{"id":14972,"date":"2024-01-03T10:48:43","date_gmt":"2024-01-03T05:18:43","guid":{"rendered":"https:\/\/envisionmathanswerkey.com\/?p=14972"},"modified":"2024-01-04T10:01:07","modified_gmt":"2024-01-04T04:31:07","slug":"envision-math-common-core-grade-7-answer-key-topic-2","status":"publish","type":"post","link":"https:\/\/envisionmathanswerkey.com\/envision-math-common-core-grade-7-answer-key-topic-2\/","title":{"rendered":"enVision Math Common Core Grade 7 Answer Key Topic 2 Analyze And Use Proportional Relationships"},"content":{"rendered":"

Go through the\u00a0enVision Math Common Core Grade 7 Answer Key<\/a>\u00a0Topic 2 Analyze And Use Proportional Relationships<\/strong>\u00a0regularly and improve your accuracy in solving questions.<\/p>\n

enVision Math Common Core 7th Grade Answers Key Topic 2 Analyze And Use Proportional Relationships<\/h2>\n

Topic 2 Essential Question<\/h3>\n

How can you recognize and represent proportional relationships and use them to solve problems?<\/p>\n

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

\"Envision
\nMixin’ It Up Drinking plenty of water each day is important. Water is necessary for everything your body does. Not drinking enough water can lead to health problems. It’s even easier to drink enough water if you like the taste. There are many ways to make water more exciting. You can drink seltzer or filtered water. You can add fruit, vegetables, herbs, or flavor enhancers. You can add more or less based on what you like. Think about this during the 3-Act Mathematical Modeling lesson.
\n\"Envision<\/p>\n

Topic 2 enVision STEM Project<\/h3>\n

\"Envision<\/p>\n

Your Task: An Essential Resource
\nAccess to fresh, clean water is important for human survival. You and your classmates will determine how much fresh water is available on Earth for people to use. You will also explore ways in which people have developed access to clean water.
\n\"Envision<\/p>\n

Topic 2 Get Ready!<\/h3>\n

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

Vocabulary<\/strong>
\nChoose the best term from the box to complete each definition.<\/p>\n

    \n
  • complex fraction<\/li>\n
  • equivalent ratios<\/li>\n
  • rate<\/li>\n
  • ratio<\/li>\n
  • terms<\/li>\n<\/ul>\n

    Question 1.
    \nThe quantities x and y in the ratio \\(\\frac{x}{y}\\) are called _________<\/p>\n

    Answer:
    \nThe quantities x and y in the ratio \\(\\frac{x}{y}\\) are called terms.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe quantities x and y in the ratio \\(\\frac{x}{y}\\) are called terms.
    \nfor example:
    \nx . y = x\/y.<\/p>\n

    Question 2.
    \n\"Envision are an example of _________<\/p>\n

    Answer:
    \n2dogs\/3 cats and 10 dogs\/15 cats are an example of Rate.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n2 dogs\/3 cats and 10 dogs\/15 cats are an example of Rate.
    \nfor example:
    \na x c = b x b.
    \na x 40 = 15 x 15.
    \na x 40 = 225.
    \na = 225\/40.<\/p>\n

    \"Analyze<\/p>\n

    Question 3.
    \nA(n) __________ is a type of ratio that has both terms expressed in different units.<\/p>\n

    Answer:
    \nA(n) ratio is a type of ratio that has both terms expressed in different units.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA(n) ratio is a type of ratio that has both terms expressed in different units.
    \nfor example:
    \n2\/5 = 2 : 5<\/p>\n

    Question 4.
    \nA(n) _________ has a fraction in its numerator, denominator, or both.<\/p>\n

    Answer:
    \nA(n) complex fraction has a fraction in its numerator, denominator, or both.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA(n) complex fraction has a fraction in its numerator, denominator, or both.
    \nfor example:
    \n3\/5.
    \nwhere 3 is a numerator.
    \n5 is a denominator.<\/p>\n

    Equivalent Ratios<\/strong><\/p>\n

    Complete each equivalent ratio.
    \nQuestion 5.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equivalent ratio is 8 boys\/14 girls.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n4 boys\/7 girls = 8 boys\/ 14 girls.
    \n4 x 2 = 8.
    \n7 x 2 = 14.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 6.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equivalent ratio is 4 tires\/ 1 car.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n16 tires\/ 4 cars.
    \n4 x 1 = 4.
    \n4 x 4 = 16.
    \nso the equivalent ratio is 4 tires\/1 car.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 7.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equivalent ratio is 40 correct\/50 total.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n8 correct\/10 total.
    \n8 x 5 = 40.
    \n10 x 5 = 50.
    \nso the equivalent ratio is 40 correct.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    \"Analyze<\/p>\n

    Question 8.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equivalent ratio is 8 pearls\/10 opals.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthere are 16 apples and 20 opals.
    \n16 pearls \/ 20 opals.
    \n2 x 8 = 16.
    \n2 x 10 = 20.
    \nso the equivalent ratio is 10 opals.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 9.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equivalent ratio is 8 pencils\/2 erasers.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n32 pencils \/ 8 erasers.
    \n4 x 8 = 32.
    \n4 x 2 = 8.
    \n32\/8 = 8\/2.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 10.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equivalent ratio is 21 balls\/ 27 bats.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthere are 7 balls and 9 bats.
    \n7 x 3 = 21.
    \n9 x 3 = 27.
    \nso 21\/27 is the equivalent ratio.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Rates<\/strong><\/p>\n

    Write each situation as a rate.
    \nQuestion 11.
    \nJohn travels 150 miles in 3 hours.<\/p>\n

    Answer:
    \nThe rate is 50 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nJohn travels 150 miles in 3 hours.
    \n150 \/ 3 = 15 miles.
    \nso the rate is 50 miles.<\/p>\n

    Question 12.
    \nCameron ate 5 apples in 2 days.<\/p>\n

    Answer:
    \nThe rate is 2.5 apples in a day.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nCameron ate 5 apples in 2 days.
    \n5\/2 = 2.5.
    \nso the rate is 2.5 apples in a day.<\/p>\n

    Equations<\/strong><\/p>\n

    Write an equation that represents the pattern in the table.
    \nQuestion 13.
    \n\"Envision<\/p>\n

    Answer:
    \nThe equation is y = 3x.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nx = 4 so y = 3 x 4.
    \nx = 5 so y = 5 x 3.
    \nx = 6 so y = 6 x 3.
    \nx = 7 so y = 7 x 3.
    \nx = 8 so y = 8 x 3.<\/p>\n

    Language Development<\/strong><\/p>\n

    Complete the graphic organizer to help you understand new vocabulary terms.
    \n\"Envision<\/p>\n

    Answer:
    \nWhen two ratios are equivalent is called proportion.
    \nThe constant of proportionality k is the constant value of the ratio of two proportional quantities y and x.
    \ny = kx.
    \nA relationship between two varying quantities where the ratio between the two quantities remains constant.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe proportion is when two ratios are equivalent.
    \ncharacteristics:
    \nwhen each ratio is simplified they should be equivalent.
    \nwhen replacing the variable with the value the equation will balance.
    \nexample:
    \n3\/5 = a\/50.
    \n3 . 50 = a x 5.
    \n150 = 5a.
    \na = 30.
    \nnon-example:
    \n4\/6 is not equal to 8\/9.
    \nexample:
    \ny = kx.
    \nnon-example:
    \ny = 3x.
    \nconstant ratio is a graph that passes through the origin.
    \nthe constant ratio of delta y\/delta x.
    \nexample:
    \ny = 2x.
    \ny = -2x.
    \nnon-example:
    \ny = 2x+3.
    \ny = x X x.<\/p>\n

    \"Analyze<\/p>\n

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

    PROJECT 2A
    \nWho do you think would win a race involving different types of animals?
    \nPROJECT: PREDICTRACE RESULTS
    \n\"Envision<\/p>\n

    PROJECT 2B
    \nWhat would it be like to travel to another planet?
    \nPROJECT: CALCULATE THE WEIGHT OF YOUR PACK
    \n\"Envision<\/p>\n

    PROJECT 2C
    \nWhat stories can you tell?
    \nPROJECT: WRITE A SHORT STORY
    \n\"Envision<\/p>\n

    PROJECT 2D
    \nIf you could play any musical instrument, what would you play? Why?
    \nPROJECT: PLAY MUSIC
    \n\"Envision<\/p>\n

    Lesson 2.1 Connect Ratios, Rates, and Unit Rates<\/h3>\n

    Explain It!<\/strong>
    \nIn a basketball contest, Elizabeth made 9 out of 25 free throw attempts. Alex made 8 out of 20 free throw attempts. Janie said that Elizabeth had a better free-throw record because she made more free throws than Alex.
    \nI can… use ratio concepts and reasoning to solve multi-step problems.
    \n\"Envision<\/p>\n

    A. Critique Reasoning Do you agree with Janie’s reasoning? Explain.<\/p>\n

    Answer:
    \nYes, I agree with Janie’s reasoning.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn a basketball contest, Elizabeth made 9 out of 25 free throw attempts.
    \nAlex made 8 out of 20 free throw attempts.
    \nJanie said that Elizabeth had a better free-throw record because she made more free throws than Alex.
    \n9\/25 = 8\/20.
    \nso I agree with Janie’s reasoning.<\/p>\n

    B. Construct Arguments Decide who had the better free-throw record. Justify your reasoning using mathematical arguments.<\/p>\n

    Answer:
    \nBoth of them had an equal record.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn a basketball contest, Elizabeth made 9 out of 25 free throw attempts.
    \nAlex made 8 out of 20 free throw attempts.
    \nJanie said that Elizabeth had a better free-throw record because she made more free throws than Alex.
    \n9\/25 = 8\/20.
    \nso both of them had an equal record.<\/p>\n

    Focus on math practices
    \nConstruct Arguments What mathematical model did you use to justify your reasoning? Are there other models you could use to represent the situation?<\/p>\n

    Answer:
    \nBoth of them had an equal record.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn a basketball contest, Elizabeth made 9 out of 25 free throw attempts.
    \nAlex made 8 out of 20 free throw attempts.
    \nJanie said that Elizabeth had a better free-throw record because she made more free throws than Alex.
    \n9\/25 = 8\/20.
    \nso both of them had an equal record.<\/p>\n

    Essential Question<\/strong>
    \nHow are ratios, rates, and unit rates used to solve problems?<\/p>\n

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

    Jennifer is a lifeguard at the same pool. She earns $137.25 for 15 hours of lifeguarding. How much does Jennifer earn per hour?
    \nJennifer earns $ ________ per hour.
    \n\"Envision<\/p>\n

    Convince Me! What do you notice about the models used to find how much each lifeguard earns per hour?<\/p>\n

    Answer:
    \nJennifer earns per hour = $9.15<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nJennifer is a lifeguard at the same pool.
    \nShe earns $137.25 for 15 hours of lifeguarding.
    \n$137.25\/15.
    \n$9.15.
    \nso Jennifer earns per hour = $9.15.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

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

    A kitchen sink faucet streams 0.5 gallon of water in 10 seconds. A bathroom sink faucet streams 0.75 gallon of water in 18 seconds. Which faucet will fill a 3-gallon container faster?<\/p>\n

    Answer:
    \nThe faucet will fill a 3-gallon container faster = bathroom sink.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA kitchen sink faucet streams 0.5 gallons of water in 10 seconds.
    \nA bathroom sink faucet streams 0.75 gallons of water in 18 seconds.
    \nbathroom sink faucet streams 0.75 gallons of water in 18 seconds.
    \nso the faucet will fill a 3-gallon container faster = bathroom sink.<\/p>\n

    KEY CONCEPT<\/strong><\/p>\n

    You can use equivalent ratios and rates, including unit rates, to compare ratios and to solve problems.
    \n\"Envision<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question How are ratios, rates, and unit rates used to solve problems?<\/p>\n

    Answer:
    \nYes, the ratios, rates, and units are used to solve problems.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \na rate is a ratio that compares two quantities with different units.
    \nunit rate is a rate that has 1 as the denominator.
    \nfor example:
    \n24 pencils cost $1.20, how much will 6 pencils cost?
    \n$1.20\/24\u00a0 = $0.05\/1 pencil.
    \n5 cents per pencil.
    \n6 pencils will cost = 6 x 5 = 30.<\/p>\n

    Question 2.
    \nUse Structure Dorian buys 2 pounds of almonds for $21.98 and 3 pounds of dried apricots for $26.25. Which is less expensive per pound? How much less expensive?
    \nComplete the tables of equivalent ratios to help you solve.
    \n\"Envision<\/p>\n

    Answer:
    \nThe less expensive per pound = $2.135.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nDorian buys 2 pounds of almonds for $21.98 and 3 pounds of dried apricots for $26.25.
    \n$21.98\/2 = $10.99.
    \n$26.25\/2 = $13.125.
    \n$13.125 – $10.99 = $2.135.
    \nso the less expensive per pound = $2.135.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 3.
    \nGeneralize How are unit rates and equivalent ratios related?<\/p>\n

