign as the integer (8) with the greater absolute value, which is positive.<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nThe temperature is 1\u00b0F at dusk. It is 8 degrees colder at dawn. What is the temperature at dawn?<\/p>\n

        Answer:
        \nThe temperature at the dawn = -7\u00b0F.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nThe temperature is 1\u00b0F at dusk.
        \nIt is 8 degrees colder at dawn.
        \n1- 8 = -7.
        \nso the temperature at the dawn = -7\u00b0F.<\/p>\n

        Question 2.
        \nKyle and Nadim are on the same space on a board game they are playing. Kyle moves back 2 spaces in one turn and moves back 3 more spaces in his second turn. Nadim has remained in the same place. What integer represents Kyle’s location relative to Nadim’s location on the game board?<\/p>\n

        Answer:
        \nThe integer represents Kyle’s location relative to Nadim’s location on the game board = -1.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nKyle and Nadim are in the same space on a board game they are playing.
        \nKyle moves back 2 spaces in one turn and moves back 3 more spaces in his second turn.
        \nNadim has remained in the same place.
        \n2 – 3 = -1.
        \nso the integer represents Kyle’s location relative to Nadim’s location on the game board = -1.<\/p>\n

        Question 3.
        \nFind the difference.
        \na. 82 \u2013 (-14)<\/p>\n

        Answer:
        \nThe difference is 96.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 82 – (-14).
        \nsubtract the numbers.
        \n82 – (-14).
        \n82 + 14 = 96.<\/p>\n

        b. -18 – (-55)<\/p>\n

        Answer:
        \nThe difference is 37.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -18 – (-55).
        \nsubtract the numbers.
        \n-18 – (-55).
        \n-18 + 55.
        \n37.<\/p>\n

        c. \u2013 17 \u2013 44<\/p>\n

        Answer:
        \nThe difference is -61.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are -17 – 44.
        \nsubtract the numbers.
        \n-17 – 44.
        \n-61.<\/p>\n

        d. 70 – (-101)<\/p>\n

        Answer:
        \nThe difference is 171.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe two numbers are 70 – (-101).
        \nsubtract the numbers.
        \n70 – (-101).
        \n70 + 101.
        \n171.<\/p>\n

        Lesson 1.5 Add and Subtract Rational Numbers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nPositive and negative rational numbers and decimals can be added and subtracted following the same rules as adding and subtracting integers.<\/p>\n

        Example<\/strong>
        \nFind -5\\(\\frac{1}{2}\\) – 1.75.
        \nConvert 1.75 to an equivalent fraction, 1 \\(\\frac{3}{4}\\)
        \n\\(-5 \\frac{1}{2}-1 \\frac{3}{4}\\)
        \n= \\(-5 \\frac{2}{4}+\\left(-1 \\frac{3}{4}\\right)\\)
        \n= \\(-6 \\frac{5}{4}\\)
        \n= \\(-7 \\frac{1}{4}\\)<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nDoug digs a hole that is 1.7 feet below ground level. He plants a bush that is 37 feet tall from the bottom of the root to the top branch. How much of the bush is above the ground?<\/p>\n

        Answer:
        \nMuch of the bush is above the ground = 38.7 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nDoug digs a hole that is 1.7 feet below ground level.
        \nHe plants a bush that is 37 feet tall from the bottom of the root to the top branch.
        \n1.7 + 37 = 38.7.
        \nso much of the bush is above the ground = 38.7feet.<\/p>\n

        Question 2.
        \nPenelope has a birdhouse that is 4 feet above the roof of her garage. She has a second birdhouse that is 5.36 feet below the roof of her garage. What is the distance between the birdhouses?<\/p>\n

        Answer:
        \nThe distance between the birdhouses = -1.36.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nPenelope has a birdhouse that is 4 feet above the roof of her garage.
        \nShe has a second birdhouse that is 5.36 feet below the roof of her garage.
        \n4 + (-5.36).
        \n4 – 5.36.
        \n-1.36.
        \nso the distance between the birdhouses = -1.36.<\/p>\n

