## 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 MATH

Win 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.
In any quiz competition, it’s important to understand the rules and scoring. Think about this during the 3-Act Mathematical Modeling lesson.

### Topic 1 enVision STEM Project

Did You Know?
The lowest recorded temperature in the world, -136°F (-93.2°C), occurred in Antarctica.

The highest recorded temperature in the world, 134°F (56.7°C), occurred in Death Valley, California.

The Celsius scale (°C) is commonly used for temperature measurement in most of the world.
Only a small number of nations, including the United States, regularly use the Fahrenheit scale (°F).

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°F range.

Your Task: How Cold is Too Cold?
There 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.

### Topic 1 Get Ready!

Review What You Know!

Vocabulary
Choose the best term from the box. Write it on the blank.

• absolute value
• Associative Property
• Commutative Property
• Distributive Property
• integers
• rational number

Question 1.
The __________ explains why a × b = b × a and a + b = b + a.

Answer:
The commutative property explains why a x b = b x a and a + b = b + a.

Explanation:
In the above-given question,
given that,
The commutative property explains why a x b = b x a and a + b = b + a.
the 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.
the property holds for addition and multiplication, but not for subtraction and division.
for example:
4 + 3 = 7.
3 + 4 = 7.
2 x 6 = 12.
6 x 2 = 12.

Question 2.
The __________ of -6 is 6, because it is 6 units from zero on the number line.

Answer:
The absolute value of -6 is 6, because it is 6 units from zero on the number line.

Explanation:
In the above-given question,
given that,
The absolute value of -6 is 6, because it is 6 units from zero on the number line.
for example:
6 and -6 are at the same distance from zero on the number line.

Question 3.
The number $$\frac{5}{3}$$ is a _________ because 5 and 3 are integers and 3 ≠ 0.

Answer:
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.

Explanation:
In the above-given question,
given that,
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
for example:
3, 5, 6, and 7 are rational numbers.
rational numbers are also integers.

Question 4.
The set of _________ consists of the counting numbers, their opposites, and zero.

Answer:
The set of integers consists of the counting numbers, their opposites, and zero.

Explanation:
In the above-given question,
given that,
The set of integers consists of the counting numbers, their opposites, and zero.
Integers are counting numbers, their opposites, and zero.
for example:
integers can either be negative, positive, or zero.
1, -1, 2, -2, and 3, -3.

Question 5.
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the _________

Answer:
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.

Explanation:
In the above-given question,
given that,
The sum of (a + b) + c is equal to the sum of a + (b + c) as explained by the distributive property.
the distributive property is sometimes called the distributive law of multiplication and division.
for example:
4(x – 3) = 20.
4x – 12 = 20.
4x – 12 + 12 = 20 + 12.
4x = 32.
x = 32/4.
x = 8.

Question 6.
If you evaluate n × (v + z) by writing it as (n × y) + (n × z), you have used the __________.

Answer:
If you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.

Explanation:
In the above-given question,
given that,
If you evaluate n x (v + z) by writing it as (n x y) + (n x z), you have used the distributive property.
for example:
4(x – 3) = 20.
4x – 12 = 20.
4x – 12 + 12 = 20 + 12.
4x = 32.
x = 32/4.
x = 8.

Add and Subtract Fractions and Decimals

Add or subtract.
Question 7.
$$2 \frac{1}{3}+6 \frac{3}{5}$$

Answer:
2(1/3) + 6(3/5) = 8.9.

Explanation:
In the above-given question,
given that,
2(1/3) + 6(3/5).
7/3 + 33/5.
2.3 + 6.6.
8.9.
2(1/3) + 6(3/5) = 8.9.

Question 8.
$$9 \frac{1}{10}-4 \frac{3}{4}$$

Answer:
9(1/10) – 4(3/4) = 4.35.

Explanation:
In the above-given question,
given that,
9(1/10) – 4(3/4).
91/10 – 19/4.
9.1 – 4.75.
4.35.
9(1/10) – 4(3/4) = 4.35.

Question 9.
19.86 + 7.091

Answer:
19.86 + 7.091 = 26.951.

Explanation:
In the above-given question,
given that,
add the numbers.
19.86 + 7.091.
19.86 + 7.091 = 26.951.

Question 10.
57 – 10.62

Answer:
57 – 10.62 = 46.38.

Explanation:
In the above-given question,
given that,
subtract the numbers.
57 – 10.62.
46.38.
57 – 10.62 = 46.38.

Multiply and Divide Fractions and Decimals

Multiply or divide.
Question 11.
4.08 × 29.7

Answer:
4.08 x 29.7 = 121.176.

Explanation:
In the above-given question,
given that,
multiply the numbers.
4.08 x 29.7 = 121.176.

Question 12.
15,183.3 ÷ 473

Answer:
15,183.3 / 473 = 32.1.

Explanation:
In the above-given question,
given that,
divide the numbers.
15,183.3 / 473 = 32.1.

Question 13.
$$\frac{15}{16} \times 9 \frac{1}{5}$$

Answer:
15/16 x 41/5 = 7.6875.

Explanation:
In the above-given question,
given that,
multiply the numbers.
15/16 x 41/5.
0.9375 x 8.2.
7.6875.
15/16 x 41/5 = 7.6875.

Question 14.
$$4 \frac{7}{9} \div 1 \frac{7}{12}$$

Answer:
4(7/9) / 1(7/12) = 2.974.

Explanation:
In the above-given question,
given that,
divide the numbers.
4(7/9) / 1(7/12).
43/9 / 19/12.
4.7 / 1.58.
2.974.
4(7/9) / 1(7/12) = 2.974.

Question 15.
Byron 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?
A. $$\frac{119}{80} \mathrm{~kg}$$
B. $$\frac{1}{8} \mathrm{~kg}$$
C. 1.4125 kg
D. $$\frac{17}{80} \mathrm{~kg}$$

Answer:
The quantity of pepper will be in each shaker = 1/8.

Explanation:
In the above-given question,
given that,
Byron has $$1 \frac{7}{10}$$ kilograms of black pepper.
He uses the pepper and splits it between 7 pepper shakers.
1(7/10).
17/10 = 1.7.

Language Development
Fill in the word map with new terms, definitions, and supporting examples or illustrations.

Answer:
Distributive property.
commutative property.
Integer.
Rational number.

Explanation:
In the above-given question,
given that,
The commutative property explains why a x b = b x a and a + b = b + a.
the 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.
the property holds for addition and multiplication, but not for subtraction and division.
for example:
4 + 3 = 7.
3 + 4 = 7.
2 x 6 = 12.
6 x 2 = 12.
The number 5/3 is a rational number because 5 and 3 are integers and 3 is not equal to 0.
for example:
3, 5, 6, and 7 are rational numbers.
rational numbers are also integers.
The set of integers consists of the counting numbers, their opposites, and zero.
Integers are counting numbers, their opposites, and zero.
for example:
integers can either be negative, positive, or zero.
1, -1, 2, -2, and 3, -3.

Pick A Project

PROJECT 1A
What is something you can make?
PROJECT: DESIGNATIANDMADE IM TO SIIT

PROJECT 1B
How old were you when petroglyphs were being painted?
PROJECT: MAKE A TIMELINE

PROJECT 1C
What makes an obstacle course fun?
PROJECT: BUILD A MODEL OF AN OBSTACLE COURSE

PROJECT 1D
What are your favorite ways to exercise ?
PROJECT: FILM AN EXERCISE VIDEO

### Lesson 1.1 Relate Integers and Their Opposites

Solve & Discuss It!
When preparing for a rocket launch, the mission control center uses the phrase “T minus” before liftoff.
…T minus 3, T minus 2, T minus 1, …
After the rocket has launched, “T plus” is used while the rocket is in flight.
…T plus 1, T plus 2, T plus 3, …
When does the rocket launch? What could “T” represent?

I can… relate integers, their opposites, and their absolute values.

Reasoning
What integers can you use to represent this situation?

Focus on math practices
Reasoning How are “T minus 4” and “T plus 4” related?

Answer:
The situation is T plus.

Explanation:
In the above-given question,
given that,
the rocket is launching.
so we have to use the “T plus”.
“T plus 4”.

Essential Question
How are integers and their opposites related?

Try It!

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?
The integer ________ represents Xavier’s climb down.
The integer 0 represents __________.

Answer:
The integer -9 represents Xavier’s climb down.
the integer 0 represents that he did not move.

Explanation:
In the above-given question,
given that,
Xavier climbs 9 feet up into an apple tree.
Xavier climbs feet down into an apple tree is -9.
the integer 0 represents that he did not move.
so the integer 0 is related to +0ft or -0ft.

Convince Me! How are the absolute values of opposite integers related?

Try It!

The temperature was 75°. At noon, the temperature increased 7°. By evening, the temperature decreased by 7°. How did the temperature change?

Answer:
The temperature change by 75° + 7 and 75° – 7.

Explanation:
In the above-given question,
given that,
The temperature was 75°. At noon, the temperature increased 7°
75° + 7.
increased means plus.
decreased means minus.
the temperature change by 75° + 7 and 75° – 7.

Try It!

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? Answer: The money does she have left =$33.

Explanation:
In the above-given question,
given that,
Shaniqua has $45 in her wallet. She spends$4 on snacks and $8 on a movie ticket.$45 – $4 =$41.
$41 –$8 = $33. we are using the minus operator. so the money does she have left =$33.

KEY CONCEPT
An integer, n, and its opposite, -n, combine to make 0.

Do You Understand?
Question 1.
Essential Question How are integers and their opposites related?

Answer:
The 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.

Explanation:
In the above-given question,
given that,
The 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.
for example:
67 is the integer and its opposite side is -67.
its opposite has to be on the other side of zero and the same amount of numbers away from zero.

Question 2.
Reasoning 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.

Answer:
Helium atom has integers are +2 and -2.

Explanation:
In the above-given question,
atom has 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.
+2, and -2.
so helium atom has integers that are +2 and -2.

Question 3.
Model with Math Explain how to use a number line to show that opposite quantities combine to make 0.

Answer:
The number line shows the + 3 and – 3.

Explanation:
In the above-given question,
given that,
there are some numbers on the number line.
-3, 0, and 3.
so the number line shows the integers +3 and -3.

Do You Know How?
Question 4.
Marcus 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?

Answer:
The Marcus have to go to return to the surface = -18 meters.

Explanation:
In the above-given question,
given that,
Marcus dives from the surface of the ocean to a reef 18 meters below sea level.
-18 represents Marcus’s location relative to the surface.
-18, 0, and 18.
so the Marcus has to go to return to the surface = -18 meters.

Question 5.
The temperature of the water in Emily’s fish tank was 78°F on Sunday. The water temperature changed by -3° 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?

Answer:
The temperature change of the water from Sunday to Tuesday = plus.
The water temperature on Tuesday = 105°F.

Explanation:
In the above-given question,
given that,
The temperature of the water in Emily’s fish tank was 78°F on Sunday.
The water temperature changed by -3° on Monday.
75°F.
and then by 3o on Tuesday.
105°F.
the temperature change of the water from Sunday to Tuesday = plus.
the water temperature on Tuesday = 105°F.

Question 6.
The scores of players on a golf team are shown in the table. The team’s combined score was 0. What was Travis’s score?

Answer:
Travis’s score = -4.

Explanation:
In the above-given question,
given that,
The scores of players on a golf team are shown in the table.
The team’s combined score was 0.
Travis’s score is -4.

Practice & Problem Solving

Leveled Practice In 7-9, write the integer that represents the situation.
Question 7.
Max spent $53 and now has no money left. He had$ _______ before his purchase.

Answer:
He had $53 before his purchase. Explanation: In the above-given question, given that, Max spent$53 and now has no money left.
$53 –$53 = 0.
so he had $53 before his purchase. Question 8. The temperature was 8°F. It dropped so that the temperature was 0°F. ________ °F represents the change in temperature. Answer: The change in temperature represents = -8°F. Explanation: In the above-given question, given that, The temperature was 8°F. It dropped so that the temperature was 0°F. 8 – 0 = 8. so the change in the temperature represents = -8°F. Question 9. An airplane descended 4,000 feet before landing. The integer that represents how many feet the airplane was above the ground before its descent is _________. Answer: The number of feet the airplane was above the ground before its descent = -4000. Explanation: In the above-given question, given that, An airplane descended 4,000 feet before landing. before landing = 4000 meters. after landing = -4000 m. so the number of feet the airplane was above the ground before its descent = -4000 m. Question 10. Carolyn says that point A and point B represent opposite integers. a. What is the opposite of the integer represented by point A? By point B? Answer: The opposite of the integer represented by point A is -7 and by point, B is 8. Explanation: In the above-given question, given that, Carolyn says that point A and point B represent opposite integers. in the number line, the numbers are -10, -5, 0, 5, and 10. so the missing numbers are points A and B. The opposite of the integer represented by point A is -7 and by point, B is 8. b. Construct Arguments Do you agree with Carolyn? Explain. Answer: Yes, Arguments agree with Carolyn. Explanation: In the above-given question, given that, Carolyn says that point A and point B represent opposite integers. in the number line, the numbers are -10, -5, 0, 5, and 10. so the missing numbers are points A and B. The opposite of the integer represented by point A is -7 and by point, B is 8. Question 11. A 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. Answer: The integer represents the yards gained = 9 yards. Explanation: In the above-given question, given that, A football team lost 9 yards during a play. The team had a combined gain or loss of 0 yards after the next play. the integer represents the yards gained = 9 yards. -9, 0, and 9. so the integer represents the yards gained = 9 yards. Question 12. A 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? Answer: The height of the car at the end of the ride = 5 m. Explanation: In the above-given question, given that, A roller coaster car goes above and below ground. The roller coaster starts at 1 meter. drops 4 meters down. rises 13 meters up. drops 6 meters. 13 – 6 = 7. so the height of the car at the end of the ride = 5 m. Question 13. Dimitri 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? Answer: The integer represents the change from the base price of the car to its final price =$1400.

Explanation:
In the above-given question,
given that,
Dimitri is buying a car.
He chooses Option 1 to add a new sound system to his car.
the base price of the car = -$700. option 1 =$1400.
so the integer represents the change from the base price of the car to its final price = $1400. Question 14. Make 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. Answer: x and y are equal to 0. Explanation: In the above-given question, given that, if |x| = 16. |y| = -16. 16 – 16 = 0. so x and y are equal to 0. Question 15. Write a situation that can be represented by the opposite of -42. Answer: The situation that can be represented by the opposite of -42 is 42. Explanation: In the above-given question, given that, the situation that can be represented by the opposite of -42 is 42. 42 is the positive integer. -42 is the negative integer. so the situation that can be represented by the opposite 0f -42 is 42. Question 16. Higher Order Thinking 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. Answer: The possible locations for Carl’s house = 3 and 5. Explanation: In the above-given question, given that, 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. 5 – 2 = 3. so the location of Ann is 0. the location of Carl is 3. the location of Beth is 5. so the possible locations for Carl’s house = 3 and 5. Assessment Practice Question 17. Which of these situations can be represented with an integer that when combined with -9 makes 0? Select all that apply. ☐ You walk down 9 flights of stairs. ☐ You climb up 9 flights of stairs. ☐ The temperature drops 9°F. ☐ You spend$9 on a book.
☐ You earn $9 from your job. Answer: You climb up 9 flights of stairs, you spend$9 on a book, and you earn $9 from your job. Explanation: In the above-given question, given that, an integer that when combined with -9 makes 0. You climb up 9 flights of stairs. you spend$9 on a book.
you earn $9 from your job. Question 18. Which of these situations can be represented by the opposite of 80? Select all that apply. ☐ An airplane descends 80 m. ☐ An elevator ascends 80 m. ☐ The cost of a train ticket drops by$80.
☐ You remove 80 songs from an MP3 player.
☐ Suzy’s grandmother is 80 years old.