    Answer:
    \nThe unit rates and equivalent ratios are related.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA unit rate is a special kind of ratio, where the second number, or the denominator, is equal to one.
    \nfor example:
    \nunit rates are miles per gallon.
    \nwhen two ratios are equivalent, the two-unit rates will always be equal.
    \nfor example:
    \n10\/4 = 2.50.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nKrystal is comparing two Internet service plans. Plan 1 costs $34.99 per month. Plan 2 costs $134.97 every 3 months. If Krystal plans to stay with one service plan for 1 year, which should she choose? How much will she save?<\/p>\n

    Answer:
    \nKrystal will choose plan 1. She will save $20.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nKrystal is comparing two Internet service plans.
    \nPlan 1 costs $34.99 per month.
    \nPlan 2 costs $134.97 every 3 months.
    \n$34.99 x 3 = 104.97.
    \n$134.97 – $104.97 = $20.
    \nso Krystal will choose plan 1.
    \nshe will save $20.<\/p>\n

    \"Analyze<\/p>\n

    Question 5.
    \nPam read 126 pages of her summer reading book in 3 hours. Zack read 180 pages of his summer reading book in 4 hours. If they continue to read at the same speeds, will they both finish the 215-page book after 5 total hours of reading? Explain.<\/p>\n

    Answer:
    \nYes, they both will finish the 215-page book after 5 total hours of reading.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nPam read 126 pages of her summer reading book in 3 hours.
    \nZack read 180 pages of his summer reading book in 4 hours.
    \n180 – 126 = 54.
    \n180 + 54 = 234.
    \nso they both will finish the 215-page book after 5 total hours of reading.<\/p>\n

    Question 6.
    \nNora and Eli are making homemade spring rolls for a party. Nora can make 8 spring rolls in 10 minutes. Eli can make 10 spring rolls in 12 minutes. If they each make 40 spring rolls, who will finish first?<\/p>\n

    Answer:
    \nThey both will finish at an equal time.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nNora and Eli are making homemade spring rolls for a party.
    \nNora can make 8 spring rolls in 10 minutes.
    \nEli can make 10 spring rolls in 12 minutes.
    \nso both of them will finish at an equal time.<\/p>\n

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

    Leveled Practice In 7-8, complete the tables of equivalent ratios to solve.
    \nQuestion 7.
    \nAfter Megan walked 5 miles, her activity tracker had Meghan’s Steps counted 9,780 steps. David’s activity tracker had counted Miles 11,928 steps after he walked 6 miles. Suppose each person’s step covers about the same distance. Who takes more steps to walk 1 mile? How many more steps?
    \n\"Envision
    \n_________ takes more steps to walk 1 mile.
    \n_______ – ________ = _________ more steps for 1 mile<\/p>\n

    Answer:
    \nDavid takes more steps to walk 1 mile = 32 steps.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAfter Megan walked 5 miles, her activity tracker had Meghan’s Steps counted 9,780 miles.
    \nDavid’s activity tracker had counted Miles 11,928 steps after he walked 6 miles.
    \n9780\/5 = 1956 miles.
    \n11928\/6 = 1988 miles.
    \n1988 – 1956 = 32.
    \nso David takes more steps to walk 1 mile = 32 steps.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 8.
    \nA package of 5 pairs of insulated gloves costs $29.45. What is the cost of a single pair of gloves?
    \n\"Envision
    \nOne pair of gloves costs ________.<\/p>\n

    Answer:
    \nThe single pair of gloves = $5.89.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA package of 5 pairs of insulated gloves costs $29.45.
    \n$29.45\/5 = $5.89.
    \n$5.89 x 1 = $5.89.
    \nso the single pair of gloves = $5.89.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 9.
    \nWhich package has the lowest cost per ounce of rice?
    \n\"Envision<\/p>\n

    Answer:
    \nThe lowest cost per ounce of rice = white rice.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe pack of white Rice is 12 ounces = $4.56.
    \nthe pack of wheat is 18 ounces = $6.30.
    \nthe pack of white rice is 7 ounces = $2.59.
    \nso the lowest cost per ounce of rice = white rice.<\/p>\n

    Question 10.
    \nA nursery owner buys 5 panes of glass to fix some damage to her greenhouse. The 5 panes cost $14.25. Unfortunately, she breaks 2 more panes while repairing the damage. What is the cost of another 2 panes of glass?<\/p>\n

    Answer:
    \nThe cost of another 2 panes of glass = $5.7.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA nursery owner buys 5 panes of glass to fix some damage to her greenhouse.
    \nThe 5 panes cost $14.25.
    \nUnfortunately, she breaks 2 more panes while repairing the damage.
    \n$14.25 \/ 5 = $2.85.
    \n$2.85 x 2 = 5.7.
    \nso the cost of another 2 panes of glass = $5.7.<\/p>\n

    Question 11.
    \nBe Precise An arts academy requires there to be 3 teachers for every 75 students and 6 tutors for every 72 students. How many tutors does the academy need if it has 120 students?<\/p>\n

    Answer:
    \nThe tutors do the academy need if it has 120 students = 7 tutors.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn arts academy requires there to be 3 teachers for every 75 students and 6 tutors for every 72 students.
    \n75 + 72 = 147.
    \n3 + 6 = 9.
    \n140\/9 = 15.5.
    \n120 \/ 7 = 17.1.
    \nso the tutors does the academy need if it has 120 students = 7 tutors.<\/p>\n

    Question 12.
    \nMake Sense and Persevere in large cities, people often take taxis to get from one place to another. What is the cost per mile of a taxi ride? How much is a 47-mile taxi ride?
    \n\"Envision<\/p>\n

    Answer:
    \nThe much is a 47-mile taxi ride = 36.30 fares.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \npeople often take taxis to get from one place to another.
    \nfor 36 miles in 25.20 fare.
    \nfor 47 miles in 36.30 fare.
    \nso the much is a 47-mile taxi ride = 36.30 fares.<\/p>\n

    Question 13.
    \nThe Largo Middle School track team needs new uniforms. The students plan to sell plush toy tigers (the school mascot) for $5. The students find three companies online that sell stuffed mascots.
    \n\"Envision
    \na. Which company has the lowest cost per tiger?<\/p>\n

    Answer:
    \nThe company that has the lowest cost per tiger = company C.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe Largo Middle School track team needs new uniforms.
    \nThe students plan to sell plush toy tigers (the school mascot) for 5$.
    \ncompany A- 12 tigers for $33.24.
    \ncompany B – 14 tigers for $44.80.
    \ncompany C – 16 tigers for $41.10.
    \n33.24\/12 = 2.77.
    \n44.80\/14 = 3.2.
    \n41.10\/16 = 2.56.
    \nso the company that has the lowest cost per tiger is company C.<\/p>\n

    b. If they use that company, how much profit will the students make for each tiger sold?<\/p>\n

    Answer:
    \nThe profit will the students make for each tiger sold = $0.84.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe Largo Middle School track team needs new uniforms.
    \nThe students plan to sell plush toy tigers (the school mascot) for 5$.
    \ncompany A- 12 tigers for $33.24.
    \ncompany B – 14 tigers for $44.80.
    \ncompany C – 16 tigers for $41.10.
    \n33.24\/12 = 2.77.
    \n44.80\/14 = 3.2.
    \n41.10\/16 = 2.56.
    \n0.21 x 4 = $0.84.
    \nso the profit will the students make for each tiger sold = $0.84.<\/p>\n

    Question 14.
    \nA contractor purchases 7 dozen pairs of padded work gloves for $103.32. He incorrectly calculates the unit price at $14.76 per pair.
    \na. What is the correct unit price?<\/p>\n

    Answer:
    \nThe correct unit price = $723.24.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA contractor purchases 7 dozen pairs of padded work gloves for $103.32.
    \nHe incorrectly calculates the unit price at $14.76 per pair.
    \n$103.32 x 7 = $723.24.
    \nso the correct unit price = $723.24.<\/p>\n

    b. Critique Reasoning What error did the contractor likely make?<\/p>\n

    Answer:
    \nThe contractor likely makes the error if he calculated the unit price wrong.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA contractor purchases 7 dozen pairs of padded work gloves for $103.32.
    \nHe incorrectly calculates the unit price at $14.76 per pair.
    \n$103.32 x 7 = $723.24.
    \nso the correct unit price = $723.24.<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong> A warehouse store sells 5.5-ounce cans of tuna in packages of 6. A package of 6 cans costs $9.24. The store also sells 6.5-ounce cans of the same tuna in packages of 3 cans for $4.68. It also sells 3.5-ounce cans in packages of 4 cans for $4.48. Which package has the lowest cost per ounce of tuna?<\/p>\n

    Answer
    \nThe package has the lowest cost per ounce of tuna = 6.5 ounces.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA warehouse store sells 5.5-ounce cans of tuna in packages of 6.
    \nA package of 6 cans costs $9.24.
    \nThe store also sells 6.5-ounce cans of the same tuna in packages of 3 cans for $4.68.
    \nIt also sells 3.5-ounce cans in packages of 4 cans for $4.48.
    \nthe package that has the lowest cost per ounce of tuna = 6.5 ounces.<\/p>\n

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

    Question 16.
    \nLena is making two dishes for an event. Each batch of her mac-and-cheese recipe calls for 6 ounces of cheese and 2 tablespoons of basil. For every two pizzas, she needs 16 ounces of cheese and 5 tablespoons of basil.
    \nPART A
    \nLena buys a 32-oz package of cheese. Does she have enough cheese to make 2 batches of mac-and-cheese and 3 pizzas? Explain.<\/p>\n

    Answer:
    \nYes, she has enough cheese to make 2 batches of mac-and-cheese and 3 pizzas.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nLena buys a 32-oz package of cheese.
    \nDoes she have enough cheese to make 2 batches of mac-and-cheese and 3 pizzas?
    \nEach batch of her mac-and-cheese recipe calls for 6 ounces of cheese and 2 tablespoons of basil.
    \n6 x 2 = 12.
    \n3 x 2 = 6.
    \nso she has enough cheese to make 2 batches of mac-and-cheese and 3 pizzas.<\/p>\n

    PART B
    \nHow many tablespoons of basil does she need? Explain your answer.<\/p>\n

    Answer:
    \nThe number of tablespoons of basil does she need = 6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nLena buys a 32-oz package of cheese.
    \nDoes she have enough cheese to make 2 batches of mac-and-cheese and 3 pizzas?
    \nEach batch of her mac-and-cheese recipe calls for 6 ounces of cheese and 2 tablespoons of basil.
    \n6 x 2 = 12.
    \n3 x 2 = 6.<\/p>\n

    Question 17.
    \nIt cost Irene $58.90 to fill her car’s gas tank with 15 gallons of gas. Select all the rates that are equivalent to $58.90 for 155 gallons of gas.
    \n\u2610 $41.80 for 11 gallons of gas
    \n\u2610 $28.80 for 8 gallons of gas
    \n\u2610 $26.60 for 7 gallons of gas
    \n\u2610 $8.55 for 2 gallons of gas
    \n\u2610 $3.80 for 1 gallon of gas<\/p>\n

    Answer:
    \nOptions A, C, and E are correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIt cost Irene $58.90 to fill her car’s gas tank with 15 gallons of gas.
    \n$58.90\/15 = 3.9.
    \n$41.80\/11 = 3.8.
    \n28.80\/8 = 3.6.
    \n$26.60\/7 = 3.8.
    \n$8.55 \/2 = 4.27.
    \n$3.80\/1 = 3.8.
    \nso options A,C, and E are correct.<\/p>\n

    Lesson 2.2 Determine Unit Rates with Ratios of Fractions<\/p>\n

    Solve & Discuss It!<\/strong>
    \nAllison and her classmates planted bean seeds at the same time as Yuki and her classmates in Tokyo did. Allison is video-chatting with Yuki about their class seedlings. Assume that both plants will continue to grow at the same rate. Who should expect to have the taller plant at the end of the school year?<\/p>\n

    I can… find unit rates with ratios of fractions and use them to solve problems.
    \n\"Envision<\/p>\n

    Look for Relationships
    \nHow can you compare the growth rates of the seedlings?<\/p>\n

    Focus on math practices
    \nBe Precise What must the students do before they can compare the heights of the plants?<\/p>\n

    Answer:
    \nThe height of Yuki’s class is 5.5 cm in 5 days.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAllison and her classmates planted bean seeds at the same time as Yuki and her classmates in Tokyo did.
    \nthe Allison class is 2.5 inches in 5 days.
    \nso the height of Yuki’s class is 5.5 cm in 5 days.<\/p>\n

    Essential Question<\/strong>
    \nWhy is it useful to write a ratio of fractions as a unit rate?<\/p>\n

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

    Sergio increases his target speed to 30 miles per hour. How many more miles does Sergio need to ride in hour to achieve this target speed?
    \n\"Envision
    \nSergio must ride _______ miles in \\(\\frac{1}{4}\\) hour to achieve this target speed, so he needs to ride an additional ______ mile per \\(\\frac{1}{4}\\) hour.<\/p>\n