        Question 3.
        \nFind the sum or difference.
        \na. -2.63 + 3\\(\\frac{1}{4}\\)<\/p>\n

        Answer:
        \nThe difference is 0.62.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-2.63 + 3\\(\\frac{1}{4}\\)
        \n-2.63 + 3 (1\/4).
        \n-2.63 + 13\/4.
        \n-2.63 + 3.25.
        \n0.62.<\/p>\n

        b. -4\\(\\frac{1}{2}\\) \u2013 (-1.07)<\/p>\n

        Answer:
        \nThe difference is -3.43.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-4\\(\\frac{1}{2}\\) \u2013 (-1.07).
        \n-4(1\/2) – (-1.07).
        \n-9\/2 + 1.07.
        \n-4.5 + 1.07.
        \n-3.43.<\/p>\n

        c. 0.74 + (-\\(\\frac{3}{5}\\))<\/p>\n

        Answer:
        \nThe difference is 0.14.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n0.74 + (-\\(\\frac{3}{5}\\)).
        \n0.74 – 3\/5.
        \n0.74 – 0.6.
        \n0.14.<\/p>\n

        d. –\\(\\frac{1}{8}\\) \u2013 0.356<\/p>\n

        Answer:
        \nThe difference is -0.481.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n–\\(\\frac{1}{8}\\) \u2013 0.356.
        \n-1\/8 – 0.356.
        \n-0.125 – 0.356.
        \n-0.481.
        \nso the difference is -0.481.<\/p>\n

        Lesson 1.6 Multiply Integers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nMultiply integers the same way you multiply whole numbers. If the signs of the factors are the same, the product is positive. If the signs of the factors are different, the product is negative.<\/p>\n

        Example<\/strong>
        \n-9 \u2022 -8 = 72
        \n-9 \u2022 8 = -72<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nMarisa buys 4 books at $13 per book. What integer represents the total change in the amount of money Marisa has?<\/p>\n

        Answer:
        \nThe integer represents the total change in the amount of money Marisa has = $52.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nMarisa buys 4 books at $13 per book.
        \n4 x $13 = $52.
        \nso the integer represents the total change in the amount of money Marisa has = $52.<\/p>\n

        Question 2.
        \nWhich expressions have a product of -18? Select all that apply.
        \n\u2610 -2 \u2022 -9
        \n\u2610 -6 \u2022 3
        \n\u2610 -3 \u2022 6
        \n\u2610 -9 \u2022 2<\/p>\n

        Answer:
        \nOptions B, C, and D are correct.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nexpressions have a product of -18.
        \n-2 x -9 = 18.
        \n-6 x 3 = -18.
        \n-3 x 6 = -18.
        \n-9 x 2 = -18.
        \nso options B, C, and D are correct.<\/p>\n

        Question 3.
        \nFind the product.
        \na. -7 \u2022 -14
        \nb. \u201315 \u2022 12
        \nc. 9 \u2022 -20
        \nd. -11 \u2022 -16<\/p>\n

        Answer:
        \nThe products are 98, -180, -180, and 176.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-7 x -14 = 98.
        \n-15 x 12 = -180.
        \n9 x -20 = -180.
        \n-11 x -16 = 176.<\/p>\n

        Lesson 1.7 Multiply Rational Numbers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nThe same rules for multiplying integers apply to multiplying rational numbers. If the signs of the factors are the same, the product will be positive. If the signs of the factors are different, the product will be negative.<\/p>\n

        Example<\/strong>
        \n-9.6 \u2022 1.8 = -17.28
        \n-9.6 \u2022 -1.8 = 17.28<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nJason spends $2.35 to buy lunch at school. If he buys a lunch on 9 days, what number represents the total change in the amount of money Jason has?<\/p>\n

        Answer:
        \nThe total change in the amount of money Jason has = $21.15.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nJason spends $2.35 to buy lunch at school.
        \nIf he buys lunch on 9 days.
        \n$2.35 x 9 = $21.15.
        \nso the total change in the amount of money Jason has = $21.15.<\/p>\n