Answer:
An airplane descends 80m, the cost of a train ticket drops by $80, and you remove 80 songs from an MP3 player. Explanation: In the above-given question, given that, the situations can be represented by the opposite of 80. An airplane descends 80m = -80m. the cost of a train ticket drops by$80 = -$80. You remove 80 songs from an MP3 player = -80. ## Lesson 1.2 Understand Rational Numbers Solve & Discuss It! Calvin 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. I can… recognize rational numbers and write them in decimal form. Answer: The width of his surfboard = 1 wide. Explanation: In the above-given question, given that, Calvin wants to customize his surfboard so that it is wider than the 82 models but narrower than the 92 models. 82 model has a wide of 221/2 and 3(1/4) of thick. 92 model has a wide of 23(1/4) and 3(1/2) thick. 23 – 22 = 1. so the width of his surfboard = 1 wide. Focus on math practices Use Structure Lindy’s surfboard is 23 inches wide. Between which two surfboard models is her custom surfboard’s width? How do you know? Answer: The two surfboard models are her custom surfboard’s width = surfboards 82 and 92. Explanation: In the above-given question, given that, Lindy’s surfboard is 23 inches wide. Between which, two surfboard models are her custom surfboard’s width. so the two surfboard models is her custom surfboard’s width = surfboards 82 and 92. Essential Question How are rational numbers written as decimals? Try It! 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? Juanita should use the decimal ________ to update her report. Answer: Juanita should use the decimal 0 to update her report. Explanation: In the above-given question, given that, the pitcher threw a total of 384 pitches and used a fastball 240 times. 380 / 240. 380 x 6 = 2304. 380 x 2 = 768. 380 x 5 = 1920. 1920 – 1920 = 0. Convince Me! How do you know that the answer is a terminating decimal? Try It! What is the decimal form of $$\frac{100}{3}, \frac{100}{5}$$ and $$\frac{100}{6}$$? Determine whether each decimal repeats or terminates. Answer: The decimal form of 100/3 = 33.3. the decimal form of 100/5 = 20. the decimal form of 100/6 = 16.6. Explanation: In the above-given question, given that, the decimal form of 100/3, 100/5, and 100/6. 100/3 = 33.3. 100/5 = 20. 100/6 = 16.6. the decimal 100/3 and 100/6 repeats. the decimal 100/5 terminates. Try It! Is -0.3 a rational number? Is 3.14144144414444… a rational number? Explain your reasoning. Answer: Yes, both are rational numbers. Explanation: In the above-given question, given that, the numbers are -0.3 and 3.14144144414444… -0.3 is a rational number. 3.14144144414444 is a rational number. KEY CONCEPT To 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. Do You Understand? Question 1. Essential Question How are rational numbers written as decimals? Answer: Rational numbers can be written as decimals in two ways. they are terminating decimal and repeating decimal. Explanation: In the above-given question, given that, the rational numbers can be written as a decimal in two ways. for example: 100/5 = 20. 20 is a terminating decimal. 100/3 = 33.3. 33.3 is a repeating decimal. Question 2. Reasoning How can you use division to find the decimal equivalent of a rational number? Answer: 100/3 = 33.3. Explanation: In the above-given question, given that, 100/3 = 33.3. 33.3 is a terminating decimal. 100/5 = 20. 20 is a repeating decimal. Question 3. Be Precise What is the difference between a terminating decimal and a repeating decimal? Answer: The difference is one decimal will stop and one decimal will continue. Explanation: In the above-given question, given that, the two terms are terminating and repeating decimals. 100/3 = 33.3. it is a repeating decimal. 100/5 = 20. 20 is a terminating decimal. Do You Know How? Question 4. What is the decimal equivalent of each rational number? a. $$\frac{7}{20}$$ Answer: 7/20 = 0.35. Explanation: In the above-given question, given that, the number is 7/20. 7/20 = 0.35. 0.35 is a terminating decimal. b. –$$\frac{23}{20}$$ Answer: -23/20 = -1.15. Explanation: In the above-given question, given that, the number is -23/20. -23/20 = -1.15. it is a terminating decimal. c. $$\frac{1}{18}$$ Answer: 1/18 = 0.055. Explanation: In the above-given question, given that, the number is 1/18. 1/18 = 0.055. 0.055 is a repeating decimal. d. –$$\frac{60}{22}$$ Answer: -60/22 = 2.727 is a repeating decimal. Explanation: In the above-given question, given that, the number is -60/22. -60/22 = 2.727. 2.727 is a repeating decimal. Question 5. There are 5,280 feet in a mile. What part of a mile, in decimal form, will you drive until you reach the exit? Answer: 5280/1000 = 5.28 miles. Explanation: In the above-given question, given that, There are 5,280 feet in a mile. 5280/1000 = 5.28. so the part of the mile in decimal form = 5.28 miles. Practice & Problem Solving Leveled Practice In 6-8, write the decimal equivalent for each rational number. Use a bar over any repeating digits. Question 6. $$\frac{2}{3}$$ Answer: 2/3 = 0.666. Explanation: In the above-given question, given that, the number is 2/3. 2/3 = 0.666. 0.666 is a repeating decimal. Question 7. $$\frac{3}{11}$$ Answer: 3/11 = 0.272727. Explanation: In the above-given question, given that, the number is 3/11. 3/11 = 0.272727. 0.272727 is a repeating decimal. Question 8. $$8\frac{4}{9}$$ Answer: 8(4/9) = repeating decimal. Explanation: In the above-given question, given that, the number is 8(4/9). 9 x 8 = 72. 72 + 4 = 76. 76/9 = 8.4444. 8.444 is a repeating decimal. Question 9. Is 1.0227 a rational number? Explain. Answer: Yes, 1.0227 is a rational number. Explanation: In the above-given question, given that, the number is 1.0227. 1.0227 is a rational number. Question 10. Which should Aaron use to convert a fraction to a decimal? Answer: Option A is correct. Explanation: In the above-given question, given that, numerator/denominator is used to convert a fraction to a decimal. so option A is correct. Question 11. Is the fraction $$\frac{1}{3}$$ equivalent to a terminating decimal or a decimal that does not terminate? Answer: 1/3 is a repeating decimal. Explanation: In the above-given question, given that, the number is 1/3. 1/3 = 0.333. so 1/3 is a repeating decimal. Question 12. Determine whether the given number belongs to each set. Answer: -34 is an Integer. Explanation: In the above-given question, given that, they have given the whole numbers, integers, and rational numbers. so -34 is an integer. Question 13. Ariel incorrectly says that 2$$\frac{5}{8}$$ is the same as 2.58. a. Convert 2$$\frac{5}{8}$$ to a decimal. Answer: 2(5/8) = 2.625. Explanation: In the above-given question, given that, Ariel incorrectly says that 2$$\frac{5}{8}$$ is the same as 2.58. the number is 2(5/8). 8 x 2 = 16. 16 + 5 = 21. 21/8 = 2.625. b. What was Ariel’s likely error? Answer: Ariel’s error was the decimal. Explanation: In the above-given question, given that, Ariel incorrectly says that 2$$\frac{5}{8}$$ is the same as 2.58. the number is 2(5/8). 8 x 2 = 16. 16 + 5 = 21. 21/8 = 2.625. Question 14. Use Structure Consider the rational number $$\frac{3}{11}$$ a. What are the values of a and b in $$a\sqrt {b}$$ when you use division to find the decimal form? Answer: 3/11 = 0.272727. Explanation: In the above-given question, given that, a = 3, and b = 11. 3/11 = 0.272727. so 3/11 is a repeating decimal. 3/11 = 0.272727. b. What is the decimal form for $$\frac{3}{11}$$? Answer: The decimal form is 0.272727. Explanation: In the above-given question, given that, a = 3, and b = 11. 3/11 = 0.272727. so 3/11 is a repeating decimal. 3/11 = 0.272727. Question 15. At a grocery store, Daniel wants to buy 3$$\frac{1}{5}$$ lb of ham. What decimal should the digital scale show? Write 3$$\frac{1}{3}$$ as a fraction and then divide. The scale should read ________ lb. Answer: The scale should read 3.3 lb. Explanation: In the above-given question, given that, At a grocery store, Daniel wants to buy 3$$\frac{1}{5}$$ lb of ham. 3(1/3). 3 x 3 = 9. 9 + 1 = 10. 10/3 = 3.333. 3.333 is a repeating decimal. so the scale should read 3.3 lb. Question 16. Reasoning 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. Answer: The number of pounds of pork she bought using a decimal = 9.47 pounds. Explanation: In the above-given question, given that, At a butcher shop, Hilda bought beef and pork. She left with 18$$\frac{8}{25}$$ pounds of meat. 18(8/25). 18 x 25 = 450. 458/25 = 18.32. the cost of the beef is 8(17/20). 20 x 8 = 160. 177/20 = 8.85. 18.32 – 8.85 = 9.47. Question 17. Be Precise Is 9.373 a repeating decimal? Is it rational? Explain your reasoning, Answer: Yes, it is a repeating decimal. Explanation: In the above-given question, given that, the number is 9.373 is a rational number. it is also a repeating decimal. Question 18. Reasoning 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? Answer: The tall will the stack be = 6.54 feet. Explanation: In the above-given question, given that, Aiden has one box that is 3$$\frac{3}{11}$$, feet tall and a second box that is 3.27 feet tall. 3(3/11). 33 + 3 = 36. 36/11 = 3.272727. 3.27 + 3.27 = 6.54, so the heght of the stack = 6.54 feets. Question 19. You 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? Answer: The decimal should I watch for on the digital pressure gauge = terminating decimal. Explanation: In the above-given question, given that, You are adding air to a tire. The air pressure in the tire should be $$32\frac{27}{200}$$ pounds per square inch. 32(27/200). 32 x 200 = 6400. 6400 + 27 = 6427. 6427/200 = 32.135. so the decimal should I watch for on the digital pressure gauge = terminating decimal. Question 20. Higher Order Thinking 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. Answer: The square box shown is big enough to fit the pizza inside. Explanation: In the above-given question, given that, Dion has a pizza with a diameter of 10$$\frac{1}{3}$$ in. 10(1/3). 30 + 1/3. 31/3 = 10.33. so the square box shown is big enough to fit the pizza inside. Assessment Practice Question 21. Which of the following mixed numbers has the same decimal value as $$110 \frac{147}{168}$$? A. $$110 \frac{49}{56}$$ B. $$110 \frac{170}{180}$$ C. $$110 \frac{56}{72}$$ D. $$110 \frac{247}{268}$$ Answer: Option A is the correct. Explanation: In the above-given question, given that, 110/ 147/168. 110/ 0.875. so option A is correct. Question 22. Select all the true statements about the negative fractions –$$\frac{4}{5}$$ and –$$\frac{5}{6}$$ –$$\frac{4}{5}$$ can be expressed as a repeating decimal. o –$$\frac{5}{6}$$ can be expressed as a repeating decimal. Both fractions can be expressed as repeating decimals. The digit that repeats is 3. The digit that repeats is 8. Answer: -5/6 can be expressed as a repeating decimal. the digit that repeats is 3. Explanation: In the above-given question, given that, the two numbers are -4/5 and -5/6. -4/5 and -5/6. -4/5 = -0.8. -5/6 = -0.8. so -5/6 can be expressed as a repeating decimal. ### Lesson 1.3 Add Integers Explore It! Rain 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. I can… add integers. A. Look for patterns in the equations in the table so you can fill in the missing numbers. Describe any relationships you notice. Answer: The missing numbers are 2, -1, -2. Explanation: In the above-given question, given that, the top of the pool is 4. the bottom of the pool is -4. we can add integers. so the missing numbers are 2, -1, and -2. B. Will the sum of 2 and (-6) be a positive or negative number? Explain. Answer: The sum of 2 and -6 be a negative number. Explanation: In the above-given question, given that, 2 + (-6). 2 – 6 = -4. the sum of two +ve numbers will always be a positive number. the sum of two -ve numbers will also be positive number. the sum of one +ve and one -ve is a negative number. so the sum of 2 and -6 be a negative number. Focus on math practices Look 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. Essential Question How do you use what you know about the absolute value to add integers? Try It! Dana recorded a temperature drop of 2o and a second temperature drop of 3°. What is the total change in temperature? _______ + ________ = _________ The sign of the sum is _______. The total change in temperature is ________°. Answer: The sign of the sum is negative. the total change in temperature is 5°. Explanation: In the above-given question, given that, Dana recorded a temperature drop of 2o and a second temperature drop of 3°. -2 – 3 = -5. so the sign of the sum is negative. the total change in temperature is 5°. Convince Me! Would the sum of two positive integers be positive or negative? Explain. Answer: The sum of two positive integers is positive. Explanation: In the above-given question, given that, the sum of two positive integers is positive. 2 + 3 = 5. so the sum is also a positive number. Try It! Find the sum for each expression. a. -66 + 42 Answer: -66 + 42 = -24. Explanation: In the above-given question, given that, the numbers are -66 and 42. add the numbers. the sum of one positive and one negative is also a negative integer. -66 + 42 = -24. b. -57 + 57 Answer: -57 + 57 = 0. Explanation: In the above-given question, given that, the numbers are -57 and 57. add the numbers. the sum of one positive and one negative is also a negative integer. -57 + 57 = 0. c. 29 + (-28) Answer: 29 + (-28) = 1. Explanation: In the above-given question, given that, the numbers are -28 and 29. add the numbers. the sum of one positive and one negative is also a negative integer. 29 + (-28) = 1. KEY CONCEPT When adding integers with the same sign, find the sum of the absolute values. (-36) + (-12) |-36| = 36 and |-12| = 12 36 + 12 = 48 So, (-36) + (-12) = -48 Use the same sign as the addends. When adding integers with different signs, find the difference of the absolute values. 18+ (-14) |18| = 18 and |-14| = 14 18 – 14 = 4 So, 18 + (-14) = 4 Use the sign of the greater absolute value. Do You Understand? Question 1. Essential Question How do you use what you know about absolute value to add integers? Answer: When adding the integers with the same sign, we will find the sum of absolute values. Explanation: In the above-given question, given that, When adding the integers with same sign, we will find the sum of absolute values. for example: (-36) + (-12) |-36| = 36 and |-12| = 12 36 + 12 = 48 So, (-36) + (-12) = -48. Question 2. Reasoning How can you tell the sign of the sum of a positive and negative integer without doing any calculations? Answer: The sum of positive and negative integers is always negative. Explanation: In the above-given question, given that, the sum of positive and negative integers is always negative. for example: -33 + 22. -11. so the sum of positive and negative integers is always negative. Question 3. Model with Math How would you use a number line to determine the sum of two negative integers? Answer: The sum of two negative integers is also a positive integer. Explanation: In the above-given question, given that, the sum of two negative integers is also a positive integer. for example: -33 + -22. -33 – 22. 55. Do You Know How? Question 4. Sarah 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. Answer: The amount Sarah paid for her bike =$205.

Explanation:
In the above-given question,
given that,
Sarah bought a bike that cost $260. She had a coupon that was worth$55 off the cost of any bike.
260 + (-55).
260 – 55.
205.
so the amount Sarah paid for her bike = $205. Question 5. A 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. Answer: The fish’s location is relative to the surface of the ocean = -85 feet. Explanation: In the above-given question, given that, A shark is swimming 60 feet below the surface of the ocean. There is a fish that is 25 feet deeper in the water. (-60) + (-25). -60 – 25 = -85. so the fish’s location relative to the surface of the ocean = -85 feet. Question 6. The high temperature one day was 30°F. Then the temperature dropped 23 degrees during the night. Does the expression 30 + (-23) represent the temperature at night? Explain. Answer: The expression 30 + (-23) represent the temperature at night = 7 degrees. Explanation: In the above-given question, given that, The high temperature one day was 30°F. Then the temperature dropped 23 degrees during the night. 30 + (-23). 30 – 23 = 7. so the expression represents the temperature at night = 7 degrees. Practice & Problem Solving Leveled Practice For 7-9, use the number lines to help find each sum. Question 7. 5 + (-3) is ________ units from 5, in the ________ direction. Answer: 5 + (-3) is ____2____ units from 5, in the _positive____ direction. Explanation: In the above-given question, given that, 5 + (-3). 5 – 3 = 2. 2 units from 5, in the positive direction. Question 8. -1 + (-3) is _______ units from -1, in the ________ direction. Answer: -1 + (-3) is __-4_____ units from -1, in the ___negative_____ direction Explanation: In the above-given question, given that, -1 + (-3). -1 – 3 = -4. -4 units from -1. in the negative direction. Question 9. In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. In City B, the temperature drops 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.M. to 10 A.M. a. What expression represents the change in temperature for City A? Answer: The expression represents the change in temperature for City A is 9° – 8° = 1°. Explanation: In the above-given question, given that, In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. 9° – 8° = 1°. so the expression represents the change in temperature for City A is 9° – 8° = 1°. b. What integer represents the change in temperature for City A? Answer: The positive integer represents the change in temperature for City A. Explanation: In the above-given question, given that, In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. 9° – 8° = 1°. so the positive integer represents the change in temperature for City A. c. What expression represents the change in temperature for City B? Answer: The expression represents the change in temperature for City B is 5° – 4° = 1°. Explanation: In the above-given question, given that, In City B, the temperature rises 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.m. to 10 A.M. 5° – 4° = 1°. so the expression represents the change in temperature for City B is 5° – 4° = 1°. d. What integer represents the change in temperature for City B? Answer: A positive integer represents the change in temperature for City B. Explanation: In the above-given question, given that, In City B, the temperature rises 5° from 8 A.M. to 9 A.M. Then the temperature drops 4° from 9 A.m. to 10 A.M. 5° – 4° = 1°. so the positive integer represents the change in temperature for City B. e. Which city has the greater change in temperature from 8 A.M. to 10 A.M.? Answer: Both the cities have an equal change in temperature. Explanation: In the above-given question, given that, In City A, the temperature rises 9° from 8 A.M. to 9 A.M. Then the temperature drops 8° from 9 A.m. to 10 A.M. 9° – 8° = 1°. 5° – 4° = 1°. so both the cities have an equal change in temperature. Question 10. An 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. a. What integer represents the airplane’s change in altitude to avoid the storm? Answer: The positive integer represents the airplane’s change in altitude to avoid the storm. Explanation: In the above-given question, given that, An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. the airplane flies up to 38,000 feet to avoid a storm. 38000 – 30000. 8000. so the positive integer represents the airplane’s change in altitude to avoid the storm. b. What integer represents the airplane’s change in altitude immediately after passing the storm? Answer: The positive integer represents the airplane’s change in altitude immediately after passing the storm. Explanation: In the above-given question, given that, An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. the airplane flies up to 38,000 feet to avoid a storm. 38000 – 30000. 8000. so the positive integer represents the airplane’s change in altitude immediately after passing the storm. c. Use Appropriate Tools Draw a number line to represent the airplane’s change in altitude. Answer: The change in altitude is 38000. Explanation: In the above-given question, given that, An airplane flying at an altitude of 30,000 feet flies up to avoid a storm. the airplane flies up to 38,000 feet to avoid a storm. 30000 + 8000 = 38000. Question 11. A 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? Answer: The diver’s present location is -95 feet below the sea. Explanation: In the above-given question, given that, A 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). -81 + (-14). -81 – 14 = -95. so the diver’s present location is -95 feet below the sea. Question 12. Rena’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? Answer: The integer represents the rowboat’s position as negative. Explanation: In the above-given question, given that, Rena’s rowboat drifts 23 feet from shore, followed by 9 more feet. The rowboat’s current position can be represented by –23 + (-9). -23 – 9 = -32. so the integer represents the rowboat’s position is negative. Question 13. Critique Reasoning 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. Answer: Yes, Adam was incorrect. Explanation: In the above-given question, given that, A submarine traveling 200 meters below the surface of the ocean increases its depth by 45 meters. 200 + 45 = 245. but Adam says that the new location of the submarine is -155 meters. so Adam was incorrect. Question 14. Kim has$45 to spend for a day at the zoo. She pays $17 for admission,$8 for lunch, and $4 for a snack. a. Model with Math Use integers to write an addition expression that represents the amount of money Kim has left. Answer: The amount of money Kim has left =$16.