    Answer:
    \nThe more miles does Sergio needs to ride in an hour to achieve this target speed = 7.5 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nSergio increases his target speed to 30 miles per hour.
    \n30\/4 = 7.5.
    \nso the more does Sergio needs to ride in an hour to achieve this target speed = 7.5 miles.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Convince Me! How does the unit rate describe Sergio’s cycling speed? How is the unit rate helpful in determining how much farther Sergio must cycle in a given amount of time each time he increases his target speed?<\/p>\n

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

    Every other weekend, Bronwyn’s brother Daniel mows the lawn. He can mow 15,000 ft2<\/sup> in \\(\\frac{3}{4}\\) hour. Who mows the lawn in less time? Explain.<\/p>\n

    Answer:
    \nThe Bronwyn’s brother Daniel mows the lawn in less time = 20000 ft.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nEvery other weekend, Bronwyn’s brother Daniel mows the lawn.
    \nHe can mow 15,000 ft2<\/sup> in \\(\\frac{3}{4}\\) hour.
    \n15000\/0.75 = 20000.
    \nso the Bronwyn,s brother Daniel mows the lawn in less time = 20000 ft.<\/p>\n

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

    Sonoma bikes 5 miles to Paige’s house. On a map, they measure that distance as \\(\\frac{5}{6}\\) cm. The same map shows that the mall is 3\\(\\frac{1}{2}\\) cm from Paige’s house. What is the actual distance between Paige’s house and the mall?<\/p>\n

    Answer:
    \nThe actual distance between Paige’s house and the mall = 2.67 cm.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nSonoma bikes 5 miles to Paige’s house.
    \nOn a map, they measure that distance as \\(\\frac{5}{6}\\) cm.
    \nThe same map shows that the mall is 3\\(\\frac{1}{2}\\) cm from Paige’s house.
    \n3(1\/2) = 7\/2.
    \n7\/2 = 3.5.
    \n5\/6 = 0.83.
    \n3.5 – 0.83 = 2.67.
    \nso the actual distance between Paige’s house and the mall = 2.67.<\/p>\n

    KEY CONCEPT<\/strong>
    \nYou can use what you know about equivalent ratios and operations with fractions to write a ratio of fractions as a unit rate.
    \nTia skateboards \\(\\frac{2}{3}\\) mile for every \\(\\frac{1}{6}\\) hour.
    \n\"Envision
    \nShe skateboards 4 miles per hour.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question Why is it useful to write a ratio of fractions as a unit rate?<\/p>\n

    Answer:
    \nThe ratio of fractions as a unit rate = 4 miles per hour.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nTia skateboards \\(\\frac{2}{3}\\) mile for every \\(\\frac{1}{6}\\) hour.
    \n2\/3\/1\/6 = 2\/3 x 6\/1\/1\/6 x 6\/1.
    \n4\/1 = 4.<\/p>\n

    Question 2.
    \nUse Structure Jacob mixes \\(\\frac{1}{3}\\) cup of yellow paint for every \\(\\frac{1}{5}\\) cup of blue paint to make green paint. How many cups of yellow paint are needed for 1 cup of blue paint? Complete the table below.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of cups of yellow paint is needed for 1 cup of blue paint = 3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nUse Structure Jacob mixes \\(\\frac{1}{3}\\) cup of yellow paint for every \\(\\frac{1}{5}\\) cup of blue paint to make green paint.
    \nso the cups of yellow paint are 3.
    \nso the number of cups of yellow paint is needed for 1 cup of blue paint = 3.
    \n1\/3 x 3.
    \n1\/5 x 1.
    \nso the number of cups of yellow paint is needed for 1 cup of blue paint = 3.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 3.
    \nConstruct Arguments How is making a table of equivalent ratios to find the unit rate similar to finding the unit rate by calculating with fractions? Use a specific example to explain your reasoning.<\/p>\n

    Answer:
    \nThe unit rates and equivalent ratios are related.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA unit rate is a special kind of ratio, where the second number, or the denominator, is equal to one.
    \nfor example:
    \nunit rates are miles per gallon.
    \nwhen two ratios are equivalent, the two-unit rates will always be equal.
    \nfor example:
    \n10\/4 = 2.50.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nClaire boarded an airplane in Richmond, VA, and flew 414 miles directly to Charleston, SC. The total flight time was \\(\\frac{3}{4}\\) hour. How fast did Claire’s airplane fly, in miles per hour?<\/p>\n

    Answer:
    \nThe fast did Claire’s airplane fly, in miles per hour = 552 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nClaire boarded an airplane in Richmond, VA, and flew 414 miles directly to Charleston, SC.
    \nThe total flight time was \\(\\frac{3}{4}\\) hour.
    \n3\/4 = 0.75.
    \n414 \/ 0.75 = 552.
    \nso the fast did Claire’s airplane fly, in miles per hour = 552 miles.<\/p>\n

    Question 5.
    \nBrad buys two packages of mushrooms. Which mushrooms cost less per pound? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nThe mushrooms cost less per pound = Cremini.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nBrad buys two packages of mushrooms.
    \nCremini is $11.25 for 2\/3 lb.
    \nChanterelle is $7.99 for 1\/2 lb.
    \n$11.25 \/ 0.6 = 18.75.
    \n$7.99 \/ 0.5 = 15.98.
    \nso the Cremini cost less pound.<\/p>\n

    Question 6.
    \nJed is baking shortbread for a bake sale. The recipe calls for 1\\(\\frac{1}{4}\\) cups of flour and \\(\\frac{1}{2}\\) stick of butter. How many cups of flour will Jed need if he uses 3 sticks of butter?<\/p>\n

    Answer:
    \nThe number of cups of flour will Jed need if he uses 3 sticks of butter = 1.5 cups.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nJed is baking shortbread for a bake sale.
    \nThe recipe calls for 1\\(\\frac{1}{4}\\) cups of flour and \\(\\frac{1}{2}\\) stick of butter.
    \n1(1\/4) = 5\/4.
    \n5\/4 = 1.25.
    \n1\/2 = 0.5.
    \n3 x 0.5 = 1.5.
    \nso the number of cups of flour will Jed need if he uses 3 sticks of butter = 1.5 cups.<\/p>\n

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

    Leveled Practice In 7-10, fill in the boxes to find the unit rate.
    \nQuestion 7.
    \n\"Envision<\/p>\n

    Answer:
    \nThe 6 cups of sugar and 1 cup of butter.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n3\/4 cups of sugar and 1\/8 cups of butter.
    \n3\/4\/1\/8.
    \n3\/4 x 8\/1.
    \n24\/6 = 6\/1.
    \nso 6 cups of sugar and 1 cup of butter.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 8.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of miles in an hour = 1.8 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n3\/5 miles in 1\/3 hour.
    \n3\/5\/1\/3.
    \n3\/5 x 3\/1.
    \n9\/5 = 1.8.
    \nso the number of miles in an hour = 1.8 miles.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 9.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of miles per gallon = 21 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n7 miles in 1\/3 gallons.
    \n7\/1\/3 = 7\/1 x 3\/1.
    \n7\/3 = 2.33.
    \nso the number of miles per gallon = 21 miles.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 10.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of pages in 1 minute = 6\/4 pages.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n3\/4 pages in 2 minutes.
    \n3\/4\/2.
    \n3\/4\/2\/1.
    \n3\/4 x 2\/1.
    \n6\/4.
    \n3\/2.
    \nso the number of pages in 1 minute = 3\/2 pages.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 11.
    \nHadley paddled a canoe \\(\\frac{2}{3}\\) mile in \\(\\frac{1}{4}\\) hour. How fast did Hadley paddle, in miles per hour?<\/p>\n

    Answer:
    \nThe fast did Hadley paddle, in miles per hour = 2.66 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nHadley paddled a canoe \\(\\frac{2}{3}\\) mile in \\(\\frac{1}{4}\\) hour.
    \n2\/3\/1\/4.
    \n2\/3 x 4\/1.
    \n8\/3 = 2.66.
    \nso the fast did Hadley paddle, in miles per hour = 2.66 miles.<\/p>\n

    Question 12.
    \nA box of cereal states that there are 90 Calories in a \\(\\frac{3}{4}\\)cup serving. How many Calories are there in 4 cups of the cereal?<\/p>\n

    Answer:
    \nThe number of Calories is there in 4 cups of cereal = 30 calories.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA box of cereal states that there are 90 calories in a 3\/4 cup serving.
    \n90\/3\/4.
    \n90\/1 x 4\/3.
    \n360\/3 = 120.
    \n120\/4 = 30.
    \nso the number of Calories is there in 4 cups of cereal = 30 calories.<\/p>\n

    Question 13.
    \nA robot can complete 8 tasks in \\(\\frac{5}{6}\\) hour. Each task takes the same amount of time.
    \na. How long does it take the robot to complete one task?<\/p>\n

    Answer:
    \nThe long does it take the robot to complete one task = 9.6 minutes.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA robot can complete 8 tasks in \\(\\frac{5}{6}\\) hour.
    \n8\/ 5\/6.
    \n8\/1 x 6\/5.
    \n48\/5 = 9.6.
    \nso the long does it take the robot to complete one task = 9.6 minutes.<\/p>\n

    b. How many tasks can the robot complete in one hour?<\/p>\n

    Answer:
    \nThe tasks can complete in one hour = 576 minutes.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA robot can complete 8 tasks in \\(\\frac{5}{6}\\) hour.
    \n8\/ 5\/6.
    \n8\/1 x 6\/5.
    \n48\/5 = 9.6.
    \n9.6 x 60 = 576.
    \nso the tasks can complete in one hour = 576 minutes.<\/p>\n

    Question 14.
    \nYou are running a fuel economy study. You want to find out which car can travel a greater distance on 1 gallon of gas.
    \n\"Envision
    \na. What is the gas mileage, in miles per gallon, for the blue car?<\/p>\n

    Answer:
    \nThe gas mileage, in miles per gallon, for the blue car = 0.04 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n1(1\/2) gallons of gas in 35(1\/2).
    \n3\/2 \/ 71\/2.
    \n3\/2 x 2\/71.
    \n6\/ 142 = 0.04.
    \nso the gas mileage, in miles per gallon, for the blue car = 0.04 miles.<\/p>\n

    b. What is the gas mileage, in miles per gallon, for the silver car?<\/p>\n

    Answer:
    \nThe gas mileage, in miles per gallon, for the silver car = 34 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n4\/5 gallons of gas in 27(1\/5) miles.
    \n4\/5 \/ 136\/5.
    \n4\/5 x 5\/136.
    \n4\/136 = 34.
    \nso the gas mileage, in miles per gallon, for the silver car = 34 miles.<\/p>\n

    c. Which car could travel the greater distance on 1 gallon of gas?<\/p>\n

    Answer:
    \nThe silver color car travels a greater distance on 1 gallon of gas.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n4\/5 gallons of gas in 27(1\/5) miles.
    \n4\/5 \/ 136\/5.
    \n4\/5 x 5\/136.
    \n4\/136 = 34.
    \nso the gas mileage, in miles per gallon, for the silver car = 34 miles.<\/p>\n

    Question 15.
    \nHenry incorrectly said the rate \\(\\frac{\\frac{1}{5} \\text { pound }}{\\frac{1}{20} \\text { quart }} \\) can be written as the unit rate 100 pound per quart.
    \na. What is the correct unit rate?<\/p>\n

    Answer:
    \nThe correct unit rate = 4 pounds.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nHenry incorrectly said the rate \\(\\frac{\\frac{1}{5} \\text { pound }}{\\frac{1}{20} \\text { quart }} \\).
    \n1\/5 \/1\/20.
    \n1\/5 x 20\/1.
    \n4.
    \nso the correct unit rate = 4 pounds.<\/p>\n

    b. Critique Reasoning What error did Henry likely make?<\/p>\n

    Answer:
    \nThe error did Henry likely make = 4 pounds.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nHenry incorrectly said the rate \\(\\frac{\\frac{1}{5} \\text { pound }}{\\frac{1}{20} \\text { quart }} \\).
    \n1\/5 \/1\/20.
    \n1\/5 x 20\/1.
    \n4.
    \nso the correct unit rate = 4 pounds.<\/p>\n

    Question 16.
    \nHigher Order Thinking<\/strong> Ari walked 2\\(\\frac{3}{4}\\) miles at a constant speed of 2\\(\\frac{1}{2}\\) miles per hour. Beth walked 1\\(\\frac{3}{4}\\) miles at a constant speed of 1\\(\\frac{1}{4}\\) miles per hour. Cindy walked for 1 hour and 21 minutes at a constant speed of 1\\(\\frac{1}{8}\\) miles per hour. List the three people in order of the times they spent walking from least time to greatest time.<\/p>\n