        Multiply.<\/strong>
        \nQuestion 2.
        \n-2\\(\\frac{2}{3}\\) \u2022 -4\\(\\frac{3}{7}\\)<\/p>\n

        Answer:
        \n-2(2\/3) x -4(3\/7) = 11.492.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-2\\(\\frac{2}{3}\\) \u2022 -4\\(\\frac{3}{7}\\)
        \n-2(2\/3) x -4(3\/7).
        \n-8\/3 x -31\/7.
        \n-2.6 x -4.42.
        \n11.492.<\/p>\n

        Question 3.
        \n-3\\(\\frac{4}{9}\\) \u2022 5\\(\\frac{2}{5}\\)<\/p>\n

        Answer:
        \n-3(4\/9) x 5(2\/5) = -18.36.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n-3\\(\\frac{4}{9}\\) \u2022 5\\(\\frac{2}{5}\\)
        \n-3(4\/9) x 5(2\/5).
        \n-31\/9 x 27\/5.
        \n-3.4 x 5.4.
        \n-18.36.<\/p>\n

        Question 4.
        \n6\\(\\frac{2}{3}\\) \u2022 -4\\(\\frac{1}{5}\\)<\/p>\n

        Answer:
        \n6(2\/3) x -4(1\/5) = 2.4.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n6\\(\\frac{2}{3}\\) \u2022 -4\\(\\frac{1}{5}\\).
        \n6(2\/3) x -4(1\/5).
        \n20\/3 – 21\/5.
        \n6.6 – 4.2.
        \n2.4.<\/p>\n

        Lesson 1.8 Divide Integers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nDivide integers the same way you divide whole numbers. The quotient is positive if the signs of the dividend and divisor are the same. The quotient is negative if the signs of the dividend and divisor are different.<\/p>\n

        Example<\/strong>
        \n-39 \u00f7 3 = -13
        \n-39 \u00f7 3 = 13<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nWhich expressions have a quotient of -4? Select all that apply.
        \n\u2610 \\(\\frac{-24}{6}\\)
        \n\u2610 -36 \u00f7 -9
        \n\u2610 -72 \u00f7 18
        \n\u2610 \\(\\frac{84}{-21}\\)<\/p>\n

        Answer:
        \nOptions A,C, and D are correct.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nthe expressions have a quotient of -4 are:
        \n-24\/6 = -4.
        \n-36\/-9 = 4.
        \n-72\/ 18 = -4.
        \n84\/-21 = -4.
        \nso options A, C, and D are correct.<\/p>\n

        Question 2.
        \nWhitney rolls a ball down a ramp that is 18 feet long. If the ball rolls down 2 feet each second, what integer represents the amount of time, in seconds, the ball takes to reach the end of the ramp?<\/p>\n

        Answer:
        \nThe integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nWhitney rolls a ball down a ramp that is 18 feet long.
        \nIf the ball rolls down 2 feet each second.
        \n18\/2 = 9.
        \nso the integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds.<\/p>\n

        Question 3.
        \nFind the quotient.
        \na. \\(\\frac{81}{-9}\\)
        \nb. -123 \u00f7 -4
        \nc. –\\(\\frac{94}{4}\\)
        \nd. 65 \u00f7 (-5)<\/p>\n

        Answer:
        \nThe quotients are\u00a0-9, 30.75, -23.5, and -13.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \na: 81\/-9 = -9.
        \n-123\/-4 = 30.75.
        \n-94\/4 = -23.5.
        \n65 \/ -5 = -13.
        \nso the quotients are -9, 30.75, -23.5, and -13.<\/p>\n

        Lesson 1.9 Divide Rational Numbers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nThe same rules for dividing integers apply to dividing all rational numbers. The quotient is positive when the numbers being divided have the same signs. The quotient is negative when the numbers being divided have different signs. Complex fractions have a fraction in the numerator, the denominator, or both. To divide by a fraction, rewrite as multiplication by its multiplicative inverse, or reciprocal.<\/p>\n