Explanation:
In the above-given question,
given that,
Kim has $45 to spend for a day at the zoo. She pays$17 for admission, $8 for lunch, and$4 for a snack.
$45 + (-$17) + (-$8) + (-$4).
$45 –$17 – $8 –$4.
$45 –$29.
$16. so the amount of money Kim has left =$16.

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. Answer: No, she does not have enough money to buy the T-shirt. Explanation: In the above-given question, given that, Kim has$45 to spend for a day at the zoo.
She pays $17 for admission,$8 for lunch, and $4 for a snack. she has left$16.
so she does not have enough money to buy the T-shirt.

Question 15.
Higher Order Thinking 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? Answer: The better deal for Samantha is Mr. Beliz charges$62 per lesson.
$62 +$62 + $62 +$62 = $248. Explanation: In the above-given question, given that, 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.
$80 +$80 + $80 =$240.
$62 +$62 + $62 +$62 = $248. so the better deal for Samantha is Mr. Beliz charges$62 per lesson.

Assessment Practice

Question 16.
A 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.

Answer:
The addition expression that represents this situation is (-10) + (-10) = -20.

Explanation:
In the above-given question,
given that,
A fish swims at 10 ft below sea level and then swims another 10 ft deeper to avoid a shark.
(-10) + (-10) = -20.
below sea level = -10.
so the addition expression that represents this situation is (-10) + (-10) = -20.

Question 17.
The temperature drops 10 degrees and then rises 10 degrees. Write an addition expression that represents this situation.

Answer:
The addition expression that represents this situation is 10 + (-10) = 0.

Explanation:
In the above-given question,
given that,
The temperature drops 10 degrees and then rises 10 degrees.
10 + (-10).
10 – 10 = 0.
so the addition expression that represents this situation is 10 + (-10) = 0.

### Lesson 1.4 Subtract Integers

Solve & Discuss It!
A 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.
I can.. subtract integers.

Make Sense and Persevere
How can you use the data to understand what happened during the day?

Focus on math practices
Reasoning 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?

Answer:
The number of books checked out and checked in that day = 45 books.

Explanation:
In the above-given question,
given that,
0 + (-45).
0 – 45 = -45.
so the number of books checked out and checked in that day = -45.

Essential Question
How is subtracting integers related to adding integers?

Try It!

On the next play, the team gained 5 yards and then lost 6 yards. What is the total change in yards?

5 – ______
= 5 + ______
= _______
The total change in yards is ________, so they had a total loss of ________ yard.

Answer:
The total change in yards is 1.
so they had a total loss of 1 yard.

Explanation:
In the above-given question,
given that,
On the next play, the team gained 5 yards and then lost 6 yards.
5 – 6 = -1.
5 + (-6) = -1.

Convince Me! Is the additive inverse of an integer always negative? Explain.

Try It!

Subtract. Use a number line to help you find the answer.

a. -4 – 6

Answer:
-4 -6 = 10.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 + (-6).
-4 – 6 = -10.

b. -6 – (-4).

Answer:
-6 – (-4) = -2.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-6 – (-4).
-6 + 4.
-2.

c. 4 – (-6)

Answer:
4 – (-6) = 10.

Explanation:
In the above-given question,
given that,
the numbers are 4 and -6.
4 – (-6).
4 + 6.
10.

d. 6 – 4

Answer:
6 – 4 = 2.

Explanation:
In the above-given question,
given that,
the numbers are 6 and -4.
6 – 4 = 2.

e. 4 – 6

Answer:
4 – 6 = -2.

Explanation:
In the above-given question,
given that,
the numbers are 4 and -6.
4 – 6 = -2.

f. -4 – (-6)

Answer:
-4 + 6 = 2.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 – (-6).
-4 + 6 = 2.

KEY CONCEPT
When subtracting integers, such as a – b, you can use the additive inverse to write subtraction as an equivalent addition expression.

Do You Understand?
Question 1.
Essential Question How is subtracting integers related to adding integers?

Answer:
Subtracting integers related to additive inverse as an equivalent addition.

Explanation:
In the above-given question,
given that,
Subtracting integers related to additive inverse as an equivalent addition.
for example:
a + (-b) = a -b.

Question 2.
Reasoning Explain how to simplify the expression -98 – 31 using the additive inverse.

Answer:
-98 + (-31) = – 129.

Explanation:
In the above-given question,
given that,
the numbers are -98 and -31.
-98 + (-31).
-129.

Question 3.
Model with Math How can you use a number line to represent the subtraction between two integers?

Answer:
-5 – 2 = -7.

Explanation:
In the above-given question,
given that,
the two numbers on the number line = -5 and -2.
-5 + (-2).
-5 – 2 = -7.

Do You Know How?
Question 4.
It was 12°C when Preston got home from school. The weather report shows a storm front moving in that will drop the temperature by 17°C. What is the expected temperature?

Answer:
The expected temperature is -5 degrees c.

Explanation:
In the above-given question,
given that,
It was 12°C when Preston got home from school.
The weather report shows a storm front moving in that will drop the temperature by 17°C.
12 – 17 = -5.
so the expected temperature is -5°C.

Question 5.
Complete the equation.
-67 – _____ = 0

Answer:
-67 – (-67) = 0.

Explanation:
In the above-given question,
given that,
-67 – (-67).
-67 + 67 = 0.

Question 6.
Find the difference.
a. 41 – 275

Answer:
41 – 275 = -234.

Explanation:
In the above-given question,
given that,
the two numbers are 41 and 275.
subtract the numbers.
41 – 275.
– 234.

b. -15 – 47

Answer:
-15 – 47 = -32.

Explanation:
In the above-given question,
given that,
the two numbers are -15 and -47.
subtract the numbers
-15 – 47.
47 – 15 = -32.

c. -72 – (-151)

Answer:
-72 – (-151) = 79.

Explanation:
In the above-given question,
given that,
the two numbers are -72 and -151.
subtract the numbers.
-72 – (-151).
-72 + 151.
79.

d. 612 – (-144)

Answer:
612 – (-144) = 756.

Explanation:
In the above-given question,
given that,
the two numbers are 612 and -144.
subtract the numbers.
612 – (-144).
612 + 144 = 756.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to solve.
Question 7.
What subtraction expression does the number line model show?

Answer:
-2 + (-9) = 7.

Explanation:
In the above-given question,
given that,
the lines are moving from -2 and -9.
-2 + (-9).
-2 – 9 = -11.

Question 8.
What is the value of the expression -9 – (-5)?
-9 – (-5)
= -9 ______ 5
= _______

Answer:
The value of the expression -9 – (-5) = -4.

Explanation:
In the above-given question,
given that,
the two numbers are -9 and -5.
subtract the numbers.
-9 – (-5) = -4.

Question 9.+e
The temperature at the beginning of the day was 6°F. The temperature dropped 9°F by the end of the day. Use the number line to find the temperature at the end of the day.

Answer:
The temperature at the end of the day = -3.

Explanation:
In the above-given question,
given that,
The temperature at the beginning of the day was 6°F.
The temperature dropped 9°F by the end of the day.
6 + (- 9) = -3.
so the temperature at the end of the day = -3.

Question 10.
Murphy 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 – 1.
a. Who is correct?

Answer:
Murphy was correct.

Explanation:
In the above-given question,
given that,
Murphy and Naryam do their math homework together.
When they find 9- (-8), they get different answers.
9 – (-8).
9 + 8 = 17.
so Murphy was correct.

b. What error likely led to the incorrect answer?

Answer:
The sign likely led to the incorrect answer.

Explanation:
In the above-given question,
given that,
Murphy and Naryam do their math homework together.
When they find 9- (-8), they get different answers.
9 – (-8).
9 + 8 = 17.

Question 11.
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature. Yesterday’s high temperature was -2°F.
a. Write an expression to represent today’s high temperature.

Answer:
The expression is -2°F + 16°F = 14°F.

Explanation:
In the above-given question,
given that,
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature.
Yesterday’s high temperature was -2°F.
so the expression is -2°F + 16°F = 14°F.

b. Reasoning is today’s high temperature positive or negative? Why?

Answer:
Today’s high temperature is positive.

Explanation:
In the above-given question,
given that,
The news reports that today’s high temperature is 16°F colder than yesterday’s high temperature.
Yesterday’s high temperature was -2°F.
so the expression is -2°F + 16°F = 14°F.

Question 12.
Max sprints forward 10 feet and then stops and sprints back 15 feet. Use subtraction to explain where Max is relative to where he started.

Answer:
He started at -5.

Explanation:
In the above-given question,
given that,
Max sprints forward 10 feet and then stops and sprints back 15 feet.
10 + (-15).
10 – 15.
-5.
so he started at -5.

Question 13.
Higher Order Thinking Use the number line at the right.

a. What subtraction equation does the number line represent?

Answer:
The subtraction equation is -2 + (-6) = -8.

Explanation:
In the above-given question,
given that,
the starting point is -2.
he then increases by -6.
-2 + (-6).
-2 -6 = -8.

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.

Answer:
0 – 2 = -2.
-4 – (- 2) = -2.

Explanation:
In the above-given question,
given that,
the subtraction equations are:
0 – 2 = -2.
-4 – (-2).
-4 + 2 = -2.

Question 14.
A 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?

Answer:
The final position of the pallet relative to where it started in the back of the truck = 5 feet.

Explanation:
In the above-given question,
given that,
A 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:
13 – 8 = 5.
so the final position of the pallet relative to where it started in the back of the truck = 5 feet.

Question 15.
Make 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.
a. Write an expression using integers to represent the difference between the elevations.

Answer:
The expression using integers to represent the difference between the elevations is 5753.

Explanation:
In the above-given question,
given that,
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.
5762 + (-9) = 5753.
so the expression using integers to represent the difference between the elevations is 5753.

b. Will the answer be written as a positive or negative integer?

Answer:
The answer will be written as a positive number.

Explanation:
In the above-given question,
given that,
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.
5762 + (-9) = 5753.
so the answer will be written as a positive number.

c. What is the difference between the highest and lowest points of the county?

Answer:
The difference between the highest and lowest points of the county is 5753.

Explanation:
In the above-given question,
given that,
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.
5762 + (-9) = 5753.
so the difference between the highest and lowest points of the county is 5753.

Assessment Practice

Question 16.
Which number line model shows the subtraction 2 – 4?

Answer:
Option D is the correct answer.

Explanation:
In the above-given question,
given that,
the number line from 0 to -2 and -2 to -6.
-2 + (-6).
-2 -6 = -8.
so option D is the correct answer.

### Lesson 1.5 Add and Subtract Rational Numbers

Solve & Discuss It!
Malik 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?

I can… add and subtract rational numbers.

Look for Relationships
How are elevation values of point A and point B related?

Focus on math practices
Reasoning What would be different about the hike from point B to point A?

Answer:
The difference between the hike from point B to point A is 120.5 meters.

Explanation:
In the above-given question,
given that,
Malik hikes Castle Trail from point A to point B.
The elevation at point A is below sea level.
120(1/2).
120 x 2 = 240.
240 + 1 = 241.
241/2 = 120.5.
so the difference between the hike from point B to point A is 120.5 meters.

Essential Question
How are adding and subtracting integers related to adding and subtracting other rational numbers?

Try It!

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?
-4$$\frac{1}{2}$$ – ________
-4$$\frac{1}{2}$$ + _______ = __________

The location of the dolphin relative to the surface of the water is ________ feet.

Answer:
The location of the dolphin relative to the surface of the water is -7.25 feet.

Explanation:
In the above-given question,
given that,
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.
-4(1/2) – 2(3/4).
-9/2 – 11/4.
-4.5 – 2.75.
-7.25.
so the location of the dolphin relative to the surface of the water is -7.25 feet.

Convince Me! How are adding and subtracting two rational numbers with different signs related to adding and subtracting two integers with different signs?

Try It!

Find the sum or difference of the rational numbers.
a. $$-2.5+\left(-5 \frac{6}{10}\right)$$

Answer:
-2.5 + (-5 x 6/10) = -8.1.

Explanation:
In the above-given question,
given that,
the sum of two numbers.
-2.5 – 5 x 6/10.
-2.5 – 56/10.
-2.5 – 5.6.
-8.1.

b. $$-4.4-\left(-1 \frac{1}{2}\right)$$

Answer:
-5.9.

Explanation:
In the above-given question,
given that,
the difference between two numbers.
-4.4 – 3/2.
-4 . 4 – 1.5.
-5.9.

c. $$-135.4+78 \frac{1}{2}$$

Answer:
-56.9.

Explanation:
In the above-given question,
given that,
-135.4 + 78(1/2).
-135.4 + 157/2.
-135.4 + 78.5.
-56.9.

Try It!

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?

Answer:
The much farther below sea level is the diver who is farthest below sea level = -15.25 feet.

Explanation:
In the above-given question,
given that,
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.
-25.5 – 40.75.
– 15.25.
so the much farther below sea level is the diver who is farthest below sea level = -15.25 feet.

KEY CONCEPT
The rules for adding and subtracting all rational numbers are the same as those for adding and subtracting integers.
The distance between any two rational numbers p and q on a number line is the absolute value of their difference.

Do You Understand?
Question 1.
Essential Question How are adding and subtracting integers related to adding and subtracting other rational numbers?

Answer:
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.

Explanation:
In the above-given question,
given that,
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
for example:
-5 – (-2).
-2 – (-5).
-2 + 5 = 3.

Question 2.
Reasoning When finding the distance between two rational numbers on a number line, does the order of the numbers you subtract matter? Explain.

Answer:
The distance between two rational numbers on a number line does the order of the numbers you subtract matter.

Explanation:
In the above-given question,
given that,
The integers are related to adding and subtracting other rational numbers is the absolute value of their difference.
for example:
-5 – (-2).
-2 – (-5).
-2 + 5 = 3.

Question 3.
Critique Reasoning 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.

Answer:
Gwen was correct.

Explanation:
In the above-given question,
given that,
Gwen says that the sum of – 1$$\frac{3}{4}$$ and 2$$\frac{1}{2}$$.
-1(3/4) and 2(1/2).
-4 + 3 = -7/4.
4 + 1 = 5/2.
-7/4 = -1.75.
5/2 = 2.5.
1(3/4) = 7/4.
7/4 = 1.75.

Do You Know How?
Question 4.
What is the distance between the top of the fishing pole and the fish?

Answer:
The distance between the top of the fishing pole and the fish = 4.

Explanation:
In the above-given question,
given that,
the top of the pole = 8(1/2).
the distance of the fish = 4(1/2).
8(1/2) – 4(1/2).
8 x 2 = 16.
16 + 1 = 17/2.
4 x 2 = 8.
8 + 1 = 9.
9/2 = 4.5.
17/2 = 8.5.
8.5 – 4.5 = 4.

Question 5.
A 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?

Answer:
The shark’s location now in relation to sea level = 35.4 meters.

Explanation:
In the above-given question,
given that,
A shark began at 172.5 meters below sea level and then swam up 137.1 meters.
172.5 – 137.1.
35.4.
so the shark’s location now in relation to sea level = 35.4 meters.

Question 6.
Find the sum or difference.
a. -12$$\frac{1}{2}$$ + 4$$\frac{1}{2}$$

Answer:
-8.

Explanation:
In the above-given question,
given that,
the two numbers.
add the numbers.
-12(1/2) + 4(1/2).
-25/2 + 9/2.
-12.5 + 4.5.
-8.

b. -0.35 – (-0.25)

Answer:
-0.10.

Explanation:
In the above-given question,
given that,
the two numbers.
subtract the numbers.
-0.35 – (-0.25).
-0.35 + 0.25.
-0.10.