    Answer:
    \nThe three people in order of the times they spent walking from least time to greatest time = Cindy, Ari, and Beth.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAri walked 2\\(\\frac{3}{4}\\) miles at a constant speed of 2\\(\\frac{1}{2}\\) miles per hour.
    \nBeth walked 1\\(\\frac{3}{4}\\) miles at a constant speed of 1\\(\\frac{1}{4}\\) miles per hour.
    \nCindy walked for 1 hour and 21 minutes at a constant speed of 1\\(\\frac{1}{8}\\) miles per hour.
    \n2(3\/4) \/ 2(1\/2) = 11\/4 \/ 5\/2.
    \n11\/4 x 2\/5 = 22\/20 = 11\/10.
    \n1(3\/4) \/ 1(1\/4) = 7\/4 \/ 5\/4.
    \n7\/4 x 4\/5 = 28\/20 = 7\/5.
    \n1\/ 21 \/ 9\/8 = 1\/21 x 8\/9 = 8\/189 = 0.04.
    \nso the constant speed from least to great is Cindly, Ari, and Beth.<\/p>\n

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

    Question 17.
    \nA blueprint shows a house with two fences. Fence A is 1\\(\\frac{4}{5}\\) inches long on the blueprint and is to be 1\\(\\frac{1}{2}\\) feet long. How long is Fence B on the blueprint?
    \n\"Envision<\/p>\n

    Answer:
    \nThe long is Fence B on the blueprint = 2 inches.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA blueprint shows a house with two fences.
    \nFence A is 1\\(\\frac{4}{5}\\) inches long on the blueprint and is to be 1\\(\\frac{1}{2}\\) feet long.
    \n1(4\/5) \/ 1(1\/2).
    \n9\/5 \/ 3\/2.
    \n9\/5 x 2\/3 = 18\/15 = 6\/3.
    \nso the long is Fence B on the blueprint = 2 inches.<\/p>\n

    Question 18.
    \nLeo reads 13 pages in \\(\\frac{1}{3}\\) hour. Use the table to find how many pages he reads in one hour.
    \n\"Envision
    \nLeo reads _________ pages in one hour.<\/p>\n

    Answer:
    \nThe number of pages he reads in one hour = 39 pages.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nLeo reads 13 pages in \\(\\frac{1}{3}\\) hour.
    \n13 \/1\/3.
    \n13\/1 x 3 \/1 = 39 \/1.
    \nso the number of pages he reads in one hour = 39 pages.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Lesson 2.3 Understand Proportional Relationships: Equivalent Ratios<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nWeight is a measure of force affected by gravity. The Moon’s gravity is less than Earth’s gravity, so objects weigh less on the Moon than on Earth. Using the information provided, how much do you think a cat will weigh on the Moon? Explain your reasoning.<\/p>\n

    I can… test for equivalent ratios to decide whether quantities are in a proportional relationship.
    \n\"Envision<\/p>\n

    Make Sense and Persevere
    \nAbout how much does a cat weigh on Earth?<\/p>\n

    Focus on math practices
    \nGeneralize How could you find the approximate weight of any object on the Moon? Explain your reasoning.<\/p>\n

    Essential Question<\/strong>
    \nHow are proportional quantities described by equivalent ratios?<\/p>\n

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

    Miles records the time it takes to download a variety of file types. How is the download time related to the file size? Explain.
    \n\"Envision
    \nThe ratios for each pair of data are ___________, so the download time and the file size are __________.<\/p>\n

    Answer:
    \nThe ratios for each pair of data are the same.
    \nso the download time and the file size are different.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMiles records the time it takes to download a variety of file types.
    \ndownloaded time \/ file size = 25\/1.25 = 20.
    \n72\/3.6 = 20.
    \n125\/6.25 = 20.
    \nso the ratios for each pair of data are the same.
    \nso the download time and the file size are different.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Convince Me!
    \nHow can you show that two quantities have a proportional relationship?<\/p>\n

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

    The table at the right shows information about regular hexagons. Is the relationship between the perimeter and the side length of the hexagons proportional? Explain.
    \n\"Envision
    \nAnswer:<\/p>\n

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

    Ginny’s favorite cookie recipe requires 1\\(\\frac{1}{3}\\) cups of sugar to make 24 cookies. How much sugar does Ginny need to make 36 of these cookies?
    \nAnswer:<\/p>\n

    KEY CONCEPT<\/strong>
    \nTwo quantities x and y have a proportional relationship if all the ratios \\(\\frac{y}{x}\\) for related pairs of x and y are equivalent.
    \nA proportion is an equation that states that two ratios are equivalent
    \n\"Envision
    \n\\(\\frac{3}{120}\\) = \\(\\frac{5}{200}\\)<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question How are proportional quantities described by equivalent ratios?<\/p>\n

    Answer:
    \nTwo quantities x and y have a proportional relationship if all the ratios y\/x for related pairs of x and y are equivalent.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nx = 120 and y = 3.
    \ny\/x = 120\/3 = 0.025.
    \nx = 200 and y = 5.
    \ny\/x = 200\/5 = 0.025.
    \nA proportion is an equation that states that two ratios are equivalent.<\/p>\n

    Question 2.
    \nLook for Relationships How do you know if a relationship between two quantities is NOT proportional?<\/p>\n

    Answer:
    \nTwo quantities x and y have a proportional relationship if all the ratios y\/x for related pairs of x and y are equivalent.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nx = 120 and y = 3.
    \ny\/x = 120\/3 = 0.025.
    \nx = 200 and y = 5.
    \ny\/x = 200\/5 = 0.025.
    \nA proportion is an equation that states that two ratios are equivalent.<\/p>\n

    Question 3.
    \nReasoning<\/strong> If the ratio \\(\\frac{y}{x}\\) is the same for all related pairs of x and y, what does that mean about the relationship between x and y?<\/p>\n

    Answer:
    \nTwo quantities x and y have a proportional relationship if all the ratios y\/x for related pairs of x and y are equivalent.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nx = 120 and y = 3.
    \ny\/x = 120\/3 = 0.025.
    \nx = 200 and y = 5.
    \ny\/x = 200\/5 = 0.025.
    \nA proportion is an equation that states that two ratios are equivalent.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nUse the table below. Do x and y have a proportional relationship? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nYes, they have a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nx = 2, 3, 5, and 8.
    \ny = 5, 7.5, 12.5, and 18.
    \ny\/x = 5\/2 = 2.5.
    \ny\/x = 7.5\/3 = 2.5.
    \ny\/x = 12.5\/5 = 2.5.
    \ny\/x = 18\/8 = 2.25.
    \nso x and y have a proportioal relationship.<\/p>\n

    Question 5.
    \nEach triangle is equilateral. Is the relationship between the perimeter and the side length of the equilateral triangles proportional? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nYes, the relationship between the perimeter and the side length of the equilateral triangles are proportional.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthere are 3 triangles.
    \nthey are 1 in, 2 in, and 3 in.
    \nso the sides of all the sides are equal in every triangle.
    \nso the relationship between the perimeter and the side length of the equilateral triangles are proportional.<\/p>\n

    Question 6.
    \nIs the relationship between the number of tickets sold and the number of hours proportional? If so, how many tickets were sold in 8 hours?
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of tickets was sold in 8 hours = 640 tickets.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number of tickets sold in the hours.
    \nh = 3, 5, 9.
    \nt = 240, 400, and 720.
    \nt\/h = 240\/3 = 80.
    \nt\/h =\u00a0 400\/5 = 80.
    \nt\/h = 720\/9 = 80.
    \n80 x 8 = 640.
    \nso the number of tickets was sold in 8 hours = 640 tickets.<\/p>\n

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

    Question 7.
    \nThe amount of seed a landscaper uses and the area of lawn covered have a proportional relationship. Complete the table.
    \n\"Envision<\/p>\n

    Answer:
    \nYes, they have a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe amount of seed a landscaper uses and the area of lawn covered have a proportional relationship.
    \nthe area covered is 50, 75, and 100 sq ft.
    \nthe seeds are 2, 3, and 4.
    \narea covered\/seed = 50\/2 = 25\/1.
    \n75\/3 = 25\/1.
    \n100\/4 = 25\/1.
    \nso they both have a proportional relationship.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 8.
    \nConstruct Arguments is the relationship between the number of slices of salami in a sandwich and the number of Calories proportional? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nNo, they don’t have a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe calories in a sandwich.
    \nslices of salami in calories are given.
    \n66\/1 = 66.
    \n96\/2 = 48.
    \n126\/3 = 42.
    \n156\/4 = 39.
    \nso they did not have a proportional relationship.<\/p>\n

    Question 9.
    \nLook for Relationships A wholesale club sells eggs by the dozen. Does the table show a proportional relationship between the number of dozens of eggs and the cost? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of dozens of eggs and the cost are proportional.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe cost of dozens of eggs.
    \ncost in dollars is 21, 28, 35, and 49.
    \ndozens are 6, 8, 10, and 14.
    \ncost\/dozen = 21\/6 =3.5.
    \n28\/8 = 3.5
    \n35\/10 = 3.5
    \n49\/14 = 3.5.
    \nso the number of dozens of eggs and the cost are proportional.<\/p>\n

    Question 10.
    \nDoes the table show a proportional relationship? If so, what is the value of y when x is 11?
    \n\"Envision<\/p>\n

    Answer:
    \nNo, they don’t show a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe values of x and y are given.
    \ny = 64\/4 = 16.
    \ny = 125\/5 = 25.
    \ny = 216\/6 = 36.
    \ny = 1000\/10 = 100.
    \ny =
    \nso they don’t have a proportional relationship.<\/p>\n

    Question 11.
    \nDoes the table show a proportional relationship? If so, what is the value of y when x is 10?
    \n\"Envision<\/p>\n

    Answer:
    \nThe value of y when x is 10 is 3.2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe value of y is 1(2\/3), 2, 2(1\/3), and 2(2\/3).
    \nthe value of x = 5, 6, 7, and 8.
    \ny\/x = 1.6\/5 = 0.32.
    \ny\/x = 2\/6 = 0.33.
    \ny\/x = 2.3\/7 = 0.32.
    \ny\/x = 2.6\/8 = 0.32.
    \ny\/x = 3.2\/10 = 0.32.
    \nso the value of y when x is 10 is 3.2.<\/p>\n

    Question 12.
    \nThe height of a building is proportional to the number of floors. The figure shows the height of a building with 9 floors.
    \n\"Envision
    \na. Reasoning Write the ratio of height of the building to the number of floors. Then find the unit rate, and explain what it means in this situation.<\/p>\n

    Answer:
    \nThe unit rate = 15.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe height of a building is proportional to the number of floors.
    \nthe number of floors = 9.
    \nthe height of the building = 135 feet.
    \n135\/9 = 15.
    \nso the unit rate is 15.<\/p>\n

    b. How tall would the building be if it had 15 floors?<\/p>\n

    Answer:
    \nThe height of the building is 9.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe height of a building is proportional to the number of floors.
    \nthe number of floors = 15.
    \nthe height of the building = 135 feet.
    \n135\/15 = 9.
    \nso the height of the building is 9.<\/p>\n

    Question 13.
    \nHigher Order Thinking<\/strong> Do the two tables show the same proportional relationship between x and y? Explain how you know.
    \n\"Envision<\/p>\n

    Answer:
    \nYes, both the tables have the proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe two tables show the same proportional relationship between x and y.
    \nthe values of x = 160, 500, and 1200.
    \nthe values of y = 360, 1125, and 2700.
    \ny\/x = 360\/160 = 2.25.
    \ny\/x = 1125\/500 = 2.25.
    \ny\/x = 2500\/1200 = 2.08.
    \ny\/x = 4.5\/2 = 2.25.
    \ny\/x = 11.25\/5 = 2.25.
    \ny\/x = 15.75\/7 = 2.25.
    \nso both the tables have a proportional relationship.<\/p>\n

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

    Question 14.
    \nThe table shows the number of cell phone towers a company will build as the number of its customers increases.
    \n\"Envision
    \nPART A
    \nIs the relationship between the number of towers and the number of customers proportional? Explain.<\/p>\n

    Answer:
    \nYes, both of them have a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number of cell phone towers a company will build as the number of its customer’s increases.
    \ntowers are 252, 300, 348, and 444.
    \ncustomers(thousands) = 5.25, 6.25, 7.25, and 9.25.
    \n252\/5.25 = 48.
    \n300\/6.25 = 48.
    \n348\/7.25 = 48.
    \n444\/9.25 = 48.
    \nso both of them have a proportional relationship.<\/p>\n

    PART B
    \nIf there are 576 towers, how many customers does the company have? Write a proportion you can use to solve.<\/p>\n

    Answer:
    \nThe number of customers does the company has = 56.1.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nif there are 576 towers.
    \nthe number of customers is 10.25.
    \n576 \/ 10.25 = 56.1.
    \nso the number of customers does the company have = 56.1.<\/p>\n