        Example<\/strong>
        \nSimplify \\(\\frac{-\\frac{3}{4}}{\\frac{15}{24}}\\)
        \n\\(-\\frac{3}{4} \\div \\frac{15}{24}=-\\frac{3}{4} \\cdot \\frac{24}{15}=-\\frac{72}{60}=-\\frac{6}{5}=-1 \\frac{1}{5}\\)<\/p>\n

        Practice<\/strong>
        \nFind the quotient.
        \nQuestion 1.
        \n\\(\\frac{8}{9}\\) \u00f7 -1\\(\\frac{4}{15}\\)<\/p>\n

        Answer:
        \n8\/9 \/ -1(4\/15) = -0.38.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \ndivide the numbers.
        \n\\(\\frac{8}{9}\\) \u00f7 -1\\(\\frac{4}{15}\\).
        \n8\/9 – 1(4\/15).
        \n0.88 – 19\/15.
        \n0.88 – 1.26.
        \n-0.38.<\/p>\n

        Question 2.
        \n\u20133.6 \u00f7 2\\(\\frac{1}{7}\\)<\/p>\n

        Answer:
        \n\u20133.6 \u00f7 2\\(\\frac{1}{7}\\) = -1.68.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \ndivide the numbers.
        \n\u20133.6 \u00f7 2\\(\\frac{1}{7}\\).
        \n-3.6 \/ 2(1\/7).
        \n-3.6 \/ 2.14.
        \n-1.68.<\/p>\n

        Question 3.
        \nA boat drops an anchor 17.5 feet to the bottom of a lake. If the anchor falls at a rate of 0.07 feet each second, how long will it take the anchor to reach the bottom of the lake?<\/p>\n

        Answer:
        \nThe long will it take the anchor to reach the bottom of the lake = 25 feet.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nA boat drops an anchor 17.5 feet to the bottom of a lake.
        \nIf the anchor falls at a rate of 0.07 feet each second.
        \n1.75 \/ 0.07 = 25.
        \nso the longer will it take the anchor to reach the bottom of the lake = 25 feet.<\/p>\n

        Lesson 1.10 Solve Problems with Rational Numbers<\/strong><\/p>\n

        Quick Review<\/strong>
        \nYou can use rational numbers to solve problems in the same way that you use whole numbers. Be sure to make sense of the problem you are solving to help you choose the correct operations and determine which values will be positive and which will be negative.<\/p>\n

        Example<\/strong>
        \nDuring a 15-day dry spell, the water level in a lake changed by -2\\(\\frac{3}{8}\\) inches. What rational number represents the average change in the water level per day?
        \n-2\\(\\frac{3}{8}\\) \u00f7 15
        \n= –\\(\\frac{19}{8}\\) \u2022 \\(\\frac{1}{15}\\)
        \n= –\\(\\frac{19}{120}\\) inch<\/p>\n

        Practice<\/strong>
        \nQuestion 1.
        \nIn 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13. What integer represents her average score for the 5 rounds?<\/p>\n

        Answer:
        \nThe integer represents her average score for the 5 rounds = 2.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nIn 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13.
        \n-3, 8, 9, -7, and 13.
        \n10\/5 = 2.
        \nso the integer represents her average score for the 5 rounds = 2.<\/p>\n

        Question 2.
        \nPeter signed up for a program that costs $10.50 per month to stream movies to his computer. He decided to cancel his service after \\(\\frac{5}{6}\\) month. He only has to pay for the amount of time he used the service. What number represents the total change in the amount of money Peter has after paying for the service?<\/p>\n

        Answer:
        \nThe number represents the total change in the amount of money Peter has after paying for the service =$8.715.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nPeter signed up for a program that costs $10.50 per month to stream movies to his computer.
        \nHe decided to cancel his service after \\(\\frac{5}{6}\\) month.
        \nHe only has to pay for the amount of time he used the service.
        \n$10.50 x 5\/6.
        \n$10.50 x 0.83.
        \n$8.715.
        \nso the number represents the total change in the amount of money Peter has after paying for the service = $8.715.<\/p>\n