Practice & Problem Solving

Leveled Practice In 7-8, complete the expressions to find the sum or difference.
Question 7.
3.2 – (-5.7)
= 3.2 + _____
= ______

Answer:
3.2 – (-5.7) = 8.9.

Explanation:
In the above-given question,
given that,
the two numbers.
subtract the numbers.
3.2 – (-5.7).
3.2 + 5.7.
8.9.

Question 8.
$$\frac{12}{13}+\left(\frac{-1}{13}\right)$$
= _____ – ______
= ______

Answer:
12/13 + (-1/13) = 0.85.

Explanation:
In the above-given question,
given that,
the two numbers.
add the numbers.
12/13 = 0.92.
1/13 = 0.07.
0.92 – 0.07 = 0.85.

Question 9.
Reasoning When Tom simplified the expression -2.6 + (-5.4), he got 2.8. What mistake did Tom likely make?

Answer:
The mistake did Tom likely make = -8.

Explanation:
In the above-given question,
given that,
-2.6 + (-5.4).
-2.6 – 5.4.
– 8.

Question 10.
The temperature in a town is 36.6°F during the day and -12.6°F at night. What is the temperature change from day to night?

Answer:
The temperature change from day to night = 24°F.

Explanation:
In the above-given question,
given that,
The temperature in a town is 36.6°F during the day and -12.6°F at night.
36.6 + (-12.6).
36.6 – 12.6.
24.
so the temperature change from day to night = 24°F.

Question 11.
Simplify each expression.
a. 50$$\frac{1}{2}$$ + (-12.3)

Answer:
50(1/2) + (-12.3) = 38.2.

Explanation:
In the above-given question,
given that,
the two numbers are 50(1/2) and -12.3.
add the numbers.
101/2 and -12.3.
50.5 + (-12.3).
50.5 – 12.3 = 38.2.

b. -50$$\frac{1}{2}$$ + (-12.3)

Answer:
-50(1/2) + (-12.3) = -62.8.

Explanation:
In the above-given question,
given that,
the two numbers are -50(1/2) and -12.3.
add the numbers.
-101/2 and -12.3.
-50.5 – 12.3.
– 62.8.

C. –50$$\frac{1}{2}$$ + 12.3

Answer:
-50(1/2) + 12.3 = -32.8.

Explanation:
In the above-given question,
given that,
the two numbers are -50(1/2) and 12.3.
add the numbers.
-101/2 + 12.3.
-50.5 + 12.3.
-38.2.

Question 12.
At 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?

Answer:
The total change in the stock market from the beginning of the day to the end of the day = 89.75 points.

Explanation:
In the above-given question,
given that,
At 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.
30(1/2) = 61/2.
61/2 = 30.5.
120(1/4) = 481/4.
481/4 = 120.25.
120.25 – 30.5 = 89.75.
so the total change in the stock market from the beginning of the day to the end of the day = 89.75 points.

Question 13.
A 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?

Answer:
The distance between the dolphin and the helicopter = 57.5 feet.

Explanation:
In the above-given question,
given that,
A 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.
75.5 – 18 = 57.5 feet.
so the distance between the dolphin and the helicopter = 57.5 feet.

Question 14.
A bird flies from its nest to the bottom of the canyon. How far did the bird fly?

Answer:
The far did the birds fly = 438.6.

Explanation:
In the above-given question,
given that,
A bird flies from its nest to the bottom of the canyon.
the top of the nest is 528(1/5).
528 x 1/5.
528 x 5 = 2640.
2641/5 = 528.2.
the height of canyon floor = -89(3/5).
-89 x 5 = 445.
-448 / 5 = -89.6.
528.2 – 89.6 = 438.6.

Question 15.
A 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?

Answer:
The fish in relation to the surface = 32.58 feet.

Explanation:
In the above-given question,
given that,
A 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.
54.96 – 22.38 = 32.58.
so the fish in relation to the surface = 32.58 feet.

Question 16.
Model with Math Write an addition expression that is represented by the number line.

Answer:
The addition expression that is represented by the number line = 0.5.

Explanation:
In the above-given question,
given that,
0 + 1 = 1.
0 + 0.5 = 0.5.
1 – 0.5 = 0.5.
so the addition expression that is represented by the number line is 1 + (-0.5).

Question 17.
The roots of a plant reach down 3$$\frac{3}{4}$$ inches below ground. How many inches is the plant above the ground?

Answer:
The number of inches is the plant above the ground = 8.75 inches.

Explanation:
In the above-given question,
given that,
The roots of a plant reach down 3$$\frac{3}{4}$$ inches below ground.
3(3/4) = 15/4.
15/4 = 3.75.
12(1/2) = 25/2.
25/2 = 12.5
12.5 – 3.75 = 8.75.
so the number of inches is the plant above the ground = 8.75 inches.

Question 18.
Higher Order Thinking
a. Simplify the expression (-13.2) + 8.1.

Answer:
-13.2 + 8.1 = -5.1.

Explanation:
In the above-given question,
given that,
the two numbers are -13.2 and 8.1.
add the numbers.
-13.2 + 8.1.
– 5.1.
-13.2 + 8.1 = -5.1.

b. How are (-13.2) + 8.1 and 13.2 + (-8.1) related? Explain without computing.

Answer:
-13.2 + 8.1 = -5.1.
13.2 – 8.1 = 5.1.

Explanation:
In the above-given question,
given that,
the two numbers are 13.2 and 8.1.
add the numbers.
they have an equal value but the sign is different.

c. Using a property of operations, what can you say about the sum of the two expressions?

Answer:
The sum of the two expressions is also positive.

Assessment Practice

Question 19.
The temperatures at sunrise and sunset are shown in the table.

PART A
Write 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.

Answer:
The properties of operations to find the value of the expression is -11.31 + 13.49.

Explanation:
In the above-given question,
given that,
the temperatures at sunrise and sunset are shown.
sunrise on day 1 is -11.31 and day 2 is -7.69.
sunset on day 1 is 13.49 and day 2 is 25.25.
-11.31 + (13.49).
-11.31 + 13.49.
2.18.
so the properties of operations to find the value of the expression is -11.31 + 13.49 is 2.18.

PART B
On which day did the temperature change more? Explain your reasoning.

Answer:
On day 2 the temperature change more.

Explanation:
In the above-given question,
given that,
the temperatures at sunrise and sunset are shown.
sunrise on day 1 is -11.31 and day 2 is -7.69.
sunset on day 1 is 13.49 and day 2 is 25.25.
-11.31 + (13.49).
-11.31 + 13.49.
2.18.
-7.69 + 25.25.
17.56.
so on day 2 the temperature change more.

Question 20.
Mischa 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.
☐ |5 – (-2.1)|
☐ |-(2.1) – (-5)|
☐ |2.1 – 5|
☐ |-(2.1) – 5|
☐ |5 + (-2.1)|

Answer:
The expression could represent the distance in meters is 5 – (-2.1).

Explanation:
In the above-given question,
given that,
Mischa dives from a platform that is 5 meters above the water.
Her dive takes her 2.1 meters below the surface of the water.
5 – (-2.1).
5 + 2.1 = 7.1.
so the expression could represent the distance in meters is 7.1.

Topic 1 Mid-Topic Checkpoint

Question 1.
Vocabulary How do you find the additive inverse of a number? Give an example of a number and its additive inverse. Lesson 1-3

Answer:
The additive inverse of a number is always negative.

Explanation:
In the above-given question,
given that,
the numbers are -4 and -6.
-4 + (-6).
-4 – 6 = -10.

Question 2.
A 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

Answer:
The positive integer represents the toy submarine’s position with respect to the surface of the water.

Explanation:
In the above-given question,
given that,
A plastic toy submarine is held 15 centimeters below the water surface in a bathtub.
The submarine is let go and rises 15 centimeters.
-15 + (15).
-15 + 15 = 0.
so the integer is zero.

Question 3.
The temperature in the late afternoon was -7.5°C. 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

Answer:
The temperature at the midnight = -21°C.

Explanation:
In the above-given question,
given that,
The temperature in the late afternoon was -7.5°C.
It dropped 5 degrees by early evening and then dropped another 8.5 degrees by midnight.
-7.5 – 5 – 8.5.
-7.5 – 13.5.
-21.
so the temperature at the midnight is -21°C.

Question 4.
The 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

Answer:
The distance has the elevator’s floor traveled = 20 feet.

Explanation:
In the above-given question,
given that,
The 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.
30 + (-10).
30 – 10.
20.
so the distance has the elevator’s floor traveled = 20 feet.

Question 5.
Greg 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

Answer:
Yes. both of them are correct.

Explanation:
In the above-given question,
given that,
Greg says that 3.3. is a rational number.
3.3 is a rational number.
Kari says 3.3 is not a terminating decimal.
3.3 is also a terminating decimal.
so both of them are correct.

Question 6.
Cece 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
A. 5.425 feet
B. 25.825 feet
C. 25$$\frac{3}{8}$$ feet
D. 5$$\frac{1}{4}$$ feet

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
Cece 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.
15(5/8) – 10.2.
125/8 – 10.2.
15.625 – 10.2.
5.425.
so option A is correct.

Topic 1 Mid-Topic Performance Task

An oceanographer, Dr. Price, is studying the types of sea life at various depths.

PART A

Dr. Price uses a table to organize the types of sea life and the positions relative to sea level of each location.

Complete each sentence.
The difference between Location A and Location B is ________ meters.
The difference between Location B and Location C is ________ meters.
The difference between Location C and Location D is ________ meters.

Answer:
The difference between Location A and Location B is 202.25 meters.
The difference between Location B and Location C is 1686.6 meters.
The difference between Location C and Location D is474.875 meters.

Explanation:
In the above-given question,
given that,
location A = -895.9.
location B = – 1098(3/20).
location C = -2784.75.
location D = – 3259(5/8).
-895.9 – (-1098(3/20).
-895.9 + 21963/20.
-895.9 + 1098.15.
202.25.
-1098.15 – (-2784.75).
-1098.15 + 2784.75.
1686.6.
-2784.75 + 3259.625.
474.875.

PART B
After observing Location B, Dr. Price returns to Location A before descending to Location C. What is the total distance she travels?

Answer:
The total distance she travels = -2987 meters.

Explanation:
In the above-given question,
given that,
location B = -1098.15.
location A = – 895.9.
location C = -2784.75.
-1098.15 + (-895.9) + (- 2784.75).
-1098.15 – 895.9 – 2784.75.
-1098.15 – 1888.85.
-2987.
so the total distance she travels = -2987 meters.

PART C
Dr. 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?

Answer:
The distance between Location D and where Dr. Price stopped to observe = -272.625 meters.

Explanation:
In the above-given question,
given that,
Dr. 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.
-3259.625 – (-2987).
-3259.625 + 2987.
-272.625.
so the distance between Location D and where Dr. Price stopped to observe = -272.625 meters.

### Lesson 1.6 Multiply Integers

Explore It!
A popular beach erodes 4 inches per year on average.
I can… multiply integers.

A. How many years will it take for the coastline to erode one foot?

Answer:
The number of years will it take for the coastline to erode one foot = 12 years.

Explanation:
In the above-given question,
given that,
A popular beach erodes 4 inches per year on average.
1 year = 4 inches.
one feet = 12 inches.
so the number of years will it take for the coastline to erode one foot = 12 years.

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?

Answer:
The erosion after 10 years = 10.

Explanation:
In the above-given question,
given that,
The number line below shows the expected change in the coastline as the years pass.
coastline this year = 0.
0 + 10 = 10.
so the erosion after 10 years = 10.

Focus on math practices
Be Precise What expression could you use to represent the change in the coastline in 5 years?

Essential Question
How do the signs of factors affect their product?

Try It!

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?
10 • ______ = ______
The change in total points is ______.

Answer:
If the player hits 10 cones = 60 points.

Explanation:
In the above-given question,
given that,
A race car game takes 6 points from a player each time the player hits a cone.
10 x 6 = 60.
so the change in total points is 60 points.

Convince Me! Could the product of a positive integer and a negative integer be positive? Explain.

Try It!

Find each product.
a. -7 • (-2)

Answer:
The product is 14.

Explanation:
In the above-given question,
given that,
the two numbers are -7 and -2.
multiply the numbers.
-7 x -2.
14.
so the product is 14.

b. 7 • (-13)

Answer:
The product is -91.

Explanation:
In the above-given question,
given that,
the two numbers are 7 and -13.
multiply the numbers.
7 x -13.
-91.
so the product is -91.

c.-6 • 8

Answer:
The product is -48.

Explanation:
In the above-given question,
given that,
the two numbers are -6 and 8.
multiply the numbers.
-6 x 8.
-48.
so the product is -48.

d.(-1) • (-1)

Answer:
The product is 1.

Explanation:
In the above-given question,
given that,
the two numbers are -1 and -1.
multiply the numbers.
-1 x -1.
1.
so the product is 1.

KEY CONCEPT
When multiplying two integers, the sign of the product depends on the sign of the factors.
If the signs of the factors are the same, the product is positive.
7 • 3 = 21
– 7 • (-3) = 21

If the signs of the factors are different, the product is negative.
-4 • 5 = -20
4 • (-5) = -20

Do You Understand?
Question 1.
Essential Question How do the signs of factors affect their product?

Answer:
The signs of factors affect their product according to the sign.

Explanation:
In the above-given question,
given that.
If the signs of the factors are the same, the product is positive.
7 x 3 = 21
– 7 x (-3) = 21
If the signs of the factors are different, the product is negative.
-4 x 5 = -20
4 x (-5) = -20

Question 2.
Construct Arguments What is the sign of the product if you multiplied three negative integers? Explain your answer.

Answer:
The product is also a negative integer.

Explanation:
In the above-given question,
given that.
the numbers are -7, -8, and -3.
-7 x -8 x -3.
-7 -24.
-168.
so the product is also a negative integer.

Question 3.
Reasoning Explain why the product of two negative integers is not negative. Use (-1)(-1) as an example.

Answer:
yes, the product of two negative integers is positive.

Explanation:
In the above-given question,
given that,
the product of two negative integers is positive.
for example:
-1 x -1 = 1.

Question 4.
Use Structure is the product the same when multiplying 22 × (-5) and multiplying (-5) × 22? Explain.

Answer:
Yes, the product is the same.

Explanation:
In the above-given question,
given that,
the numbers are 22 and -5.
22 x -5 = 110.
the numbers are -5 and 22.
-5 x 22 = 110.
so both the product is the same.

Do You Know How?
Question 5.
Represent 2 • (-3) on the number line.

Answer:
The number is -6.

Explanation:
In the above-given question,
given that,
the numbers are 2 and -3.
2 x -3 = -6.
so the number is -6.

Question 6.
Which of these products is negative? Select all that apply.
☐ -8 • (-3)
☐ -2 • 8
☐ 0 • (-2)
☐ 15 • (-5)
☐ -8 • (-9)

Answer:
-2 x 8 = -16.
15 x -5 = -75.

Explanation:
In the above-given question,
given that,
the product is negative.
the numbers are – 2 and 8.
-2 x 8 = -16.
15 x -5 = -75.

Question 7.
Find each product.
a. -9 • (-4)
b.-7 • 12
c. 8 • (-8)
d. 9 • 15

Answer:
The products are 36, -84, -64, and 135.

Explanation:
In the above-given question,
given that,
-9 x -4 = 36.
12 x -7 = -84.
8 x -8 = 64.
15 x 9 = 135.
so the products are 36, -84, -64, and 135.

Question 8.
A 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. Answer: The numbers are$20 x $30. Explanation: In the above-given question, given that, A game show contestant starts a game by answering two questions incorrectly. Each incorrect answer costs the contestant$600.
the product of two integers is $600.$20 x $30. so the numbers are$20 and $30. Practice & Problem Solving In 9-14, multiply. Question 9. (-6) • (-2) Answer: The product is 12. Explanation: In the above-given question, given that, the numbers are -6 and -2. -6 x -2. 12. so the product is 12. Question 10. 4 • (-8) Answer: The product is -32. Explanation: In the above-given question, given that, the numbers are -8 and 4. -8 x 4. -32. so the product is -32. Question 11. 7 • (-5) Answer: The product is -35. Explanation: In the above-given question, given that, the numbers are -5 and -7. -5 x 7. -35. so the product is -35. Question 12. -5 • 2 Answer: The product is -10. Explanation: In the above-given question, given that, the numbers are -5 and 2. -5 x 2. -10. so the product is -10. Question 13. -1 • (-24) Answer: The product is 24. Explanation: In the above-given question, given that, the numbers are -1 and -24. -1 x -24. 24. so the product is 24. Question 14. (5) • (-9) • (-2) Answer: The product is 90. Explanation: In the above-given question, given that, the numbers are 5, -9, and -2. -9 x -2. 18. 18 x 5 = 90. so the product is 90. Question 15. A 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? Answer: The total change in yards from where the team started = 18 yards. Explanation: In the above-given question, given that, A football team lost the same number of yards on each of 3 consecutive plays. 6 x 3 = 18. so the total change in yards from where the team started = 18 yards. Question 16. a. Find the product. -41 • (-1) Answer: The product is 41. Explanation: In the above-given question, given that, the numbers are -41 and -1. multiply the numbers. -41 x -1 = 41. so the product is 41. b. Construct Arguments Describe how you use the properties of multiplication to find the product. Answer: The properties of multiplication depend on the sign. Explanation: In the above-given question, given that, If the signs of the factors are the same, the product is positive. 7 x 3 = 21 – 7 x (-3) = 21 If the signs of the factors are different, the product is negative. -4 x 5 = -20 4 x (-5) = -20 Question 17. Alex is working to simplify 5 • (-8) • 2. a. What is the product? Answer: The product is -80. Explanation: In the above-given question, given that. the numbers are 5, 2, and -8. multiply the numbers. 5 x -8 x 2 = -80. so the product is -80. b. Suppose Alex found the opposite of the correct product. Describe an error he could have made that resulted in that product. Answer: The opposite of the correct product is 80. Explanation: In the above-given question, given that, the numbers are 5, 2, and -8. multiply the numbers. 5 x -8 x 2 = -80. so the product is -80. Question 18. Which product is greater, (-4) • (-6) or (-7) • (-8)? Explain. Answer: -7 x -8 = 56. Explanation: In the above-given question, given that, the product of two numbers. -4 x -6 = 24. -7 x -8 = 56. so 56 is the greatest product. Question 19. Make 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? Answer: The integer represents Cecilia’s movement on the board for those 9 turns = 54 times. Explanation: In the above-given question, given that, While playing a board game, Cecilia had to move back 6 spaces 9 times. 6 x 9 = 54. so the integer represents Cecilia’s movement on the board for those 9 turns = 54 times. Question 20. Anya makes withdrawals from and deposits into her bank account. a. What integer represents the change in the amount in her account if Anya withdraws$12 once each day for four days?