    Question 15.
    \nSelect all true statements about the table at the right.
    \n\"Envision
    \n\u2610 The table shows a proportional relationship.
    \n\u2610 When x is 20, y is 2.55.
    \n\u2610 All the ratios \\(\\frac{y}{x}\\) for related pairs of x and y are equivalent to 8.
    \n\u2610 The unit rate of \\(\\frac{y}{x}\\) for related pairs of x and y is \\(\\frac{1}{8}\\).
    \n\u2610 If the table continued, when x is 30, y would be 3.75.<\/p>\n

    Answer:
    \nOptions A, B, and E are correct.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe values of x and y are shown.
    \ny\/x = 1.5\/12 = 0.125.
    \ny\/x = 2.25\/18 = 0.125.
    \ny\/x = 2.75\/22 = 0.125.
    \ny\/x = 3.25\/26 = 0.125.
    \nso options A, B, and E are correct.<\/p>\n

    Lesson 2.4 Describe Proportional Relationships: Constant of Proportionality<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nJamal can run 1 mile in 5.05 minutes. If Jamal maintains this pace during a 5-kilometer (5K) race, he expects to break the course record of 15.25 minutes. Is Jamal’s expectation reasonable? Explain.<\/p>\n

    Answer:
    \nYes, Jamal’s expectation is reasonable.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nJamal can run 1 mile in 5.05 minutes.
    \nIf Jamal maintains this pace during a 5-kilometer (5K) race, he expects to break the course record of 15.25 minutes.
    \n15.25\/5 = 3.05.
    \nso Jamal’s expectation is reasonable.<\/p>\n

    I can… use the constant of proportionality in an equation to represent a proportional relationship
    \n\"Envision<\/p>\n

    Be Precise How can you convert 5 kilometers to miles?<\/p>\n

    Focus on math practices
    \nReasoning Assuming that Jamal runs at a constant rate, how does his pace describe the time it takes him to finish a race of any length?<\/p>\n

    Essential Question<\/strong>
    \nHow can you represent a proportional relationship with an equation?<\/p>\n

    Answer:
    \nYes, Jamal’s expectation is reasonable.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nJamal can run 1 mile in 5.05 minutes.
    \nIf Jamal maintains this pace during a 5-kilometer (5K) race, he expects to break the course record of 15.25 minutes.
    \n15.25\/5 = 3.05.
    \nso Jamal’s expectation is reasonable.<\/p>\n

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

    Maria made two batches of fruit punch. The table at the right shows how many quarts of juice she used for each batch. Write an equation that relates the proportional quantities.
    \n\"Envision
    \nThe constant of proportionality is __________.
    \nAn equation that represents this proportional relationship is y = ________ x.<\/p>\n

    Answer:
    \nThe constant of proportionality is 1.6.
    \nAn equation that represents this proportional relationship is y = kx.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMaria made two batches of fruit punch.
    \nthe grape juice contains 8 and 16.
    \napple juice contains 5 and 10.
    \ny\/x = 8\/5 = 1.6.
    \ny\/x = 16\/10 = 1.6.
    \nan equation that represents this proportional relationship is y = kx.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Convince Me! How does the equation change if the amount of grape juice is the independent variable, x, and the amount of apple juice is the dependent variable, y?<\/p>\n

    Answer:
    \nAn equation that represents this proportional relationship is y = kx.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMaria made two batches of fruit punch.
    \nthe grape juice contains 8 and 16.
    \napple juice contains 5 and 10.
    \ny\/x = 8\/5 = 1.6.
    \ny\/x = 16\/10 = 1.6.
    \nan equation that represents this proportional relationship is y = kx.<\/p>\n

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

    A florist sells a dozen roses for $35.40. She sells individual roses for the same unit cost. Write an equation to represent the relationship between the number of roses, x, and the total cost of the roses, y. How much would 18 roses cost?<\/p>\n

    Answer:
    \nThe cost of 18 roses is $1.96.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA florist sells a dozen roses for $35.40.
    \nShe sells individual roses for the same unit cost.
    \n$35.40\/18 = 1.96.
    \nso the cost of 18 roses is $1.96.<\/p>\n

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

    Balloon A is released 5 feet above the ground. Balloon B is released at ground level. Both balloons rise at a constant rate.
    \n\"Envision
    \nWhich situation can you represent using an equation of the form y = kx? Explain.<\/p>\n

    Answer:
    \nBalloon A does not have a proportional relationship.
    \nBalloon B has a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nBalloon A is released 5 feet above the ground.
    \nBalloon B is released at ground level.
    \ny = kx.
    \nk = y\/x.
    \nk = 9\/1 = 9.
    \nk = 13\/2 = 6.5.
    \nk = 17\/3 = 5.6.
    \nk = 4\/1 = 4.
    \nk = 8\/2 = 4.
    \nk = 12\/3 = 4.<\/p>\n

    KEY CONCEPT<\/strong>
    \nTwo proportional quantities x and y are related by a constant multiple, or the constant of proportionality, k.
    \nYou can represent a proportional relationship using the equation y = kx.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question How can you represent a proportional relationship with an equation?<\/p>\n

    Answer:
    \nTwo proportional quantities x and y are related by a constant multiple, or the constant of proportionality, K.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nTwo proportional quantities x and y are related by a constant multiple, or the constant of proportionality, k.
    \nYou can represent a proportional relationship using the equation y = kx.<\/p>\n

    Question 2.
    \nGeneralize<\/strong> How can you use an equation to find an unknown value in a proportional relationship?<\/p>\n

    Answer:
    \nThe unknown value in a proportional relationship is y = kx.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nTwo proportional quantities x and y are related by a constant multiple, or the constant of proportionality, k.
    \nYou can represent a proportional relationship using the equation y = kx.<\/p>\n

    Question 3.
    \nReasoning<\/strong> Why does the equation y = 3x + 5 NOT represent a proportional relationship?<\/p>\n

    Answer:
    \nThe equation y = 3x + 5 not represent a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe equation y = 3x + 5.
    \nwhere k = 3.
    \nso the equation does not represent a proportional relationship.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nDetermine whether each equation represents a proportional relationship. If it does, identify the constant of proportionality.
    \na. y = 0.5x – 2<\/p>\n

    Answer:
    \nThe constant of proportionality = 0.5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe equation is y = 0.5x – 2.
    \nthe constant of proportionality = 0.5.<\/p>\n

    b. y = 1,000x<\/p>\n

    Answer:
    \nThe constant of proportionality = 1000.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe equation is y = kx.
    \nwhere k = 1000.
    \nso the constant of proportionality = 1000.<\/p>\n

    c. y = x + 1<\/p>\n

    Answer:
    \nThe constant of proportionality = x.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe equation is y = kx.
    \nwhere k = x.
    \nso the constant of proportionality = x.<\/p>\n

    Question 5.
    \nThe manager of a concession stand estimates that she needs 3 hot dogs for every 5 people who attend a baseball game. If 1,200 people attend the game, how many hot dogs should the manager order?<\/p>\n

    Answer:
    \nThe number of hot dogs should the manager order = 80.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe manager of a concession stand estimates that she needs 3 hot dogs for every 5 people who attend a baseball game.
    \ny = kx.
    \ny = 5 x 3.
    \ny = 15.
    \n1200 \/ 15 = 80.
    \nso the number of hot dogs should the manager order = 80.<\/p>\n

    Question 6.
    \nA half dozen cupcakes cost $15. What constant of proportionality relates the number of cupcakes and total cost? Write an equation that represents this relationship.<\/p>\n

    Answer:
    \nThe equation represents the relationship = $15.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA half dozen cupcakes cost $15.
    \nthe constant of proportionality relates to the number of cupcakes and the total cost is y = kx.
    \ny = 15x.
    \nso the equation represents the relationship = $15.<\/p>\n

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

    Question 7.
    \nWhat is the constant of proportionality in the equation y = 5x?<\/p>\n

    Answer:
    \nThe constant of proportionality in the equation is 5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe constant of proportionality in the equation is 5.
    \ny = 5x.<\/p>\n

    Question 8.
    \nWhat is the constant of proportionality in the equation y = 0.41x?<\/p>\n

    Answer:
    \nThe constant of proportionality in the equation y = 0.41x.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe equation is y = 0.41x.
    \nk = 0.41.
    \nso the constant of proportionality in the equation y = 0.41x.<\/p>\n

    Question 9.
    \nThe equation P= 3s represents the perimeter P of an equilateral triangle with side length s. Is there a proportional relationship between the perimeter and the side length of an equilateral triangle? Explain.<\/p>\n

    Answer:
    \nYes, there is a proportional relationship between the perimeter and the side length of an equilateral triangle.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe equation P= 3s represents the perimeter P of an equilateral triangle with side length s.
    \nthe triangle has 3 sides and all the sides has equal length.
    \np = 3 x s.
    \nso there is a proportional relationship between the perimeter and the side length of an equilateral triangle.<\/p>\n

    Question 10.
    \nModel with Math In a chemical compound, there are 3 parts zinc for every 16 parts copper, by mass. A piece of the compound contains 320 grams of copper. Write and solve an equation to determine the amount of zinc in the chemical compound.<\/p>\n

    Answer:
    \nThe equation to solve the amount of zinc in the chemical compound is y = 960.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIn a chemical compound, there are 3 parts zinc for every 16 parts copper, by mass.
    \nA piece of the compound contains 320 grams of copper.
    \ny = kx.
    \ny = 3 x 320.
    \ny = 960.
    \nso the equation to solve the amount of zinc in the chemical compound is y = 960.<\/p>\n

    Question 11.
    \nThe weight of 3 eggs is shown. Assuming the three eggs are all the same weight, find the constant of proportionality.
    \n\"Envision<\/p>\n

    Answer:
    \nThe constant of proportionality = 40.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe weight of 3 eggs is shown.
    \nthe constant of proportionality is:
    \n120\/3 = 40.
    \nso the constant of proportionality = 40.<\/p>\n

    Question 12.
    \nThe height of a stack of DVD cases is proportional to the number of cases in the stack. The height of 6 DVD cases is 114 mm.
    \na. Write an equation that relates the height, y, of a stack of DVD cases and the number of cases, x, in the stack.<\/p>\n

    Answer:
    \nThe equation that relates the height is 19 mm.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe height of a stack of DVD cases is proportional to the number of cases in the stack.
    \nThe height of 6 DVD cases is 114 mm.
    \n114\/6 = 19.
    \nso the equation that relates the height is 19.<\/p>\n

    b. What would be the height of 13 DVD cases?<\/p>\n

    Answer:
    \nThe height of 13 DVD cases = 8.769 mm.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe height of a stack of DVD cases is proportional to the number of cases in the stack.
    \nThe height of 13 DVD cases is 114 mm.
    \n114\/13 = 8.769.
    \nso the height of 13 DVD cases = 8.769 mm.<\/p>\n

    Question 13.
    \nAnn’s car can travel 228 miles on 6 gallons of gas.
    \na. Write an equation to represent the distance, y, in miles Ann’s car can travel on x gallons of gas.<\/p>\n

    Answer:
    \nThe equation to represent the distance is 38.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAnn’s car can travel 228 miles on 6 gallons of gas.
    \ny = kx.
    \n228 = k . 6.
    \nk = 228\/6.
    \nk = 38.
    \nso the equation to represent the distance is 38.<\/p>\n

    b. Ann’s car used 7 gallons of gas during a trip. How far did Ann drive?<\/p>\n

    Answer:
    \nThe far did Ann drive = 32.57.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAnn’s car used 7 gallons of gas during a trip.
    \ny = kx.
    \n228\/7 = 32.57.
    \nso the far did Ann drive = 32.57.<\/p>\n

    Question 14.
    \nThe value of a baseball player’s rookie card began to increase once the player retired in 1996. The value has increased by $2.52 each year since then.
    \n\"Envision
    \na. How much was the baseball card worth in 1997? In 1998? In 1999?<\/p>\n

    Answer:
    \nThe baseball card was worth in 1997, 1998, and 1999 is $9.98, $12.50, and $15.02.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe value of a baseball player’s rookie card began to increase once the player retired in 1996.
    \nThe value has increased by $2.52 each year since then.
    \nthe value is $7.46 in 1996.
    \n$7.46 + $2.52 = $9.98 in 1997.
    \n$9.98 + $2.52 = $12.50 in 1998.
    \n$12.50 + $2.52 = $15.02 in 1999.<\/p>\n

    b. Construct Arguments Why is there not a proportional relationship between the years since the player retired and the card value? Explain.<\/p>\n

    Answer:
    \nThe value has increased by $2.52 each year.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe value of a baseball player’s rookie card began to increase once the player retired in 1996.
    \nThe value has increased by $2.52 each year since then.
    \nthe value is $7.46 in 1996.
    \n$7.46 + $2.52 = $9.98 in 1997.
    \n$9.98 + $2.52 = $12.50 in 1998.
    \n$12.50 + $2.52 = $15.02 in 1999.
    \nso there is no proportional relationship.<\/p>\n

    Question 15.
    \nHigher Order Thinking<\/strong> A car travels 2\\(\\frac{1}{3}\\) miles in 3\\(\\frac{1}{2}\\) minutes at a constant speed.
    \na. Write an equation to represent the distance the car travels, d, in miles for m minutes.<\/p>\n