        Question 3.
        \nMaggie spent $4.05 on cheese and fruit at the farmer’s market. She bought \\(\\frac{1}{8}\\) pound of apples, \\(\\frac{1}{4}\\) pound of pears, and 1.25 pounds of bananas. If fruit cost $0.80 per pound, how much did Maggie spend on cheese?<\/p>\n

        Answer:
        \nMaggie spends on cheese = 5.675 pounds.<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \nMaggie spent $4.05 on cheese and fruit at the farmer’s market.
        \nShe bought \\(\\frac{1}{8}\\) pound of apples, \\(\\frac{1}{4}\\) pound of pears, and 1.25 pounds of bananas.
        \nIf fruit cost $0.80 per pound.
        \n$4.05 + 1\/8 + 1\/4 + 1.25 = 5.3 + 0.125 + 0.25.
        \n5.3 + 0.375.
        \n5.675.
        \nso the Maggie spend on cheese = 5.675 pounds.<\/p>\n

        Topic 1 Fluency Practice<\/h3>\n

        Crisscrossed<\/strong>
        \nFind each sum, difference, product, or quotient. Write your answers in the cross-number puzzle below. Each digit and negative sign in your answers goes in its own box.
        \nI can… add, subtract, multiply, and divide integers.
        \n\"Envision<\/p>\n

        Across
        \n2. 248 + (-1,027)
        \n5. 818 – (-1,021)
        \n6. -516 + 774
        \n8. 242 + (-656)
        \n9. 2,087 + (-1,359)
        \n10. 631 \u2013 897
        \n11. \u2013342 + 199
        \n12. -49 \u2022 -27
        \n13. \u2013321 \u2013 987
        \n14. 2,988 \u00f7 -3
        \n15. 2,580 \u00f7 6
        \n16. 4,592 \u00f7 -82
        \n17. 48 \u2022 -27
        \n18. -24 \u2022 83
        \n21. -118 + 1,201
        \n22. -45 \u2022 -59<\/p>\n

        Answer:<\/p>\n

        Explanation:
        \nIn the above-given question,
        \ngiven that,
        \n248 – 1027 =<\/p>\n

        Down
        \n1. 246 + 173
        \n2. 22 \u2022 -22
        \n3. 726 \u2013 (-219)
        \n4. 501 – 699
        \n7. -10,740 \u00f7 15
        \n8. 6,327 \u00f7 -9
        \n10. 144 \u2022 -16
        \n11. 15 \u2022 -67
        \n12. 7,164 \u00f7 4
        \n13. \u201333 \u2022 63
        \n14. -2,695 \u00f7 55
        \n17. -1,032 \u2013 (-285)
        \n18. 512 \u2013 720
        \n19. \u2013729 + 951
        \n20. \u201317 \u2022 -25<\/p>\n","protected":false},"excerpt":{"rendered":"

        Go through the\u00a0enVision Math Common Core Grade 7 Answer Key\u00a0Topic 1 Rational Number Operations\u00a0regularly and improve your accuracy in solving questions. enVision Math Common Core 7th Grade Answers Key Topic 1 Rational Number Operations Topic 1 Essential Question How can the properties of operations be used to solve problems involving integers and rational numbers? 3-ACT …<\/p>\n","protected":false},"author":1,"featured_media":25561,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/envisionmathanswerkey.com\/wp-content\/uploads\/2021\/09\/Envision-Math-Common-Core-Grade-7-Answer-Key-Topic-1-Rational-Number-Operations.png","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/15180"}],"collection":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/comments?post=15180"}],"version-history":[{"count":4,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/15180\/revisions"}],"predecessor-version":[{"id":28200,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/15180\/revisions\/28200"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media\/25561"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=15180"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=15180"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=15180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}