Answer:
The integer represents the change in the amount in her account = $48. Explanation: In the above-given question, given that, Anya makes withdrawals and deposits into her bank account.$12 x 4 = $48. so the integer represents the change in the amount in her account =$48.

b. What integer represents the change in the amount in her account if Anya deposits $12 once each day for four days? Answer: The integer represents the change in the amount in her account =$48.

Explanation:
In the above-given question,
given that,
Anya makes withdrawals and deposits into her bank account.
$12 x 4 =$48.
so the integer represents the change in the amount in her account = $48. c. Look for Relationships Explain the difference between the integer for the withdrawals and the integer for the deposits. Answer: The difference is zero. Explanation: In the above-given question, given that, Anya makes withdrawals and deposits into her bank account.$12 x 4 = $48. so the integer represents the change in the amount in her account =$48.
$48 –$48 = 0.

Question 21.
Higher Order Thinking 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?

Answer:
At 14 seconds the elevator of the miners is 210 feet.

Explanation:
In the above-given question,
given that,
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.
14 x 4 = 56.
14 x 15 = 210.
so the elevator of the miners is 210 feet.

Assessment Practice

Question 22.
A number line is shown.

Write a multiplication equation that is represented by the number line.

Answer:
The equation is -35 x 2 = -70.

Explanation:
In the above-given question,
given that,
the number line is moving from 0 to -35.
-35 to -70.
-35 x 2 = -70.
so the equation is -35 x 2 = -70.

Question 23.
Which of these expressions have the same product as (-6) • 7? Select all that apply.
☐ (-3) • 14
☐ 16 • (-3)
☐ -6 • (-7)
☐ 7 • (-6)
☐ 14 • (-3)

Answer:
-3 x 14, 7 x -6 and 14 x -3.

Explanation:
In the above-given question,
given that,
the product is -6 x 7 = -42.
-3 x 14 = -42.
7 x -6 = -42.
14 x -3 = -42.

### Lesson 1.7 Multiply Rational Numbers

Solve & Discuss It!
Stella 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.

I can… multiply rational numbers.

Focus on math practices
Be 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?

Answer:
The total area is the blue region of the flag = 0.2152.

Explanation:
In the above-given question,
given that,
The blue region of the flag is $$\frac{7}{13}$$ the width.
$$\frac{2}{5}$$ the length of the flag.
7/13 x 2/5.
0.538 x 0.4.
0.2152.
so the total area is the blue region of the flag = 0.2152.

Essential Question
How is multiplying rational numbers like multiplying integers?

Try It!

Meghan’s bank account is charged $9.95 per month for an online newspaper subscription. How could you represent the change in her account balance after three months of charges? _____ • -9.95 = ______ After three months, the change in her account balance is$ ________.

Answer:
The change in her account balance after three months of charges = -20.05.

Explanation:
In the above-given question,
given that,
Meghan’s bank account is charged $9.95 per month for an online newspaper subscription.$9.95 x -30 = -20.05.
so the change in her account balance after three months of charges = -20.05.

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.

Try It!

Find each product.
a. -5.3 • (-2.6)

Answer:
The product is 13.78.

Explanation:
In the above-given question,
given that,
the numbers are -5.3 and -2.6.
multiply the numbers.
-5.3 x – 2.6.
13.78.

b. $$-\frac{3}{5} \cdot 4 \frac{1}{6}$$

Answer:
-3/5 x 4(1/6) = -2.46.

Explanation:
In the above-given question,
given that,
the numbers are -3/5 and 25/6.
-0.6 x 4.1.
-2.46.
the product is -2.46.

c. 0.2 • (-1.78)

Answer:
0.2 x – 1.78 = -0.356.

Explanation:
In the above-given question,
given that,
the numbers are 0.2 x -1.78.
multiply the numbers.
0.2 x -1.78 = -0.356.

d. -2.5 • (70)

Answer:
The product is -175.

Explanation:
In the above-given question,
given that,
the numbers are 70 x -2.5.
multiply the numbers.
-2.5 x 70 = -175.

KEY CONCEPT
The same rules for multiplying integers apply to multiplying all rational numbers. When multiplying two rational 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.

Do You Understand?
Question 1.
Essential Question How is multiplying rational numbers like multiplying integers?

Answer:
The same rules for multiplying integers apply to multiplying all rational numbers.

Explanation:
In the above-given question,
given that,
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.

Question 2.
How do you multiply a decimal greater than 0 and a fraction less than 0?

Answer:
The product is negative.

Explanation:
In the above-given question,
given that,
the decimal greater than 0 is 1.5.
a fraction less than 0 is – 2.2.
1.5 x – 2.2 = -3.3.
so the product is negative.

Question 3.
Model with Math How does this number line represent multiplication of a negative number by a positive number? Explain.

Answer:
-2/3 x -1/3 = 0.108.

Explanation:
In the above-given question,
given that,
the two numbers are -2/3 and -1/3.
multiply the numbers.
-2/3 x -1/3.
-0.6 x – 0.33.
0.108.
-2/3 x -1/3 = 0.108.

Do You Know How?
Question 4.
Use the number line to find the product

Answer:
The product is -4.5.

Explanation:
In the above-given question,
given that,
the numbers are 3 and -1(1/2).
multiply the numbers.
3 x -3/2.
3 x – 1.5.
-4.5.
so the product is -4.5.

Question 5.
Which of these products is positive? Select all that apply.
☐ -0.2 • (12.5)
☐ $$\frac{1}{12}$$ • $$\left(-6 \frac{1}{2}\right)$$
☐ 3.2 • $$\left(-\frac{1}{900}\right)$$
☐ -3$$\frac{1}{2}$$ • 0
☐ -4.7 • (-1)

Answer:
-4.7 x -1 = 4.7.

Explanation:
In the above-given question,
given that,
the numbers are -4.7 and -1.
multiply the product.
-4.7 x -1 = 4.7.
so the product is 4.7.

Question 6.
Find the product.
a. -3.1 • (-2.9)

Answer:
The product is 8.99.

Explanation:
In the above-given question,
given that,
the numbers are -3.1 and -2.9.
multiply the product.
-3.1 x -2.9 = 8.99.
so the product is 8.99.

b. 1$$\frac{1}{2}$$ • $$\left(-\frac{5}{3}\right)$$

Answer:
1(1/2) x -5/3 = -2.4.

Explanation:
In the above-given question,
given that,
the numbers are 3/2 and -5/3.
multiply the product.
3/2 x – 5/3.
1.5 x – 1.6.
-2.4.
so the product is -2.4.

c. -3$$\frac{1}{2}$$ • 0.5

Answer:
-3(1/2) x 0.5 = – 1.75.

Explanation:
In the above-given question,
given that,
the numbers are -7/2 and 0.5.
multiply the product.
-7/2 x 0.5.
-3.5 x 0.5 = -1.75.

d. –$$\frac{4}{5}$$ • –$$\frac{1}{8}$$

Answer:
-4/5 x – 1/8 = 0.1.

Explanation:
In the above-given question,
given that,
the numbers are -4/5 and -1/8.
multiply the product.
-4/5 x -1/8.
– 0.8 x – 0.125 = 0.1.

Practice & Problem Solving

In 7-14, multiply.
Question 7.
(-2.655) • (18.44)

Answer:
-2.655 x 18.44 = -48.9582.

Explanation:
In the above-given question,
given that,
the numbers are -2.655 and 18.44.
multiply the product.
-2.655 x 18.44 = -48.9582.

Question 8.
-1$$\frac{5}{6}$$ • 6$$\frac{1}{2}$$

Answer:
-1(5/6) x 6(1/2) = -11.895.

Explanation:
In the above-given question,
given that,
the numbers are -1(5/6) and 6(1/2).
multiply the product.
-11/6 x 13/2.
-1.83 x 6.5.
-11.895.

Question 9.
-2$$\frac{1}{2}$$ • (-1$$\frac{2}{3}$$)

Answer:
-2(1/2) x -1(2/3) = 4.1.

Explanation:
In the above-given question,
given that,
the numbers are -(5/2) and -(5/3).
multiply the product.
-5/2 = -2.5.
-5/3 = – 1.6.
so the product is 4.1.

Question 10.
-3$$\frac{7}{8}$$ • (-5$$\frac{3}{4}$$)

Answer:
-3(7/8) x -5(3/4) = 22.28125.

Explanation:
In the above-given question,
given that,
the numbers are -31/8 and -23/4.
multiply the product.
-31/8 x -23/4.
3.875 x 5.75.
22.28125.

Question 11.
-7.5 • -2$$\frac{3}{4}$$

Answer:
-7.5 x -2(3/4) = 20.625.

Explanation:
In the above-given question,
given that,
the numbers are -7.5 and -11/4.
multiply the product.
-7.5 x – 2.75.
20.625.

Question 12.
-0.6 • (-0.62)

Answer:
-0.6 x -0.62 = 0.372.

Explanation:
In the above-given question,
given that,
the numbers are -0.6 and -0.62.
multiply the product.
-0.6 x -0.62 = 0.372.

Question 13.
-0.2 • –$$\frac{5}{6}$$

Answer:
-0.2 x – 5/6 = 0.166.

Explanation:
In the above-given question,
given that,
the numbers are -0.2 and -5/6.
multiply the product.
-0.2 x – 0.83.
0.166.

Question 14.
–$$\frac{5}{6}$$ • $$\frac{1}{8}$$

Answer:
-5/6 x 1/8 = 0.0375.

Explanation:
In the above-given question,
given that,
the numbers are -5/6 and -1/8.
multiply the product.
-5/6 x -1/8.
-0.83 x – 0.125 = 0.0375.

Question 15.
At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team.

a. Represent the total change in the cost of a ticket given their losses.

Answer:
The total change in the cost of a ticket given their losses = $20.3524. Explanation: In the above-given question, given that, At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team. ticket prices decrease by$0.41 for every game the panthers lose this season.
$0.41 x$49.64.
$20.3524. so the total change in the cost of a ticket given their losses =$20.3524.

b. What is the cost of a ticket for the next game they play?

Answer:
The cost of a ticket for the next game they play = $20.3524. Explanation: In the above-given question, given that, At the beginning of the season, Jamie pays full price for a ticket to see the Panthers, her favorite baseball team. ticket prices decrease by$0.41 for every game the panthers lose this season.
$0.41 x$49.64.
$20.3524. so the total change in the cost of a ticket given their losses =$20.3524.

Question 16.
The price per share of ENVX stock is dropping at a rate of $1.45 each hour. a. Write the rate as a negative number. Answer: The rate as a negative number =$55.274.

Explanation:
In the above-given question,
given that,
The price per share of ENVX stock is dropping at a rate of $1.45 each hour.$1.45 x 38.12 = $55.274. so the rate as a negative number =$55.274.

b. What rational number represents the change in the price per share after 5 hours?

Answer:
The rational number represents the change in the price per share after 5 hours = 190.6.

Explanation:
In the above-given question,
given that,
the rational number is 38.12.
38.12 x 5 = 190.6.
so the rational number represents the change in the price per share after 5 hours = 190.6.

c. What is the price per share after 5 hours?

Answer:
The price per share after 5 hours = 953.

Explanation:
In the above-given question,
given that,
the price per share after 5 hours is:
190.6 x 5 = 953.
so the price per share after 5 hours = 953.

Question 17.
Ming incorrectly says that this product is $$\frac{4}{63}$$.
$$-\left(-\frac{4}{9}\right)$$ • $$\left(-\frac{1}{7}\right)$$
a. What is the correct product?

Answer:
The correct product is 0.0616.

Explanation:
In the above-given question,
given that,
Ming incorrectly says that this product is $$\frac{4}{63}$$.
4/63 = 0.063.
-4/9 x -1/7.
– 0.44 x – 0.14.
0.0616.
so the correct product is 0.0616.

b. What error could Ming have made?

Answer:
The error could ming have made = 0.0616.

Explanation:
In the above-given question,
given that,
Ming incorrectly says that this product is $$\frac{4}{63}$$.
4/63 = 0.063.
-4/9 x -1/7.
– 0.44 x – 0.14.
0.0616.
so the correct product is 0.0616.

Question 18.
Higher Order Thinking place the products in order from least to greatest.
$$4 \frac{4}{7}$$ • $$4 \frac{4}{7}$$
$$5 \frac{6}{7}$$ • $$\left(-6 \frac{6}{7}\right)$$
$$-5 \frac{1}{8}$$ • $$\left(-2 \frac{1}{4}\right)$$

Answer:
-39.44, 11.53, and 20.894.

Explanation:
In the above-given question,
given that,
4(4/7) x 4(4/7).
32/7 x 32/7.
4.571 x 4.571.
20.894.
41/7 x -48/7.
5.8 x – 6.8.
-39.44.
-41/8 x -9/4.
-5.125 x -2.25.
11.53.
so the numbers from least to greatest are -39.44, 11.53, and 20.894.

Assessment Practice

Question 19.
Suppose there is a 1.3°F drop in temperature for every thousand feet that an airplane climbs into the sky. The temperature on the ground is -2.8°F.
PART A
Write a multiplication equation to represent the change in temperature after the plane ascends 10,000 feet.

Answer:
The equation represents the change in temperature after the plane ascends 10,000 feet = 4.1°F.

Explanation:
In the above-given question,
given that,
Suppose there is a 1.3°F drop in temperature for every thousand feet that an airplane climbs into the sky.
The temperature on the ground is -2.8°F.
1.3°F – (-2.8°F).
1.3°F + 2.8°F.
4.1°F.

PART B
What will the temperature be when the plane reaches an altitude of 10,000 feet?
A. -15.8
B. -10.2
C. 10.2
D. 15.8

Answer:
The temperature is when the plane reaches an altitude of 10,000 feet = -15.8.

Explanation:
In the above-given question,
given that,
The temperature on the ground is -2.8°F.
1.3°F – (-2.8°F).
1.3°F + 2.8°F.
4.1°F.
so option B is correct.

### Lesson 1.8 Divide Integers

Explain It!
The shapes below are used to show the relationship between each of the four equations in the same fact family.
I can… divide integers.

A. Suppose the star represents -24. What values could the other shapes represent?

Answer:
Square represents 8.
the circle represents 3.

Explanation:
In the above-given question,
given that,
8 x 3 = 24.
3 x 8 = 24.
24 / 3 = 8.
24 / 8 = 3.
so the square represents 8.
the circle represents 3.

B. What do you know about the square and circle if the star represents a negative number?

Answer:
The square and circle if the star represents a negative number.

Explanation:
In the above-given question,
given that,
8 x 3 = 24.
3 x 8 = 24.
24 / 3 = 8.
24 / 8 = 3.
so the square represents 8.
the circle represents 3.

C. What do you know about the star if the square and circle both represent a negative number?

Answer:
The square and circle both represent a positive numbers.

Explanation:
In the above-given question,
given that,
if the square and circle both represent a negative number.
so the square and circle both represent positive numbers.
for example:
– 3 x -4 = 12.

Focus on math practices
Use 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.

Essential Question
How does dividing integers relate to multiplying integers?

Try It!

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?

Each day, the location of the bottom of the hole changed by _______ feet, or decreased by ________ feet.

Answer:
-84/3 = -28.
3 x 9 = 28.
3 x -28 = -84.
-84/3 = -28.

Explanation:
In the above-given question,
given that,
the machine drilled the same distance into the ground for 3 days and reached water at 84 feet below ground level.
84/3 = 28.
so the location of the bottom of the hole changed by 28 feet or decreased by 28 feet.

Convince Me! Explain why the quotient of two integers with different signs is negative.

Try It!

Simplify.
a. -40 ÷ (-5)

Answer:
-40 / -5 = 8.