    Answer:
    \nThe equation to represent the distance the car travels = 0.6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA car travels 2\\(\\frac{1}{3}\\) miles in 3\\(\\frac{1}{2}\\) minutes at a constant speed.
    \n2(1\/3) in 3(1\/2).
    \n7\/3 in 7\/2.
    \ny = kx.
    \nd = k m.
    \nk = d\/m.
    \nk = 7\/3 \/ 7\/2.
    \nk = 7\/3 x 2\/7.
    \nk = 14\/21.
    \nk\u00a0 = 0.6.<\/p>\n

    b. Write an equation to represent the distance the car travels, d, in miles for h hours.<\/p>\n

    Answer:
    \nThe equation to represent the distance the car travels, d, in miles for h hours is k = d\/m.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA car travels 2\\(\\frac{1}{3}\\) miles in 3\\(\\frac{1}{2}\\) minutes at a constant speed.
    \n2(1\/3) in 3(1\/2).
    \n7\/3 in 7\/2.
    \ny = kx.
    \nd = k m.
    \nk = d\/m.<\/p>\n

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

    Question 16.
    \nFor every ten sheets of stickers you buy at a craft store, the total cost increases $20.50.
    \n\"Envision
    \nAn equation that relates the number of sheets purchased, x, and the total cost, y, of the stickers is y = _______x.
    \nUse the equation you wrote to complete the table.<\/p>\n

    Answer:
    \nThe number of sheets purchased, x, and the total cost, y, of the stickers is y = kx.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nFor every ten sheets of stickers, you buy at a craft store, the total cost increases by $20.50.
    \ny = kx.
    \ny = k 2.
    \n$20.50\/2 = k.
    \nk = 10.25.
    \n$10.25\/7 = 1.46.
    \n$26.65\/ 12 = 2.22.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 17.
    \n600,000 gallons of water pass through a given point along a river every minute. Which equation represents the amount of water, y, that passes through the point in x minutes?
    \nA. x = 10,000y
    \nB. y = 600,000x
    \nC. y = 10,000x
    \nD. y = 600,000 + x<\/p>\n

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

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe amount of water, y, that passes through the point in x minutes.
    \ny = kx.
    \nwhere k = 600000.
    \ny = 600000x.
    \nso option B is correct.<\/p>\n

    Topic 2 Mid-Topic Checkpoint<\/h3>\n

    Question 1.
    \nVocabulary How do you know if a situation represents a proportional relationship? Lesson 2-3<\/p>\n

    Answer:
    \nTwo proportional quantities x and y are related by a constant multiple, or the constant of proportionality, K.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nTwo proportional quantities x and y are related by a constant multiple, or the constant of proportionality, k.
    \nYou can represent a proportional relationship using the equation y = kx.<\/p>\n

    Question 2.
    \nAna runs \\(\\frac{3}{4}\\) mile in 6 minutes. Assuming she runs at a constant rate, what is her speed, in miles per hour? Lessons 2-1 and 2-2
    \n\"Envision<\/p>\n

    Answer:
    \nThe speed in miles per hour = 8.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAna runs \\(\\frac{3}{4}\\) mile in 6 minutes.
    \ny = kx.
    \n6 = kx 0.75.
    \nk = 6\/0.75.
    \nk = 8.
    \nso the speed in miles per hour = 8.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 3.
    \nA trail mix recipe includes granola, oats, and almonds. There are 135 Calories in a \\(\\frac{1}{4}\\) cup serving of the trail mix. How many Calories are in 2 cups of trail mix? Lesson 2-2<\/p>\n

    Answer:
    \nThe number of calories is in 2 cups of trail mix = 1080 calories.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA trail mix recipe includes granola, oats, and almonds.
    \nThere are 135 Calories in a \\(\\frac{1}{4}\\) cup serving of the trail mix.
    \n135 \/ 1\/4.
    \n135 x 4 = 540.
    \n540 x 2 =
    \nso the number of calories are in 2 cups of trail mix = 1080 calories.<\/p>\n

    Question 4.
    \nDoes this table show a proportional relationship? If so, what is the constant of proportionality? Lessons 2-3 and 2-4
    \n\"Envision
    \nA. Yes; \\(\\frac{1}{5}\\)
    \nB. Yes; 5
    \nC. Yes; 10
    \nD. The quantities are not proportional.<\/p>\n

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

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \n40\/8 = 5.
    \n50\/10 = 5.
    \n60\/12 = 5.
    \n70\/14 = 5.
    \nso option B is correct.<\/p>\n

    Question 5.
    \nJanet, Rosi, and Tanya buy postcards from their favorite souvenir shop. Janet buys 3 postcards for $1.05. Rosi buys 5 postcards for $1.75. Tanya buys 8 postcards for $2.80. Are the cost, y, in dollars and the number of postcards, x, proportional? Explain your answer. Write an equation to represent this relationship, if possible. Lesson 2-4<\/p>\n

    Answer:
    \nYes, they are in a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nJanet, Rosi, and Tanya buy postcards from their favorite souvenir shop.
    \nJanet buys 3 postcards for $1.05. Rosi buys 5 postcards for $1.75. Tanya buys 8 postcards for $2.80.
    \n1.05\/3 = 0.35.
    \n1.75\/5 = 0.35.
    \n2.80\/8 = 0.35.
    \nso they are in a proportional relationship.<\/p>\n

    Question 6.
    \nFour movie tickets cost $30.00. Five concert tickets cost $36.50. Does the movie or the concert cost less per ticket? How much less? Lesson 2-1<\/p>\n

    Answer:
    \nThe less per ticket = $6.5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nFour movie tickets cost $30.00.
    \nFive concert tickets cost $36.50.
    \n$36.50 – $30.00 = 6.5.
    \nso the cost is less per ticket = $6.5.<\/p>\n

    Topic 2 Mid-Topic Performance Task<\/h3>\n

    The school’s theater club is building sets that will make ordinary students look like giants. The actors need a door, a table, and a stool that will make them look almost twice as tall.
    \nPART A
    \nThe heights of objects in the set are proportional to the actual heights of objects. Complete the table.
    \n\"Envision
    \nThe constant of proportionality is _________.<\/p>\n

    Answer:<\/p>\n

    PART B
    \nHow can the stage manager use the constant of proportionality to find the dimensions of any new props the director requires?
    \nAnswer:<\/p>\n

    PART C
    \nWhat should the height of a can of fruit juice used as a prop be if its actual height is 7 inches? Show your work.
    \nAnswer:<\/p>\n

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

    3-Act Mathematical Modeling: Mixin’ It Up
    \n\"Envision<\/p>\n

    ACT 1<\/strong>
    \nQuestion 1.
    \nAfter watching the video, what is the first question that comes to mind?<\/p>\n

    Answer:
    \nWhen we are drinking first we have to shake it and we have to drink.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMixin ti up.
    \nthat means when we are drinking we have to shake it and we have to use it.
    \neither it is a syrup or any other drink.<\/p>\n

    Question 2.
    \nWrite the Main Question you will answer.<\/p>\n

    Answer:
    \nWhen we are drinking first we have to shake it and we have to drink.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMixin ti up.
    \nthat means when we are drinking we have to shake it and we have to use it.
    \neither it is a syrup or any other drink.<\/p>\n

    Question 3.
    \nConstruct Arguments Predict an answer to this Main Question. Explain your prediction.
    \n\"Envision<\/p>\n

    Answer:
    \nThere is one pineapple and 5 apples.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthere are 5 apples and 1 pineapple.
    \nso 1\/5 is the unit rate.<\/p>\n

    Question 4.
    \nOn the number line below, write a number that is too small to be the answer. Write a number that is too large. Too small
    \n\"Envision<\/p>\n

    Answer:
    \nThe number that is too small is 0.01.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers on the number line are 0.01, 0.02, and …..
    \nthe number that is too small is 0.01.
    \nthe number that is too large is 1.
    \nso the number that is too small is 0.01.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 5.
    \nPlot your prediction on the same number line.<\/p>\n

    Answer:
    \nThe number that is too small is 0.01.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe numbers on the number line are 0.01, 0.02, and …..
    \nthe number that is too small is 0.01.
    \nthe number that is too large is 1.
    \nso the number that is too small is 0.01.<\/p>\n

    ACT 2<\/strong>
    \nQuestion 6.
    \nWhat information in this situation would be helpful to know? How would you use that information?
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 7.
    \nUse Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
    \nAnswer:<\/p>\n

    Question 8.
    \nModel with Math Represent the situation using mathematics. Use your representation to answer the Main Question.
    \nAnswer:<\/p>\n

    Question 9.
    \nWhat is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.
    \n\"Envision
    \nAnswer:<\/p>\n

    ACT 3<\/strong>
    \nQuestion 10.
    \nWrite the answer you saw in the video.
    \nAnswer:<\/p>\n

    Question 11.
    \nReasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 12.
    \nMake Sense and Persevere Would you change your model now that you know the answer? Explain.
    \n\"Envision
    \nAnswer:<\/p>\n

    Reflect<\/strong>
    \nQuestion 13.
    \nModel with Math Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?
    \n\"Envision
    \nAnswer:<\/p>\n

    Question 14.
    \nCritique Reasoning Choose a classmate’s model. How would you adjust that model?
    \nAnswer:<\/p>\n

    SEQUEL<\/strong>
    \nQuestion 15.
    \nUse Structure A classmate usually adds 6 drops to 16 ounces of water. Use your updated model to predict the number of drops she would use for the large container.
    \n\"Envision
    \nAnswer:<\/p>\n

    Lesson 2.5 Graph Proportional Relationships<\/h3>\n

    Explore It!<\/strong>
    \nThe graph shows the time it takes Jacey to print T-shirts for her school’s math club.<\/p>\n

    I can… use a graph to determine whether two quantities are proportional.
    \n\"Envision<\/p>\n

    A. Use the points on the graph to complete the table. Are the quantities proportional? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nYes, they are proportional.<\/p>\n

    Explanation:
    \nIn the bove-given question,
    \ngiven that,
    \nThe graph shows the time it takes Jacey to print T-shirts for her school’s math club.
    \nthe number of t-shirts is 1 the time in minutes is 5.
    \n2 x 5 = 10.
    \n3 x 5 = 15.
    \n4 x 5 = 20.
    \n5 x 5 = 25.
    \nso they are proportional.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    B. Start at (1,5). As you move from one point to the next on the graph, how does the x-coordinate change? How does the y-coordinate change?<\/p>\n

    Answer:
    \nThe x and y coordinates do not change.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nStart at (1,5). As you move from one point to the next on the graph.
    \n1 x 5 = 5.
    \nso the coordinates x and y are not changed.<\/p>\n

    C. Write the points for 0 T-shirts and for 5 T-shirts as ordered pairs. Graph the points and draw the line that passes through all six points.<\/p>\n

    Answer:
    \nThe graph increases.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points for 0 t-shirts and for 5 T-shirts as ordered pairs.
    \nthe point is (0,5).
    \nso the graph increases automatically.<\/p>\n

    Focus on math practices
    \nReasoning Suppose that after printing 4 T-shirts it takes Jacey 4 minutes to change the ink cartridge. Would this point for 5 T-shirts lie on the line you drew in Part C? Explain.<\/p>\n

    Answer:
    \nThe point (5,15) lies on the line.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nafter printing 4 T-shirts it takes Jacey 4 minutes to change the ink cartridge.
    \n5,15 lies on the line.<\/p>\n

    Essential Question<\/strong>
    \nWhat does the graph of a proportional relationship look like?<\/p>\n

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

    Each \\(\\frac{1}{4}\\) cup serving of cereal has 3 grams of protein. How can you use the graph at the right to determine whether the quantities are proportional and to find how many grams of protein are in 1 cup of the cereal?
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of grams of protein is in 1 cup of cereal = 9 grams.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nEach \\(\\frac{1}{4}\\) cup serving of cereal has 3 grams of protein.
    \n1\/4 x 3.
    \nfor 1 cup of cereal is 9 grams of protein.
    \n1 x 9 = 9.
    \nso the number of grams of protein is in 1 cup of cereal = 9 grams.<\/p>\n

    Convince Me! How can you find the constant of proportionality from the coordinates of one point on the graph?<\/p>\n

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

    Suppose the graph of Mr. Brown’s Road Trip is extended. Find the ordered pair with an x-coordinate of 7. What does this point represent in the situation?
    \nIf the graph is extended, it will pass through the point (7, _______). This means
    \nMr. Brown drives _______ miles in _______ hours.<\/p>\n

    Answer:<\/p>\n

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

    Draw two graphs that pass through the point (2, 3), one that represents a proportional relationship and one that does not. Label your graphs as Proportional or NOT Proportional.
    \n\"Envision<\/p>\n