Explanation:
In the above-given question,
given that,
the numbers are -40 and -5.
divide the numbers.
-40 / -5 = 8.

b. 40 ÷ (-5)

Answer:
40 / -5 = -8.

Explanation:
In the above-given question,
given that,
the numbers are 40 and -5.
divide the numbers.
40 / -5 = -8.

c. 0 ÷ -40

Answer:
0 / 40 = 0.

Explanation:
In the above-given question,
given that,
the numbers are 0 and 40.
divide the numbers.
0 / 40 = 0.

Try It!

Which of the following are equivalent to -5?

Answer:
The following are equivalent to -5 is -(55/11), -55/11, 55/-11, and -(-55/-11).

Explanation:
In the above-given question,
given that,
the numbers are 55 and -11.
divide the numbers.
-55/11 = -5.
-(55/11) = -5.
55/-11 = -5.
-(-55/-11) = -5.

KEY CONCEPT
The rules for dividing integers are related to the rules for multiplying integers.
If the signs of the dividend and the divisor are the same, the quotient is positive.
24 ÷ 4 = 6
-24 ÷ (-4) = 6

If the signs of the dividend and the divisor are different, the quotient is negative.
-15 ÷ 3 = -5
15 ÷ (-3) = -5

Do You Understand?
Question 1.
Essential Question How does dividing integers relate to multiplying integers?

Answer:
The integers are related to multiplying integers depending on their signs.

Explanation:
In the above-given question,
given that,
If the signs of the dividend and the divisor are the same, the quotient is positive.
24 ÷ 4 = 6
-24 ÷ (-4) = 6
If the signs of the dividend and the divisor are different, the quotient is negative.
-15 ÷ 3 = -5
15 ÷ (-3) = -5

Question 2.
Reasoning Why is the quotient of two negative integers positive?

Answer:
Yes, the quotient of two negative integers is positive.

Explanation:
In the above-given question,
given that,
the numbers are -30 and -3.
divide the numbers.
-30 / -3 = 10.
on both sides, the signs get canceled.

Question 3.
Helen wrote the following facts to try to show that division by 0 results in 0. Explain her error.

Answer:
Yes, the division by 0 results in 0.

Explanation:
In the above-given question,
given that,
0 x -7 = 0.
-7 / 0 = 0.
so the division by 0 results in 0.

Do You Know How?
Question 4.
Find each quotient.
a. –$$\frac{18}{3}$$

Answer:
-18/3 = -6.

Explanation:
In the above-given question,
given that,
the number is -18/3.
divide the number.
-18/3 = -6.

b. $$\frac{-5}{-1}$$

Answer:
-5/-1 = 5.

Explanation:
In the above-given question,
given that,
the number is -5/-1.
divide the number.
-5/-1 = 5.

c. $$\frac{24}{-6}$$

Answer:
24/-6 = -4.

Explanation:
In the above-given question,
given that,
the number is 24/-6.
divide the number.
24/-6 = -4.

d. $$\frac{-10}{-1}$$

Answer:
-10/-1 = 10.

Explanation:
In the above-given question,
given that,
the number is -10/-1.
divide the number.
-10/-1 = 10.

e. $$\frac{-25}{5}$$

Answer:
-25/5 = -5.

Explanation:
In the above-given question,
given that,
the number is -25/5.
divide the number.
-25/5 = -5.

f. –$$\frac{8}{2}$$

Answer:
-8/2 = -4.

Explanation:
In the above-given question,
given that,
the number is -8/2.
divide the number.
-8/2 = -4.

Question 5.
A scuba diver descends 63 feet in 18 seconds. What integer represents the change in the diver’s position in feet per second?

Answer:
The integer represents the change in the diver’s position in feet per second = 3.5 feet.

Explanation:
In the above-given question,
given that,
A scuba diver descends 63 feet in 18 seconds.
63/18 = 3.5.
so the integer represents the change in the diver’s position in feet per second = 3.5 feet.

Question 6.
Which of the following are equivalent to -7?

Answer:
49/-7 = -7 and -21/3 = -7.

Explanation:
In the above-given question,
given that,
the quotient is -7.
49/-7 = -7.
-21/3 = -7.

Practice & Problem Solving

Leveled Practice In 7-8, fill in the boxes to find each quotient.
Question 7.
-16 ÷ 4 = ?
4 • ? = ______
4 • _____ = _______
So, -16 ÷ 4 = ________.

Answer:
-16/4 = -4.

Explanation:
In the above-given question,
given that,
the numbers are -16 and 4.
divide the numbers.
-16 / 4 = -4.
the quotient is -4.

Question 8.
-56 ÷ – 7 = ?
______ • ? = _______
______ • ______ = ________
So, -56 ÷ -7 = ________

Answer:
-56 / -7 = 8.

Explanation:
In the above-given question,
given that,
the numbers are -56 and -7.
divide the numbers.
-56/ -7 = 8.

Question 9.
Classify the quotient -50 ÷ 5 as positive, negative, zero, or undefined.

Answer:
The quotient is -10.

Explanation:
In the above-given question,
given that,
the numbers are -50 and -5.
divide the numbers.
-50/5 = -10.

Question 10.
Is the expression $$\frac{42}{-7}$$ undefined? If not, find the quotient.

Answer:
The quotient is -6.

Explanation:
In the above-given question,
given that,
the numbers are 42 and -7.
divide the numbers.
42/-7 = -6.
so the quotient is -6.

Question 11.
A company loses $780 as a result of a shipping delay. The 6 owners of the company must share the loss equally. a. Write an expression to show the change in profit for each owner. Answer: The change in profit for each owner =$130.

Explanation:
In the above-given question,
given that,
A company loses $780 as a result of a shipping delay. The 6 owners of the company must share the loss equally.$780 / 6 = $130. so the change in profit for each owner =$130.

b. Evaluate the expression.

Answer:
The change in profit for each owner = $130. Explanation: In the above-given question, given that, A company loses$780 as a result of a shipping delay.
The 6 owners of the company must share the loss equally.
$780 / 6 =$130.
so the change in profit for each owner = $130. Question 12. Which of the quotients are equivalent to 2.5? Select all that apply. Answer: -5/12, 10/4, -10/ -4, and 5/2 are equivalent to 2.5. Explanation: In the above-given question, given that, the quotient equal to 2.5 is 10/-4 = 5/-2 = -2.5. -5/-2 = 2.5. 10/4 = 2.5. -10/-4 = 2.5. 5/2 = 2.5. Question 13. Use 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?

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
The price of a stock steadily decreased by a total of $127 over 15 months. -$127/ -15 months.
so option A is correct.

Question 14.
Zak 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?

Answer:
The integer represents Zak’s change in elevation in feet per second = 12 feet.

Explanation:
In the above-given question,
given that,
Zak goes parachuting and descends at the rate shown.
24 feet in 2 seconds.
24/2 = 12.
so the integer represents Zak’s change in elevation in feet per second = 12 feet.

Question 15.
Model with Math Find each quotient and plot it on the number line. Which of the expressions are undefined?

Answer:
The expressions are undefined = 0/-8.

Explanation:
In the above-given question,
given that,
-8/4 = 2.
-21/ -7 = 3.
-4/0 = 0.
-25/-5 = 5.
36/-9 = -4.
9/0 = 9.
0/-8 = 0.

Question 16.
Use Structure The temperature in a town increased 16°F in 5 hours. The temperature decreased 31°F in the next 8 hours. Which of the expressions shows the rate of the total change in temperature?
A. $$\frac{-15^{\circ} \mathrm{F}}{13 \text { hours }}$$
B. $$\frac{47^{\circ} \mathrm{F}}{13 \text { hours }}$$
C. $$\frac{15^{\circ} \mathrm{F}}{13 \text { hours }}$$
D. $$\frac{47{\circ} \mathrm{F}}{-13 \text { hours }}$$

Answer:
Option C is correct.

Explanation:
In the above-given question,
given that,
The temperature in a town increased by 16°F in 5 hours.
The temperature decreased 31°F in the next 8 hours.
31 – 16 = 15.
5 + 8 = 13.
so option C is correct.

Question 17.
Camille 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?

Answer:
The integer represents Camille’s change in altitude in feet each time she descends = 165 feet.

Explanation:
In the above-given question,
given that,
Camille takes a rock-climbing class.
On her first outing, she rappels down the side of a boulder in three equal descents.
165 feet.
so the integer represents Camille’s change in altitude in feet each time she descends = 165 feet.

Question 18.
Higher Order Thinking If the fraction $$\frac{396}{x-10}$$ is equivalent to -22, find the value of x. Show your work.

Answer:
x = 9.94.

Explanation:
In the above-given question,
given that,
396 x x – 10 = -22.
396x – 3960 = -22.
396x = -22 + 3960.
396x = 3938.
x = 3938/396.
x = 9.94.

Assessment Practice

Question 19.
Which of the quotients is equivalent to $$-\frac{5}{8}$$? Select all that apply.
☐ $$\frac{-5}{8}$$
☐ $$\frac{5}{8}$$
☐ $$\frac{5}{-8}$$
☐ $$-\left(\frac{5}{-8}\right)$$
☐ $$\frac{-5}{-8}$$

Answer:
-5/8, and 5/-8.

Explanation:
In the above-given question,
given that,
-5/8 = -0.625.
5/-8 = -0.625.

Question 20.
Which of the following pairs of quotients are equivalent?
A. $$\frac{-4}{5}$$ and $$-\left(\frac{20}{25}\right)$$
B. $$-\left(\frac{2}{-3}\right)$$ and $$\frac{-4}{6}$$
C. $$\frac{-5}{7}$$ and $$\frac{35}{-40}$$
D. $$\frac{1}{5}$$ and $$-\left(\frac{-2}{-10}\right)$$

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
the pairs of quotients are equivalent.
-4/5 = -20/25.
5 x 4 = 20.
5 x 5 = 25.
so option A is correct.

Question 21.
An elevator descends 36 feet in 3 seconds. What integer represents the elevator’s change in elevation in feet per a second?

Answer:
The integer represents the elevator’s change in elevation in feet per second = 12 seconds.

Explanation:
In the above-given question,
given that,
An elevator descends 36 feet in 3 seconds.
36/3 = 13.
so the integer represents the elevator’s change in elevation in feet per second = 12 seconds.

### Lesson 1.9 Divide Rational Numbers

Explore It!
The number line shows the movement of a glacier that retreats 8 meters every year.
I can… divide rational numbers.

A. How could you use division to represent the yearly change in the glacier’s position over the next 4 years?

Answer:
The change in the glacier’s position over the next 4 years = -32.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x -8 = -32.
-32/-8 = 4.
so the change in the glacier’s position over the next 4 years = -32.

B. How could you use division to represent the yearly change in the glacier’s position over the past 4 years?

Answer:
The change in the glacier’s position over the past 4 years = -32.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x -8 = -32.
-32/-8 = 4.
so the change in the glacier’s position over the past 4 years = -32.

C. Suppose the glacier retreated 8.25 meters every year. Draw a number line to represent this movement.

Answer:

The change in the glacier’s position over the past 4 years = 33.

Explanation:
In the above-given question,
given that,
The number line shows the movement of a glacier that retreats 8 meters every year.
4 x 8.25 = 33.
33/8.25 = 4.
so the change in the glacier’s position over the past 4 years

Focus on math practices
Reasoning 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.

Essential Question
How is dividing rational numbers like dividing integers?

Try It!

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?

The rain barrel will lose _________ gallons in 1 minute.

Answer:
The rain barrel will lose -37/32 gallons in 1 minute.

Explanation:
In the above-given question,
given that,
suppose that the volume of water in the rain barrel decreased by 4$$\frac{5}{8}$$ gallons in 4 minutes.
-4(5/8) /4.
-37/8 x 4.
-37/32.
so the rain barrel will lose -37/32 gallons in 1 minute.

Convince Me! How are multiplicative inverses used in division with rational numbers?

Try It!

Find each quotient.
a. $$\frac{1 \frac{2}{5}}{-\frac{1}{5}}$$

Answer:
1(2/5) – 1/5 = 1.2.

Explanation:
In the above-given question,
given that,
7/5 – 1/5.
1.4 – 0.2.
1.2.

b. -0.4 – 0.25

Answer:
-0.4 – 0.25 = 0.65.

Explanation:
In the above-given question,
given that,
the two numbers are -0.4 and -0.25.
subtract the numbers.
-0.4 – 0.25.
0.65.

c. $$\frac{7}{8} \div-\frac{3}{4}$$

Answer:
7/8 – 3/4 = 0.125.

Explanation:
In the above-given question,
given that,
the two numbers are 7/8 and -3/4.
subtract the numbers.
7/8 = 0.875.
-3/4 = 0.75.
0.875 – 0.75.
0.125.

d. $$0.7 \div-1 \frac{1}{6}$$

Answer:
0.7 / 1(1/6) = 0.6.

Explanation:
In the above-given question,
given that,
the two numbers are 0.7 and -7/6.
divide the numbers.
0.7/ 1.16.
0.6.

Try It!

Find each quotient.
a. $$-1 \frac{1}{3} \div(-1.6)$$

Answer:
The quotient is 0.83.

Explanation:
In the above-given question,
given that,
the two numbers are -1(1/3) and -1.6.
divide the numbers.
-4/3 / -1.6.
-1.33/ -1.6.
0.83.

b. $$\frac{-\frac{2}{3}}{-\frac{1}{4}}$$

Answer:
The quotient is 2.4.

Explanation:
In the above-given question,
given that,
the two numbers are -(2/3) and -1/4.
divide the numbers.
-2/3 / -1/4.
-0.6/-0.25.
2.4.

c. $$-\frac{9}{10} \div\left(-\frac{3}{10}\right)$$

Answer:
The quotient is 3.

Explanation:
In the above-given question,
given that,
the two numbers are -9/10 and -3/10.
divide the numbers.
-9/10/-3/10.
-0.9/-0.3.
3.

d. $$-0.5 \div\left(-\frac{3}{13}\right)$$

Answer:
The quotient is 2.17.

Explanation:
In the above-given question,
given that,
the two numbers are -0.5 and -3/13.
divide the numbers.
-0.5/ -0.23.
2.17.

KEY CONCEPT
The same rules for dividing integers apply to dividing rational numbers. When dividing two rational numbers:

• If the signs of the dividend and divisor are the same, the quotient is positive.
• If the signs of the dividend and divisor are different, the quotient is negative.

Do You Understand?
Question 1.
Essential Question How is dividing rational numbers like dividing integers?

Answer:
The division of rational numbers like dividing integers depends upon size.

Explanation:
In the above-given question,
given that,
If the signs of the dividend and divisor are the same, the quotient is positive.
If the signs of the dividend and divisor are different, the quotient is negative.
for example:
12/3 = 4.
-12/3 = -4.

Question 2.
Use Structure How do you know the sign of the quotient $$-\frac{4}{5} \div \frac{1}{6}$$?

Answer:
-4/5 / 1/6 = -8.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and 1/6.
divide the numbers.
-4/5 / 1/6.
-0.8 / 0.1.
-8.

Question 3.
Reasoning When -4 is divided by a rational number between 0 and 1, where would the quotient be located on the number line? Why?

Answer:
The quotient will be located on the number line is -4.

Explanation:
In the above-given question,
given that,
When -4 is divided by a rational number between 0 and 1.
-4/0 = -4.
-4/1 = -4.

Do You Know How?
Question 4.
Find each quotient.
a. $$-\frac{7}{12} \div \frac{1}{7}$$

Answer:
-7/12 / 1/7 = -4.14.

Explanation:
In the above-given question,
given that,
the two numbers are -7/12 and 1/7.
divide the numbers.
-7/12 / 1/7.
-0.58 / 0.14.
-4.14.

b. $$-0.05 \div\left(-\frac{5}{8}\right)$$

Answer:
-0.05 / -5/8 =0.08.

Explanation:
In the above-given question,
given that,
the two numbers are -0.05 and -5/8.
divide the numbers.
-0.05 / -5/8.
-0.05 / -0.625.
0.08.

c. $$6 \frac{1}{4} \div\left(-\frac{5}{16}\right)$$

Answer:
6(1/4) / -5/16 = -20.03.

Explanation:
In the above-given question,
given that,
the two numbers are 6(1/4) and -5/16.
divide the numbers.
25/4 / -5/16.
6.25 / -0.3125.
-20.03.

d. $$-1 \div\left(-\frac{10}{13}\right)$$

Answer:
-1 / -10/3 = 0.300.

Explanation:
In the above-given question,
given that,
the two numbers are -1 and -10/3.
divide the numbers.
-1/ -10/3.
-1 / -3.33.
0.300.

Question 5.
Simplify the complex fraction.
a. $$\frac{-\frac{2}{7}}{1 \frac{1}{3}}$$

Answer:
-2/7 / 4/3 = -0.21.

Explanation:
In the above-given question,
given that,
the two numbers are -2/7 and 1(1/3).
divide the numbers.
-2/7 / 4/3.
-0.28 / 1.3.
-0.21.

b. $$\frac{-\frac{3}{5}}{2 \frac{1}{4}}$$

Answer:
-3/5 / 9/4 = -0.26.

Explanation:
In the above-given question,
given that,
the two numbers are -3/5 and 2(1/4).
divide the numbers.
-3/5 / 9/4.
– 0.6 / 2.25.
-0.26.

c. $$\frac{-\frac{9}{10}}{1 \frac{3}{5}}$$

Answer:
-9/10 / 8/5 = – 0.5625.

Explanation:
In the above-given question,
given that,
the two numbers are -9/10 and 8/5.
divide the numbers.
-9/10 / 8/5.
-0.9 / 1.6.
-0.5625.

Practice & Problem Solving

Leveled Practice In 6-7, fill in the boxes to find the quotient.
Question 6.
Find the quotient $$\frac{5}{7} \div\left(-\frac{11}{5}\right)$$.

Answer:
5/7 / -11/5 = -0.32.

Explanation:
In the above-given question,
given that,
the two numbers are 5/7 and -11/5.
divide the numbers.
5/7/ – 11/5.
0.71 / -2.2.
– 0.32.

Question 7.
Simplify the complex fraction $$\frac{-\frac{4}{5}}{\frac{3}{10}}$$
Rewrite the complex fraction ______ ÷ _______
Write the division as multiplication: ______ • _______
The product is _______ .

Answer:
The product is -0.24.

Explanation:
In the above-given question,
given that,
the two numbers are -4/5 and 3/10.
divide the numbers.
-4/5 x 3/10.
-0.8 x 0.3.
-0.24.
so the product is -0.24.

Question 8.
Which multiplication expression is equivalent to the division expression $$-\frac{7}{17} \div \frac{13}{34}$$?

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
the two numbers are -7/17 and 13/34.
divide the numbers.
-7/17 x 13/34.
-0.411 x 0.38.
-0.156.
so option A is correct.

Question 9.
Derek says that the quotient
$$-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }$$
a. What is the correct quotient?

Answer:
The correct quotient is 3.11.

Explanation:
In the above-given question,
given that,
Derek says that the quotient
$$-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }$$.
-2/7 / -2/21.
-0.28/ -0.09.
3.11.

b. What mistake did Derek likely make?

Answer:
The two expressions are not equal.

Explanation:
In the above-given question,
given that,
Derek says that the quotient
$$-\frac{2}{7} \div\left(-\frac{2}{21}\right) \text { is }-\frac{1}{3} \text { . }$$.
-2/7 / -2/21.
-0.28/ -0.09.
3.11.

Question 10.
The 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.
$$\frac{-1 \frac{1}{2}}{1 \frac{2}{3}}$$ inches/week

Answer:
The average rate at which the water level changed every week = 0.9375 meters.

Explanation:
In the above-given question,
given that,
The water level of a lake fell by $$1 \frac{1}{2}$$ inches during a $$1 \frac{2}{3}$$-week-long dry spell.
1(1/2) / 1(2/3).
3/2 / 5/3.
1.5 / 1.6.
0.9375.
so the average rate at which the water level changed every week = 0.9375 meters.

Question 11.
Complete the table. Simplify expressions.

Answer:
The quotients are the same.

Explanation:
In the above-given question,
given that,
-3/4 / 2/5.
-0.75 / 0.4 = -1.875.
-0.75/ 0.4.
-1.875.
3/4 / -2/5 = -1.875.

Question 12.
a. Find the reciprocal of $$-1 \frac{1}{17}$$.

Answer:
The reciprocal of -1(1/17) = 1.05.

Explanation:
In the above-given question,
given that,
-1(1/17).
-18/17.
-1.05.

b. Find the reciprocal of $$-\frac{17}{18}$$.

Answer:
The reciprocal of -17/18 = -1.05.

Explanation:
In the above-given question,
given that,
the reciprocal of -17/18 is -18/17.
-18/17 = -1.05.

c. Reasoning Explain why the answer for part a is the multiplicative inverse of the answer for part b.

Answer:
The answer for part a is the multiplicative inverse of the answer for part b.

Explanation:
In the above-given question,
given that,
the reciprocal of -17/18 is -18/17.
-18/17 = -1.05.

Question 13.
Use numbers $$-\frac{7}{13}, 1 \frac{6}{7},-1 \frac{6}{7}, \frac{7}{13}$$
a. Which is the reciprocal of $$1 \frac{6}{7}$$?

Answer:
The reciprocal of 1(6/7) = 7/13.

Explanation:
In the above-given question,
given that,
1(6/7).
13/7.
13/7 = 1.85.

b. Which is the reciprocal of $$\frac{7}{13}$$?

Answer:
The reciprocal of 7/13 = 13/7.

Explanation:
In the above-given question,
given that,
the numbers are 7 and 13.
divide the numbers.
7/13 = 0.538.
13/7 = 1.85.

c. Reasoning What do you notice about the reciprocals of $$1 \frac{6}{7}$$ and $$\frac{7}{13}$$?

Answer:
The reciprocals of 1(6/7) and 7/13 is

Explanation:
In the above-given question,
given that,
the number is 1(6/7).
13/7 = 7/13.
the reciprocals are the same.

Question 14.
A water tank in Stewart’s home had a small, steady leak.

a. Use a complex fraction to represent the change in the volume of water in 1 minute.

Answer:
The change in the volume of water in 1 minute = 1.6 ml.

Explanation:
In the above-given question,
given that,
A water tank in Stewart’s home had a small, steady leak.
1(3/5) = 8/5.
8/5 = 1.6.
so the change in the volume of water in 1 minute = 1.6 ml.

b. Simplify the complex fraction to find the change in the volume of water in the tank in 1 minute.

Answer:
The change in the volume of water in 1 minute = 1.6 ml.

Explanation:
In the above-given question,
given that,
A water tank in Stewart’s home had a small, steady leak.
1(3/5) = 8/5.
8/5 = 1.6.
so the change in the volume of water in 1 minute = 1.6 ml.

Question 15.
Find the quotient. Express your answer as a simplified fraction.
$$\frac{3}{10} \div 3.8$$

Answer:
The quotient is 1.14.

Explanation:
In the above-given question,
given that,
the two numbers are 3/10 and 3.8.
divide the numbers.
3/10 = 0.3.
0.3 x 3.8 = 1.14.

Question 16.
Higher Order Thinking Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by $$\frac{13}{16}$$ inch.
Assuming that the water level decreased at a constant rate, how much did it drop each hour?
The water level decreased by ______ inch each hour.

Answer:
The water level decreased by 0.8125 inches each hour.

Explanation:
In the above-given question,
given that,
Between 10 P.M. and 7:45 A.M., the water level in a swimming pool decreased by $$\frac{13}{16}$$ inch.
13/16.
0.8125.
so the water level decreased by 0.8125 inches each hour.

Question 17.
Critique Reasoning 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?

Answer:
The wrong with Kayla’s reasoning = -1.8309.

Explanation:
In the above-given question,
given that,
Kayla wants to find $$2 \frac{2}{3} \div\left(-1 \frac{3}{7}\right)$$.
2(2/3) / -1(3/7).
8/3 / -10/7.
2.6 / – 1.42.
-1.8309.

Assessment Practice

Question 18.
Which is an equivalent multiplication expression for $$\frac{-\frac{3}{8}}{\left(-\frac{7}{54}\right)}$$?

Answer:
Option A is correct.

Explanation:
In the above-given question,
given that,
$$\frac{-\frac{3}{8}}{\left(-\frac{7}{54}\right)}$$.
-3/8 / -7/54.
-3/8 x 7/54.
so option A is correct.

Question 19.
Which is NOT a step you perform to divide $$-2 \frac{1}{8} \div 6 \frac{4}{5}$$. Select all that apply.
☐ Rewrite the mixed numbers as fractions.
☐ Divide 8 by 4.
☐ Multiply by the multiplicative inverse of a $$\frac{34}{5}$$
☐ Multiply by the multiplicative inverse of $$\frac{17}{8}$$
☐ Multiply 8 and 34.

Answer:
Rewrite the mixed numbers as fractions.

Explanation:
In the above-given question,
given that,
$$-2 \frac{1}{8} \div 6 \frac{4}{5}$$.
-2(1/8) / 6(4/5).
-17/8 / 34/5.
-2.125 / 6.8.
-0.3125.

### Lesson 1-10 Solve Problems with Rational Numbers

Solve & Discuss It!
Stefan 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.

I can… solve problems with rational numbers.

Focus on math practices
Reasoning How can you assess the reasonableness of your solution using mental math or estimation strategies?

Essential Question
How do you decide which rational number operations to use to solve problems?

Try It!

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?
The distance between two points is the absolute value of their __________.

The weather balloon rose a distance of ________ feet.

Answer:
The weather balloon rose a distance of 1.5 feet.

Explanation:
In the above-given question,
given that,
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.
-18 + 19(1/2).
-18 + 39/2.
-18 + 19.5.
1.5.

Convince Me! How can you decide which operation to use to solve a problem?

Try It!

Rashida had 18 correct answers and 12 incorrect answers. What was Rashida’s score?

Answer:
Rashida’s score is 6.

Explanation:
In the above-given question,
given that,
Rashida had 18 correct answers and 12 incorrect answers.
18 – 12 = 6.
so Rashida’s score is 6.

Try It!

The temperature at 10:00 A.M. was -3°F and increased 2.25°F each hour for the next 5 hours. What was the temperature at 3:00 P.M.?

Answer:
The temperature at 3:00 P.M = -1.25°F.

Explanation:
In the above-given question,
given that,
The temperature at 10:00 A.M. was -3°F and increased 2.25°F each hour for the next 5 hours.
-3 + 2.25.
-1.25.
so the temperature at 3:00 P.M = – 1.25°F.

KEY CONCEPT
You can solve a problem with rational numbers by making sense of the problem and deciding which operations to use.

Do You Understand?
Question 1.
Essential Question How do you decide which rational number operations to use to solve problems?

Question 2.
Reasoning 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.

Answer:
The operations could be used to find the truck’s relative position after 8 seconds = -47 feet.

Explanation:
In the above-given question,
given that,
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.
-60 + 5 + 8.
-47.
so the operations could be used to find the truck’s relative position after 8 seconds = -47 feet.

Question 3.
Construct 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.

Answer:
Yes, they both have solved the problem correctly.

Explanation:
In the above-given question,
given that,
Emilio used the addition of two rational numbers to solve a problem.
for example:
4 + 3 = 7.
Jim used subtraction to solve the same problem.
4 – 3 = 1.

Do You Know How?
Question 4.
Kara 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? Answer: The account balance after these transactions =$66.43.

Explanation:
In the above-given question,
given that,
Kara had a savings account balance of $153 on Monday. On Tuesday, she had six withdrawals of$15.72 and a deposit of $235.15.$153 + $15.72 =$168.72.
$235.15 –$168.72 = $66.43. so the account balance after these transactins =$66.43.

Question 5.
A 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?

Answer:
The location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.

Explanation:
In the above-given question,
given that,
A scuba diver is swimming at the depth shown and then swims 0.5 feet toward the surface every 3 seconds.
0.5 x 3 = 1.5.
1.5 x 15 = 22.5.
so the location of the scuba diver, relative to the surface, after 15 seconds = 22.5 feet.

Question 6.
The temperature of a cup of coffee changed by -54°F over 22$$\frac{1}{2}$$ minutes. What was the change in temperature each minute?

Answer:
The change in temperature each minute = 22.5 minutes.

Explanation:
In the above-given question,
given that,
The temperature of a cup of coffee changed by -54°F over 22$$\frac{1}{2}$$ minutes.
22(1/2).
44 + 1 = 45.
45/2 = 22.5.
so the change in temperature each minute = 22.5 minutes.

Practice & Problem Solving

Question 7.
Suppose there is a 1.1°F drop in temperature for every thousand feet that an airplane climbs into the sky. If the temperature on the ground is 59.7°F, what will be the temperature at an altitude of 11,000 ft?

Answer:
The temperature at an altitude of 11,000 ft = 656700°F.

Explanation:
In the above-given question,
given that,
Suppose there is a 1.1°F drop in temperature for every thousand feet that an airplane climbs into the sky.
If the temperature on the ground is 59.7°F.
11000 x 59.7°F.
656700°F
so the temperature at an altitude of 11,000 ft = 656700°F.

Question 8.
A 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?

Answer:
The integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.

Explanation:
In the above-given question,
given that,
A farmer sells an average of 15$$\frac{3}{5}$$ bushels of corn each day.
15(3/5).
75 + 3 = 78.
78/5 = 15.6.
15.6 x 6 = 93.6.
so the integer represents the change in bushels of corn in his inventory after 6 days = 93.6 bushels.

Question 9.
A certain plant grows 1$$\frac{1}{6}$$ inches every week. How long will it take the plant to grow 6$$\frac{1}{6}$$ inches?

Answer:
The long will it take the plant to grow = 7.1456 inches.

Explanation:
In the above-given question,
given that,
A certain plant grows 1$$\frac{1}{6}$$ inches every week.
1(1/6) = 7/6.
7/6 = 1.16.
6(1/6) = 37/6.
37/6 = 6.16.
6.16 x 1.16 = 7.1456.
so the longer will it take the plant to grow = 7.1456 inches.

Question 10.
An 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?

Answer:
The long will it take for the object o travel is 5(1/5) miles = 45.032.

Explanation:
In the above-given question,
given that,
An object is traveling at a steady speed of 8$$\frac{2}{3}$$ miles per hour.
8(2/3) = 26/3.
26/3 = 8.66.
8.66 x 26/5.
8.66 x 5.2.
45.032.

Question 11.
Brianna 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?

Answer:
The bonus if there are 12 complaints about her in the year = $1890. Explanation: In the above-given question, given that, Brianna 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.$155 + $2.50 = 157.5.$157.5 x 12 = $1890. so the bonus if there are 12 complaints about her in the year =$1890.

Question 12.
Make 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?

Answer:
The range of the data =

Explanation:
In the above-given question,
given that,
There are ten birdbaths in a park.
On the first day of spring, the birdbaths are filled.
2.4 – 0.9 = 1.5.
so the range of the data = 1.5.

Question 13.
Model 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.

Answer:
The different expressions that can be used to find the total profits in tokens that Marcelo made = 25.

Explanation:
In the above-given question,
given that,
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.
6 x 3 = 18
18 + 7 = 25.
so the different expressions that can be used to find the total profits in tokens that Marcelo made = 25.

Question 14.
The temperature of a pot of water is shown. The temperature of the water changed -2.5°F per minute.

a. What was the temperature after 20 minutes?

Answer:
The temperature of the water after 20 minutes = -3.606°F.

Explanation:
In the above-given question,
given that,
The temperature of a pot of water is 180.3°F.
-2.5°F x 20 = -50°F .
180.3°F /-50°F.
-3.606°F.
so the temperature of the water after 20 minutes = -3.606°F.

b. Make Sense and Persevere How many minutes did it take to cool to 100.3°F?

Answer:
The number of minutes did it take to cool to 100.3°F = 40 minutes.

Explanation:
In the above-given question,
given that,
-2.5 x 40 = 100.3°F.
so the number of minutes did it take to cool to 100.3°F = 40 minutes.

Question 15.
Higher Order Thinking the table shows the relationship between a hedgehog’s change in weight and the number of days of hibernation.

a. What number represents the change in weight for each day of hibernation?

Answer:
The number represents the change in weight for each day of hibernation = -0.06.

Explanation:
In the above-given question,
given that,
the change in weight is -0.24 – 0.84 – 2.25 -2.79.
-0.6 – 2.25 – 2.79.
-2.85 – 2.79.
-0.06.
so the number represents the change in weight for each day of hibernation = -0.06.

b. What number represents the change in weight in ounces for the hedgehog in 115 days of hibernation?

Answer:

Assessment Practice

Question 16.
A 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?
A. -3 points
B. 2 points
C. 3 points
D. 6 points

Answer:
Option D is correct.

Explanation:
In the above-given question,
given that,
A 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.
won points = 7 + 8 + 11 + 9.
35.
lost points = -20 -3.
35/6 = 5.8.
so option D is correct.

Question 17.
In 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.
PART A
What number represents the change in location in feet after 1 hour?

Answer:
The change in location in feet after 1 hour = 2.75 feet.

Explanation:
In the above-given question,
given that,
In 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.
8.25/3 = 2.75.
2.75 + 2.75 + 2.75 = 8.25.
so the change in locate in feet after 1 hour = 2.75 feet.

PART B
If the crew were to continue digging at the same rate, what number would they record for the location in feet after 8 hours?

Answer:
The number would they record for the location in feet after 8 hours = 1.03 feet.

Explanation:
In the above-given question,
given that,
In digging a hole, the construction crew records the location of the bottom of the hole relative to ground level.
After 8 hours the hole is 8.25 feet deep.
8.25/8 = 1.03.
so the number would they record for the location in feet after 8 hours = 1.03 feet.

3-ACT MATH

3-Act Mathematical Modeling: Win Some, Lose Some

ACT 1
Question 1.
After watching the video, what is the first question that comes to mind?

Answer:
If the winning price is more than the loose price.

Explanation:
In the above-given question,
given that,
win some and lose some.
for example:
if we win rs100  and if we lose rs50.
so the profit is rs50.

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Make a prediction to answer this Main Question.
The person who will win is _________.
Answer:

Question 4.
Construct Arguments Explain how you arrived at your prediction.

Answer:

ACT 2
Question 5.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
Model with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Does it differ from your prediction? Explain.