    Answer:
    \nThe point (2,3) passes through the point.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe point (2,3) is a proportional relationship.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    KEY CONCEPT<\/strong>
    \nThe graph of a proportional relationship is a straight line through the origin.
    \n\"Envision
    \nThis graph shows that the total cost of tickets at a county fair is proportional to the number of tickets purchased. For each point (x, y) on the line except (0, 0), \\(\\frac{y}{x}\\) = 5, which is the constant of proportionality.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question What does the graph of a proportional relationship look like?<\/p>\n

    Answer:
    \nThe graph shows the proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThis graph shows that the total cost of tickets at a county fair is proportional to the number of tickets purchased.
    \nfor each point (x,y) except (0,0).
    \ny\/x = 5.
    \nso the constant of proportionality is 5.<\/p>\n

    Question 2.
    \nReasoning<\/strong> Why will the graph of every proportional relationship include the point (0, 0)?<\/p>\n

    Answer:
    \nYes, the graph of every proportional relationship includes the point (0,0).<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nfor every graph, the point (0,0) is included.
    \nthe points are (0,0), (1,5), (2, 10), (3, 15).
    \nso the graph of every proportional relationship includes the point (0,0).<\/p>\n

    Question 3.
    \nConstruct Arguments<\/strong> Makayla plotted two points, (0, 0) and (3, 33), on a coordinate grid. Noah says that she is graphing a proportional relationship. Is Noah correct? Explain.<\/p>\n

    Answer:
    \nYes, it is a graphing a proportional.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points are (0,0) and (3, 33).
    \nso it is a graphing a proportional.<\/p>\n

    Do You Know How?<\/strong>
    \nFor 4-7, use the information below.<\/p>\n

    Martin and Isabelle go bowling. Each game costs $10, and they split that cost. Martin has his own bowling shoes, but Isabelle pays $3 to rent shoes.
    \nQuestion 4.
    \nComplete the graphs below. Martin’s
    \n\"Envision<\/p>\n

    Answer:
    \nThe points ($3, $10).<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMartin and Isabelle go bowling. Each game costs $10, and they split that cost.
    \nMartin has his own bowling shoes, but Isabelle pays $3 to rent shoes.
    \nthe point (3, 10) is not a proportional relationship.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Question 5.
    \nWhich graph shows a proportional relationship? Explain why.<\/p>\n

    Answer:
    \nThe point (1,5) shows a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \npoints 1,5, and 2,10 are proportional to each other.
    \nthe point (1,5) and (2,10) shows a proportional relationship.<\/p>\n

    Question 6.
    \nChoose one point on the graph of the proportional relationship and explain what this point means in terms of the situation.<\/p>\n

    Answer:
    \nThe point (1,5) shows a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \npoints 1,5, and 2,10 are proportional to each other.
    \nthe point (1,5) and (2,10) shows a proportional relationship.<\/p>\n

    Question 7.
    \nWhat equation represents the proportional relationship?<\/p>\n

    Answer:
    \nThe point (1,5) shows a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \npoints 1,5, and 2,10 are proportional to each other.
    \nthe point (1,5) and (2,10) shows a proportional relationship.<\/p>\n

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

    Question 8.
    \nFor each graph shown, tell whether it shows a proportional relationship. Explain your reasoning.
    \n\"Envision<\/p>\n

    Answer:
    \nGraph a shows the proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe graph (2,2) and (4,4) shows the proportional relationship.
    \nso graph a shows the proportional relationship.<\/p>\n

    Question 9.
    \nThe graph shows the number of boxes a machine packages over time. Is the relationship proportional? How many boxes does the machine package in 4 minutes?
    \n\"Envision<\/p>\n

    Answer:
    \nThe number of boxes does the machine package in 4 minutes = 100 boxes.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe graph shows the points.
    \nthe points are (0,0), (2,50), (4,100).
    \nin 4 minutes the number of boxes that can be package is 100 boxes.
    \nso the number of boxes does the machine package in 4 minutes = 100 boxes.<\/p>\n

    Question 10.
    \nDoes the graph show a proportional relationship? If so, use the graph to find the constant of proportionality.
    \n\"Envision<\/p>\n

    Answer:
    \nYes, the graph shows a proportional relationship.
    \nThe constant of proportionality is 25.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points (0,25), (1, 50), (2, 75), (3 , 100), (4,125), (5, 175).
    \nso the graph shows a proportional relationship.
    \nso the constant of proportionality is 25.<\/p>\n

    Question 11.
    \nThe graph shows a proportional relationship between the cups of flour a baker uses and the number of cookies made.
    \n\"Envision
    \na. Use Structure What does the point (0, 0) represent in the situation?<\/p>\n

    Answer:
    \nThe points (0,0) represents a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points are (0,0), (1, 18), (2, 35), (3, 50), (4, 75) and (6, 125).
    \nso the points (0,0) represents a proportional relationship.<\/p>\n

    b. What does the point (1, 18) represent?<\/p>\n

    Answer:
    \nThe point (1, 18) is a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points are (0,0), (1, 18), (2, 35), (3, 50), (4, 75) and (6, 125).
    \nso the points (1,18) represents a proportional relationship.<\/p>\n

    Question 12.
    \nThe points (0.5, \\(\\frac{1}{10}\\)) and (7, 1\\(\\frac{2}{5}\\)) are on the graph of a proportional relationship.
    \na. What is the constant of proportionality?<\/p>\n

    Answer:
    \nThe graph is a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe points (0.5, \\(\\frac{1}{10}\\)) and (7, 1\\(\\frac{2}{5}\\)).
    \npoints (0.5, 0.1) and (7, 1.4).
    \nso the graph shows a proportional relationship.<\/p>\n

    b. Name one more point on the graph.<\/p>\n

    Answer:
    \nPoint (4,75) is a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points are (0.5, 0.1), (7, 1.4), and (4, 75).
    \nthe point (4, 75) is a proportional relationship.<\/p>\n

    c. Write an equation that represents the proportional relationship.<\/p>\n

    Answer:
    \nThe point (3, 50) represents the proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points are (0.5, 0.1), (7, 1.4), and (4, 75).
    \nthe point (3, 50) represents the proportional relationship.<\/p>\n

    Question 13.
    \nHigher Order Thinking<\/strong> Denmark uses the kroner as its currency. Before a trip to Denmark, Mia wants to exchange $1,700 for kroner.
    \n\"Envision
    \na. Does Bank A or Bank B have the better exchange rate? Explain.<\/p>\n

    b. How many more kroner would Mia get if she exchanged her $1,700 at the bank with the better exchange rate?
    \nAnswer:<\/p>\n

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

    Question 14.
    \nDoes the graph at the right show a proportional relationship between x and y? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nNo, the graph does not show a proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe points are (0,1), (2, 2), (3, 2), and (4, 5).
    \nthe point (2, 2) shows the relationship.
    \nbut the graph does not show a proportional relationship.<\/p>\n

    Question 15.
    \nThe graph at the right shows the relationship between rainfall during the growing season and the growth of a type of plant. Which statements about the graph are true?
    \n\"Envision
    \n\u2610 The point (1, 10) shows the constant of proportionality.
    \n\u2610 The constant of proportionality is \\(\\frac{5}{7}\\).
    \n\u2610 The graph does not show a proportional relationship.
    \n\u2610 The graph is a straight line through the origin.
    \n\u2610 The point (28, 20) means the type of plant grows 20 mm when it rains 28 cm.<\/p>\n

    Answer:
    \nThe constant of proportionality is (5\/7).<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe constant of proportionality is 5\/7.
    \n2 x 5 = 10, 2 x 7 = 14.
    \n4 x 5 = 20, and 4 x 7 = 28.
    \n6 x 5 = 30, and 6 x 7 = 42.
    \nso the constant of proportionality is (5\/7).<\/p>\n

    Lesson 2.6 Apply Proportional Reasoning to Solve Problems<\/h3>\n

    Solve & Discuss It!<\/strong>
    \nXander and Pedro are at an ice cream social. For every scoop of ice cream, Xander uses cup of fruit topping. Pedro uses one more tablespoon of fruit topping than the number of scoops. If Xander and Pedro each use the same amount of fruit topping, how many scoops of ice cream does each use?<\/p>\n

    I can… determine whether a relationship is proportional and use representations to solve problems.<\/p>\n

    Look for Relationships
    \nThere are 2 tablespoons in \\(\\frac{1}{8}\\) cup.<\/p>\n

    Focus on math practices
    \nReasoning for which person, Xander or Pedro, is the relationship between the quantities of ice cream and fruit topping proportional? Explain.
    \n\"Envision<\/p>\n

    Essential Question<\/strong>
    \nHow can proportional reasoning help solve a problem?<\/p>\n

    Answer:
    \nYes, proportional reasoning help in solving a problem.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThere are 2 tablespoons in \\(\\frac{1}{8}\\) cup.
    \n2 x 1\/8 = 4.
    \nso the constant of proportionality is 4.<\/p>\n

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

    After selling half their card collections, DeShawn and Stephanie each buy 9 new cards. What is the ratio of the number of cards DeShawn has to the number Stephanie has?<\/p>\n

    Answer:
    \nThe number of cards DeShawn has to the number Stephanie has = 9 : 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAfter selling half their card collections, DeShawn and Stephanie each buy 9 new cards.
    \nso the ratio is 9 : 2.
    \nso the number of cards DeShawn has to the number Stephanie has = 9 : 2.<\/p>\n

    Convince Me! Why did the ratio stay the same in the Example but change in the Try It?<\/p>\n

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

    A florist makes bouquets that include 50 white flowers and 7 red flowers. If the florist orders 1,050 white flowers and 140 red flowers, there will be leftover flowers. How can the florist adjust the order so there are no leftover flowers?<\/p>\n

    Answer:
    \nThe florist adjusts the order so there are no leftover flowers = 20.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA florist makes bouquets that include 50 white flowers and 7 red flowers.
    \nIf the florist orders 1,050 white flowers and 140 red flowers.
    \n1050 : 50 = 20.
    \n140 : 7 = 20.
    \nso there are no leftover flowers.<\/p>\n

    KEY CONCEPT<\/strong>
    \nThink about how two quantities are related before you decide to use proportional reasoning to solve a problem.
    \n\"Envision
    \nYou cannot use proportional reasoning to solve this problem because Josh’s age is not a constant multiple of Evie’s age.<\/p>\n

    Do You Understand?<\/strong>
    \nQuestion 1.
    \nEssential Question How can proportional reasoning help solve a problem?<\/p>\n

    Answer:
    \nProportional reasoning can help in solving a problem.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \ntwo quantities are related before you decide to use proportional reasoning to solve a problem.
    \nfor example:
    \nJosh is 4 years older than Evie.
    \nJosh is 3 times as old as Evie.
    \nwhen Evie is 2 years old, Josh is 6 years old.<\/p>\n

    Question 2.
    \nUse Appropriate Tools How can knowing how to represent proportional relationships in different ways be useful in solving problems?<\/p>\n

    Answer:
    \nProportional reasoning can help in solving a problem.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \ntwo quantities are related before you decide to use proportional reasoning to solve a problem.
    \nfor example:
    \nJosh is 4 years older than Evie.
    \nJosh is 3 times as old as Evie.
    \nwhen Evie is 2 years old, Josh is 6 years old.<\/p>\n

    Question 3.
    \nReasoning<\/strong> How many ways are there to adjust two quantities so that they are in a given proportional relationship? Explain your reasoning.<\/p>\n

    Answer:
    \nThere are 2 ways to adjust two quantities so that they are in given proportional reasoning to solve a problem.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \ntwo quantities are related before you decide to use proportional reasoning to solve a problem.
    \nfor example:
    \nJosh is 4 years older than Evie.
    \nJosh is 3 times as old as Evie.
    \nwhen Evie is 2 years old, Josh is 6 years old.<\/p>\n

    Do You Know How?<\/strong>
    \nQuestion 4.
    \nA recipe calls for 15 oz of flour for every 8 oz of milk.
    \na. Is the relationship between ounces of flour and ounces of milk proportional? Explain.<\/p>\n

    Answer:
    \nNo, there is no proportional relationship.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA recipe calls for 15 oz of flour for every 8 oz of milk.
    \n15\/8.
    \nso there is no proportional relationship.<\/p>\n

    b. If you use 15 oz of milk, how much flour should you use?<\/p>\n

    Answer:
    \nWe can use either 3 or 5 oz of flour.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nif we use 15 oz of milk, we can use either 3 or 5 oz of flour.
    \n15\/3 = 5.
    \nso we can use either 3 or 5 oz of flour.<\/p>\n

    Question 5.
    \nA food packing company makes a popular fruit cocktail. To ensure a good mixture of fruit, there are 3 cherry halves for every 8 white grapes in a jar. An inspector notices that one jar has 12 cherry halves and 20 white grapes. What can be done to fix the error?<\/p>\n

    Answer:
    \nThe error is 6.6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA food packing company makes a popular fruit cocktail.
    \nTo ensure a good mixture of fruit, there are 3 cherry halves for every 8 white grapes in a jar.
    \nAn inspector notices that one jar has 12 cherry halves and 20 white grapes.
    \n12\/8 = 3\/2.
    \n20\/3 = 6.6.
    \nthe error is 6.6.<\/p>\n