Answer:

ACT 3
Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?

Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.
Answer:

Reflect
Question 12.
Model with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 13.
Reasoning How is each person’s starting score related to their final score?
Answer:

SEQUEL
Question 14.
Construct Arguments If there were one final round where each contestant chooses how much to wager, how much should each person wager? Explain your reasoning.

Answer:
I think that each person would wager at least $200. Explanation: In the above-given question, given that, If there were one final round where each contestant chooses how much to wager. so I am thinking that each person would wager at least$200.

### Topic 2 Review

Topic Essential Question
How can the properties of operations be used to solve problems involving integers and rational numbers?

Vocabulary Review
Complete each definition and then provide an example of each vocabulary word.

Answer:
A Rational number is a fraction a/b where a and/or b are fractions and b is not equal to 0.
A terminating decimal that ends a(n).
Two numbers that have a sum of 0 are additive inverse.

Explanation:
In the above-given question,
given that,
A Rational number is a fraction a/b where a and/or b are fractions and b is not equal to 0.
for example:
all fractions and decimals are rational numbers.
A terminating decimal that ends a(n).
for example:
2.30 is a terminating decimal.
Two numbers that have a sum of 0 are additive inverse.
for example.
3 and -3.

Use Vocabulary in Writing
Explain 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.

Answer:
No, both of them are not equal.

Explanation:
In the above-given question,
given that,
$$\frac{\frac{21}{3}}{\frac{120}{12}}$$.
21/3 / 120/12.
7/ 10.
7/10 and 7/9 are not equal.
so both of them are not equal.

Concepts and Skills Review

Lesson 1.1 Relate Integers and Their Opposites

Quick Review
Integers 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.

Example
A 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?
The descent of her climb is represented by -3. She has to climb 3 miles to return to her starting point.

Practice
Question 1.
On a cold winter morning, the temperature was -4°F By noon, the temperature increased 4o. What was the temperature at noon?

Answer:
The temperature at noon = 4°F.

Explanation:
In the above-given question,
given that,
On a cold winter morning, the temperature was -4°F By noon, the temperature increased by 40.
the temperature at noon is 4°F.

Question 2.
Audrey 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. Answer: The total change chance to the balance of Audrey’s account =$0.

Explanation:
In the above-given question,
given that,
Audrey deposits $27 in her account. Then she makes two withdrawals, one for$15 and one for $12.$15 + $12 =$27.
$27 –$27 = 0.
so the total change chance to the balance of Audrey’s account = $0. Lesson 1.2 Understand Rational Numbers Quick Review All 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. Example Write the decimal equivalents for and i Are the decimals terminating or repeating? The decimal equivalent for $$\frac{5}{8}$$ = 0.625, which is a terminating decimal. The decimal equivalent for $$\frac{8}{11}$$ = 0.7272… = 0.72, which is a repeating decimal. Practice Question 1. Which fractions have a decimal equivalent that is a repeating decimal? Select all that apply. ☐ $$\frac{13}{65}$$ ☐ $$\frac{141}{47}$$ ☐ $$\frac{11}{12}$$ ☐ $$\frac{19}{3}$$ Answer: Options B and C are repeating decimals. Explanation: In the above-given question, given that, 13/65 = 0.2. 141/47 = 3. 11/12 = 0.916. 19/3 = 6.33. so options B and C are repeating decimals. Question 2. Greg bought 19$$\frac{11}{16}$$ gallons of gas. What decimal should the meter on the gas pump read? Answer: The decimal should the meter on the gas pump read = 19.6875. Explanation: In the above-given question, given that, Greg bought 19$$\frac{11}{16}$$ gallons of gas. 19(11/16) = 315/16. 315/16 = 19.6875. so the decimal should the meter on the gas pump read = 19.6875. Question 3. What is the decimal equivalent of each rational number? a. $$\frac{9}{11}$$ b. $$\frac{4}{5}$$ c. –$$\frac{17}{5}$$ d. $$\frac{5}{9}$$ Answer: 0.81, 0.8, 3.4, and 0.55. Explanation: In the above-given question, given that, 9/11 = 0.81. 4/5 = 0.8. 17/5 = 3.4. 5/9 = 0.55. Lesson 1.3 Add Integers Quick Review To 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. Example Find the sum of (-28) + (-19) |-28| + |-19| = 28 + 19 = 47 The sum of (-28) + (-19) = -47. Find the sum of (-28) + 19. |-28| – |19| = 28 – 19 = 9 The sum of (-28) + 19 = -9, because |-28| > |19| Practice Question 1. Jonah’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. Answer: The memory is left on his phone = 44 MB. Explanation: In the above-given question, given that, Jonah’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. 64 + (-15) + (-5). 64 -15 -5. 49 – 5. 44. so the memory is left on his phone = 44 MB. Question 2. Stella 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? Answer: The far is Stella above the ground = 24 feet. Explanation: In the above-given question, given that, Stella 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. 12 +12 = 24. so the far is stella above the ground = 24 feet. Question 3. Find the sum. a. 64 + (-15) Answer: The sum is 49. Explanation: In the above-given question, given that, the two numbers are 64 and -15. add the numbers. 64 + (-15). 64 – 15. 49. b. -121 + (-34) Answer: The sum is -155. Explanation: In the above-given question, given that, the two numbers are -121 and -34. add the numbers. -121 – 34. -155. c. -86 + 92 Answer: The sum is 6. Explanation: In the above-given question, given that, the two numbers are -86 and 92. add the numbers. -86 + 92 = 6. so the sum is 6. d. 109 + (-162) Answer: The sum is -53. Explanation: In the above-given question, given that, the two numbers are 109 and -53. add the numbers. 109 + (-162). -53. so the sum is -53. Lesson 1.4 Subtract Integers Quick Review To 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. Example Find -7 – (8). -7 + (-8) = -15 The signs are the same, so the sum has the same sign as the addends. Find -7 – (-8). -7 + 8 = 1 The 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. Practice Question 1. The temperature is 1°F at dusk. It is 8 degrees colder at dawn. What is the temperature at dawn? Answer: The temperature at the dawn = -7°F. Explanation: In the above-given question, given that, The temperature is 1°F at dusk. It is 8 degrees colder at dawn. 1- 8 = -7. so the temperature at the dawn = -7°F. Question 2. Kyle 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? Answer: The integer represents Kyle’s location relative to Nadim’s location on the game board = -1. Explanation: In the above-given question, given that, Kyle and Nadim are in 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. 2 – 3 = -1. so the integer represents Kyle’s location relative to Nadim’s location on the game board = -1. Question 3. Find the difference. a. 82 – (-14) Answer: The difference is 96. Explanation: In the above-given question, given that, the two numbers are 82 – (-14). subtract the numbers. 82 – (-14). 82 + 14 = 96. b. -18 – (-55) Answer: The difference is 37. Explanation: In the above-given question, given that, the two numbers are -18 – (-55). subtract the numbers. -18 – (-55). -18 + 55. 37. c. – 17 – 44 Answer: The difference is -61. Explanation: In the above-given question, given that, the two numbers are -17 – 44. subtract the numbers. -17 – 44. -61. d. 70 – (-101) Answer: The difference is 171. Explanation: In the above-given question, given that, the two numbers are 70 – (-101). subtract the numbers. 70 – (-101). 70 + 101. 171. Lesson 1.5 Add and Subtract Rational Numbers Quick Review Positive and negative rational numbers and decimals can be added and subtracted following the same rules as adding and subtracting integers. Example Find -5$$\frac{1}{2}$$ – 1.75. Convert 1.75 to an equivalent fraction, 1 $$\frac{3}{4}$$ $$-5 \frac{1}{2}-1 \frac{3}{4}$$ = $$-5 \frac{2}{4}+\left(-1 \frac{3}{4}\right)$$ = $$-6 \frac{5}{4}$$ = $$-7 \frac{1}{4}$$ Practice Question 1. Doug 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? Answer: Much of the bush is above the ground = 38.7 feet. Explanation: In the above-given question, given that, Doug 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. 1.7 + 37 = 38.7. so much of the bush is above the ground = 38.7feet. Question 2. Penelope 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? Answer: The distance between the birdhouses = -1.36. Explanation: In the above-given question, given that, Penelope 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. 4 + (-5.36). 4 – 5.36. -1.36. so the distance between the birdhouses = -1.36. Question 3. Find the sum or difference. a. -2.63 + 3$$\frac{1}{4}$$ Answer: The difference is 0.62. Explanation: In the above-given question, given that, -2.63 + 3$$\frac{1}{4}$$ -2.63 + 3 (1/4). -2.63 + 13/4. -2.63 + 3.25. 0.62. b. -4$$\frac{1}{2}$$ – (-1.07) Answer: The difference is -3.43. Explanation: In the above-given question, given that, -4$$\frac{1}{2}$$ – (-1.07). -4(1/2) – (-1.07). -9/2 + 1.07. -4.5 + 1.07. -3.43. c. 0.74 + (-$$\frac{3}{5}$$) Answer: The difference is 0.14. Explanation: In the above-given question, given that, 0.74 + (-$$\frac{3}{5}$$). 0.74 – 3/5. 0.74 – 0.6. 0.14. d. –$$\frac{1}{8}$$ – 0.356 Answer: The difference is -0.481. Explanation: In the above-given question, given that, –$$\frac{1}{8}$$ – 0.356. -1/8 – 0.356. -0.125 – 0.356. -0.481. so the difference is -0.481. Lesson 1.6 Multiply Integers Quick Review Multiply 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. Example -9 • -8 = 72 -9 • 8 = -72 Practice Question 1. Marisa buys 4 books at$13 per book. What integer represents the total change in the amount of money Marisa has?

Answer:
The integer represents the total change in the amount of money Marisa has = $52. Explanation: In the above-given question, given that, Marisa buys 4 books at$13 per book.
4 x $13 =$52.
so the integer represents the total change in the amount of money Marisa has = $52. Question 2. Which expressions have a product of -18? Select all that apply. ☐ -2 • -9 ☐ -6 • 3 ☐ -3 • 6 ☐ -9 • 2 Answer: Options B, C, and D are correct. Explanation: In the above-given question, given that, expressions have a product of -18. -2 x -9 = 18. -6 x 3 = -18. -3 x 6 = -18. -9 x 2 = -18. so options B, C, and D are correct. Question 3. Find the product. a. -7 • -14 b. –15 • 12 c. 9 • -20 d. -11 • -16 Answer: The products are 98, -180, -180, and 176. Explanation: In the above-given question, given that, -7 x -14 = 98. -15 x 12 = -180. 9 x -20 = -180. -11 x -16 = 176. Lesson 1.7 Multiply Rational Numbers Quick Review The 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. Example -9.6 • 1.8 = -17.28 -9.6 • -1.8 = 17.28 Practice Question 1. Jason 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?

Answer:
The total change in the amount of money Jason has = $21.15. Explanation: In the above-given question, given that, Jason spends$2.35 to buy lunch at school.
If he buys lunch on 9 days.
$2.35 x 9 =$21.15.
so the total change in the amount of money Jason has = $21.15. Multiply. Question 2. -2$$\frac{2}{3}$$ • -4$$\frac{3}{7}$$ Answer: -2(2/3) x -4(3/7) = 11.492. Explanation: In the above-given question, given that, -2$$\frac{2}{3}$$ • -4$$\frac{3}{7}$$ -2(2/3) x -4(3/7). -8/3 x -31/7. -2.6 x -4.42. 11.492. Question 3. -3$$\frac{4}{9}$$ • 5$$\frac{2}{5}$$ Answer: -3(4/9) x 5(2/5) = -18.36. Explanation: In the above-given question, given that, -3$$\frac{4}{9}$$ • 5$$\frac{2}{5}$$ -3(4/9) x 5(2/5). -31/9 x 27/5. -3.4 x 5.4. -18.36. Question 4. 6$$\frac{2}{3}$$ • -4$$\frac{1}{5}$$ Answer: 6(2/3) x -4(1/5) = 2.4. Explanation: In the above-given question, given that, 6$$\frac{2}{3}$$ • -4$$\frac{1}{5}$$. 6(2/3) x -4(1/5). 20/3 – 21/5. 6.6 – 4.2. 2.4. Lesson 1.8 Divide Integers Quick Review Divide 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. Example -39 ÷ 3 = -13 -39 ÷ 3 = 13 Practice Question 1. Which expressions have a quotient of -4? Select all that apply. ☐ $$\frac{-24}{6}$$ ☐ -36 ÷ -9 ☐ -72 ÷ 18 ☐ $$\frac{84}{-21}$$ Answer: Options A,C, and D are correct. Explanation: In the above-given question, given that, the expressions have a quotient of -4 are: -24/6 = -4. -36/-9 = 4. -72/ 18 = -4. 84/-21 = -4. so options A, C, and D are correct. Question 2. Whitney 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? Answer: The integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds. Explanation: In the above-given question, given that, Whitney rolls a ball down a ramp that is 18 feet long. If the ball rolls down 2 feet each second. 18/2 = 9. so the integer represents the amount of time in seconds, the ball takes to reach the end of the ramp = 9 seconds. Question 3. Find the quotient. a. $$\frac{81}{-9}$$ b. -123 ÷ -4 c. –$$\frac{94}{4}$$ d. 65 ÷ (-5) Answer: The quotients are -9, 30.75, -23.5, and -13. Explanation: In the above-given question, given that, a: 81/-9 = -9. -123/-4 = 30.75. -94/4 = -23.5. 65 / -5 = -13. so the quotients are -9, 30.75, -23.5, and -13. Lesson 1.9 Divide Rational Numbers Quick Review The 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. Example Simplify $$\frac{-\frac{3}{4}}{\frac{15}{24}}$$ $$-\frac{3}{4} \div \frac{15}{24}=-\frac{3}{4} \cdot \frac{24}{15}=-\frac{72}{60}=-\frac{6}{5}=-1 \frac{1}{5}$$ Practice Find the quotient. Question 1. $$\frac{8}{9}$$ ÷ -1$$\frac{4}{15}$$ Answer: 8/9 / -1(4/15) = -0.38. Explanation: In the above-given question, given that, divide the numbers. $$\frac{8}{9}$$ ÷ -1$$\frac{4}{15}$$. 8/9 – 1(4/15). 0.88 – 19/15. 0.88 – 1.26. -0.38. Question 2. –3.6 ÷ 2$$\frac{1}{7}$$ Answer: –3.6 ÷ 2$$\frac{1}{7}$$ = -1.68. Explanation: In the above-given question, given that, divide the numbers. –3.6 ÷ 2$$\frac{1}{7}$$. -3.6 / 2(1/7). -3.6 / 2.14. -1.68. Question 3. A 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? Answer: The long will it take the anchor to reach the bottom of the lake = 25 feet. Explanation: In the above-given question, given that, A 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. 1.75 / 0.07 = 25. so the longer will it take the anchor to reach the bottom of the lake = 25 feet. Lesson 1.10 Solve Problems with Rational Numbers Quick Review You 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. Example During 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? -2$$\frac{3}{8}$$ ÷ 15 = –$$\frac{19}{8}$$ • $$\frac{1}{15}$$ = –$$\frac{19}{120}$$ inch Practice Question 1. In 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13. What integer represents her average score for the 5 rounds? Answer: The integer represents her average score for the 5 rounds = 2. Explanation: In the above-given question, given that, In 5 rounds of a game, Jill scored -3, 8, 9, -7, and 13. -3, 8, 9, -7, and 13. 10/5 = 2. so the integer represents her average score for the 5 rounds = 2. Question 2. Peter 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?

Answer:
The number represents the total change in the amount of money Peter has after paying for the service =$8.715. Explanation: In the above-given question, given that, Peter 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.
$10.50 x 5/6.$10.50 x 0.83.
$8.715. so the number represents the total change in the amount of money Peter has after paying for the service =$8.715.

Question 3.
Maggie 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?

Answer:
Maggie spends on cheese = 5.675 pounds.

Explanation:
In the above-given question,
given that,
Maggie 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.
\$4.05 + 1/8 + 1/4 + 1.25 = 5.3 + 0.125 + 0.25.
5.3 + 0.375.
5.675.
so the Maggie spend on cheese = 5.675 pounds.

### Topic 1 Fluency Practice

Crisscrossed
Find 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.
I can… add, subtract, multiply, and divide integers.

Across
2. 248 + (-1,027)
5. 818 – (-1,021)
6. -516 + 774
8. 242 + (-656)
9. 2,087 + (-1,359)
10. 631 – 897
11. –342 + 199
12. -49 • -27
13. –321 – 987
14. 2,988 ÷ -3
15. 2,580 ÷ 6
16. 4,592 ÷ -82
17. 48 • -27
18. -24 • 83
21. -118 + 1,201
22. -45 • -59

Answer:

Explanation:
In the above-given question,
given that,
248 – 1027 =

Down
1. 246 + 173
2. 22 • -22
3. 726 – (-219)
4. 501 – 699
7. -10,740 ÷ 15
8. 6,327 ÷ -9
10. 144 • -16
11. 15 • -67
12. 7,164 ÷ 4
13. –33 • 63
14. -2,695 ÷ 55
17. -1,032 – (-285)
18. 512 – 720
19. –729 + 951
20. –17 • -25