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

    Multimedia In 6 and 7, determine whether you can use proportional reasoning and then solve.
    \nQuestion 6.
    \nIf Hector is 8 years old and Mary is 3 years old, how old will Mary be when Hector is 16?<\/p>\n

    Answer:
    \nMary is 6 years old.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nIf Hector is 8 years old and Mary is 3 years old.
    \n8 x 2 = 16.
    \n3 x 2 = 6.
    \n16\/2 = 8.
    \n6\/2 = 3.
    \nso Mary is 6 years old.<\/p>\n

    Question 7.
    \nMarco needs to buy some cat food. At the nearest store, 3 bags of cat food cost $15.75. How much would Marco spend on 5 bags of cat food?<\/p>\n

    Answer:
    \nThe Marco spend of 5 bags of cat food = $26.25.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMarco needs to buy some cat food.
    \nAt the nearest store, 3 bags of cat food cost $15.75.
    \n$15.75 \/ 3 = 5.25.
    \n5.25 x 5 = 26.25.
    \nso the Marco spend on 5 bags of cat food = $26.25.<\/p>\n

    Question 8.
    \nAn architect makes a model of a new house with a patio made with pavers. In the model, each paver in the patio is \\(\\frac{1}{3}\\) in. long and a \\(\\frac{1}{6}\\) in. wide. The actual dimensions of the pavers are shown.
    \n\"Envision
    \na. What is the constant of proportionality that relates the length of a paver in the model and the length of an actual paver?<\/p>\n

    Answer:
    \nThe length of a paper in the model and the length of the actual paver = 0.03125: 0.0528.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn architect makes a model of a new house with a patio made with pavers.
    \neach paver in the patio is \\(\\frac{1}{3}\\) in. long and a \\(\\frac{1}{6}\\) in. wide.
    \narea of the paver = l x b.
    \narea = 1\/3 x 1\/6.
    \narea = 0.33 x 0.16.
    \narea = 0.0528.
    \nactual paver area = l x b.
    \narea = 1\/4 x 1\/8.
    \narea = 0.25 x 0.125.
    \narea = 0.03125.
    \nso the ratio is 0.03125 : 0.0528.<\/p>\n

    b. What is the constant of proportionality that relates the area of a paver in the model and the area of an actual paver? Explain your reasoning.<\/p>\n

    Answer:
    \nThe constant of proportionality is 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nAn architect makes a model of a new house with a patio made with pavers.
    \neach paver in the patio is \\(\\frac{1}{3}\\) in. long and a \\(\\frac{1}{6}\\) in. wide.
    \narea of the paver = l x b.
    \narea = 1\/3 x 1\/6.
    \narea = 0.33 x 0.16.
    \narea = 0.0528.
    \nactual paver area = l x b.
    \narea = 1\/4 x 1\/8.
    \narea = 0.25 x 0.125.
    \narea = 0.03125.
    \n2 x 4 = 8.
    \n3 x 2 = 6.
    \nso the proportionality of constant is 2.<\/p>\n

    Question 9.
    \nReasoning<\/strong> The table lists recommended amounts of food to order for 25 guests. Nathan is hosting a graduation party for 40 guests. There will also be guests stopping by for a short time. For ordering purposes, Nathan will count each of the 45 “drop-in” guests as half a guest. How much of each food item should Nathan order?
    \n\"Envision<\/p>\n

    Answer:
    \nThe amount of each food item should Nathan order is 24 pieces of fried chicken, 3.66 pounds of Delimeats, and 1.75 pounds of Lasagna.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe table lists recommended amounts of food to order for 25 guests.
    \n24 pieces of fried chicken, 3.66 pounds of Deli meats, and 1.75 pounds of Lasagna.<\/p>\n

    Question 10.
    \nEmily and Andy each go to a hardware store to buy wire. The table shows the relationship between the cost and the length of wire.
    \n\"Envision
    \na. Emily needs 24 feet of wire. How much will she spend on wire?<\/p>\n

    Answer:
    \nThe amount she will spend on wire = $5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nEmily needs 24 feet of wire.
    \n580\/23.2 = 25.
    \n$25 \/ 5 = 5.
    \nso the amount she will spend on wire = $5.<\/p>\n

    b. Andy needs 13 yards of wire. How much will he spend on wire?<\/p>\n

    Answer:
    \nThe amount he will spend on wire = $0.024.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nEmily needs 24 feet of wire.
    \n580\/23.2 = 25.
    \n1 inch = 0.027 yards.
    \n0.027 x 12 = 0.324.
    \n0.324\/13 = 0.024.<\/p>\n

    Question 11.
    \nMake Sense and Persevere The weights of Michael’s and Brittney’s new puppies are shown in the table and graph. Whose dog gains weight more quickly? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nThe dog gains weight more quickly = Brittney’s puppy.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe weights of Michael’s and Brittney’s new puppies are shown in the table in the graph.
    \nthe weight of 155\/1 = 155.
    \n310\/2 = 155.
    \n465\/3 = 155.
    \n8.6\/1 = 8.6.
    \n17.2\/2 = 8.6.
    \nso the weight of the Brittney’s puppy weighs more.<\/p>\n

    Question 12.
    \nHigher Order Thinking<\/strong> Marielle’s painting has the dimensions shown. The school asks her to paint a larger version that will hang in the cafeteria. The larger version will be twice the width and twice the height. Is the area of the original painting proportional to the area of the larger painting? If so, what is the constant of proportionality?
    \n\"Envision<\/p>\n

    Answer:
    \nThe constant of proportionality = 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMarielle’s painting has the dimensions shown.
    \nThe larger version will be twice the width and twice the height.
    \narea = length x width.
    \narea = 24 x 18.
    \narea = 432 sq inches.
    \nso the constant of proportionality = 2.<\/p>\n

    Assessment Practice<\/p>\n

    Question 13.
    \nThe ratio of orange juice concentrate to water that Zoe used to make orange juice yesterday was 3: 7. She used 14 ounces of water in the juice yesterday. Today she wants to make twice as much orange juice.
    \nPART A
    \nIs the relationship between the amount of orange juice concentrate c and the amount of water w proportional? Explain.<\/p>\n

    Answer:
    \nThe amount of orange juice concentrate c and the amount of water w is proportional.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe ratio of orange juice concentrate to water that Zoe used to make orange juice yesterday was 3: 7.
    \nShe used 14 ounces of water in the juice yesterday.
    \n3 : 7 = 6 : 14.
    \n3 x 2 = 6.
    \n7 x 2 = 14.
    \nso the amount of orange juice concentrate c and the amount of water w is proportional.<\/p>\n

    PART B
    \nHow much orange juice concentrate does she need?
    \nA. 6 ounces
    \nB. 12 ounces
    \nC. 18 ounces
    \nD. 28 ounces<\/p>\n

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

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe ratio of orange juice concentrate to water that Zoe used to make orange juice yesterday was 3: 7.
    \nShe used 14 ounces of water in the juice yesterday.
    \n3 : 7 = 6 : 14.
    \n3 x 2 = 6.
    \n7 x 2 = 14.
    \nso option A is correct.<\/p>\n

    Topic 2 Review<\/h3>\n

    Topic Essential Question
    \nHow can you recognize and represent proportional relationships and use them to solve problems?<\/p>\n

    Vocabulary Review<\/strong>
    \nComplete each definition and then provide an example of each vocabulary word.<\/p>\n

    \"Envision<\/p>\n

    Answer:
    \n1. Proportional relationship.
    \n2. Constant of proportionality.
    \n3. True proportion.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe definitions of proportional relationship, constant of proportionality, and true proportion.
    \n(1,5), (2, 10), (3, 15) and (4,20) are the examples of proportional relationship.
    \ny x 12.5 = 90.
    \ny = 90\/12.5.
    \ny = 72 is the example of constant of proportionality.
    \n2\/10 = 1\/5.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n

    Use Vocabulary in Writing
    \nA sunflower grows 8 feet in 92 days. Assuming that it grows at a constant rate, explain how you could use this information to find the number of days it took for the sunflower to grow 6 feet. Use vocabulary words in your explanation.<\/p>\n

    Answer:
    \nThe number of days it took for the sunflower to grow 6 feet = 15.3 days.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA sunflower grows 8 feet in 92 days.
    \ny = kx.
    \n6 = k . 92.
    \nk = 92\/6.
    \nk = 15.3.
    \nso the number of days it took for the sunflower to grow 6 feet = 15.3 days.<\/p>\n

    Concepts and Skills Review<\/strong><\/p>\n

    Lessons 2.1 And 2.2 Connect Ratios, Rates, and Unit Rates Determine Unit Rates with Ratios of Fractions<\/strong><\/p>\n

    Quick Review<\/strong>
    \nA ratio is a relationship in which for every x units of one quantity there are y units of another quantity. A rate is a ratio that relates two quantities with different units. A unit rate relates a quantity to 1 unit of another quantity. You can use what you know about equivalent fractions and calculating with fractions to write a ratio of fractions as a unit rate.<\/p>\n

    Example<\/strong>
    \nIn 3 days, a typical robin can eat up to 42.6 feet of earthworms. Write a rate to relate the number of feet of earthworms to the number of days. Then find the unit rate.
    \n\\(\\frac{42.6 \\text { feet }}{3 \\text { days }}=\\frac{14.2 \\text { feet }}{1 \\text { day }}\\)
    \nA robin eats 14.2 feet of earthworms per day.<\/p>\n

    Practice<\/strong>
    \nQuestion 1.
    \nMealworms are a healthy food for wild songbirds. Adam buys a 3.5-oz container of mealworms for $8.75. Marco buys a 3.75-oz container of mealworms for $9.75. Which container is the better deal?<\/p>\n

    Answer:
    \nThe better deal = Marco’s deal.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nMealworms are healthy food for wild songbirds.
    \nAdam buys a 3.5-oz container of mealworms for $8.75.
    \nMarco buys a 3.75-oz container of mealworms for $9.75.
    \n8.75 \/ 3.5 = 2.5.
    \n9.75 \/ 3.75 = 2.6.
    \nso the Marco container is better.<\/p>\n

    Question 2.
    \nA painter mixes 2\\(\\frac{1}{2}\\) pints of yellow paint with 4 pints of red paint to make a certain shade of orange paint. How many pints of yellow paint should be mixed with 10 pints of red paint to make this shade of orange?<\/p>\n

    Answer:
    \nThe number of yellow paint should be mixed = 20.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nA painter mixes 2\\(\\frac{1}{2}\\) pints of yellow paint with 4 pints of red paint to make a certain shade of orange paint.
    \n2(1\/2) = 5\/2.
    \n5\/2 = 2.5.
    \n2.5 x 2 = 5.
    \n10 x 2 = 20.
    \nso the number of yellow paint should be mixed = 20.<\/p>\n

    Lesson 2.3 Understand Proportional Relationships: Equivalent Ratios<\/strong><\/p>\n

    Quick Review<\/strong>
    \nTwo quantities x and y have a proportional relationship if the ratios for every related pair of x and y are equivalent. You can write a proportion to show that two ratios have the same value.<\/p>\n

    Example<\/strong>
    \nDoes the table show a proportional relationship between x and y? Explain.
    \n\"Envision
    \nBecause every ratio is equivalent, there is a proportional relationship between x and y.<\/p>\n

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

    In 1 and 2, use the table below that shows information about squares.
    \n\"Envision
    \nQuestion 1.
    \nAre the perimeter and the side length of squares proportional? Explain.<\/p>\n

    Answer:
    \nThe perimeter and the side length of squares are proportional.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe side lengths are 2, 4, and 6.
    \nthe perimeters are 8, 16, and 24.
    \n2 x 4 = 8.
    \n4 x 4 = 16.
    \n6 x 4 = 24.
    \nso the perimeter and the side length of squares are proportional.<\/p>\n

    Question 2.
    \nWrite and solve a proportion to find the perimeter of a square when its side length is 12.<\/p>\n

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

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe side length is 12.
    \n12 x 4 = 48.
    \nso the perimeter is 12 cm.<\/p>\n

    Lesson 2.4 Describe Proportional Relationships: Constant of Proportionality<\/strong><\/p>\n

    Quick Review<\/strong>
    \nThe equation y = kx describes a proportional relationship between two quantities x and y, where k is the constant of proportionality.k= for any related pair of x and y except when x = 0.<\/p>\n

    Example<\/strong>
    \nThe table shows the wages Roger earned for the hours he worked. What equation relates the wages, w, and the number of hours, h?
    \n\"Envision
    \nFind the constant of proportionality, k.
    \n\\(\\frac{27}{3}\\) = 9 \\(\\frac{45}{5}\\) = 9 \\(\\frac{54}{6}\\) = 9
    \nWrite the equation in the form y = kx.
    \nW = 9h<\/p>\n

    Practice<\/strong>
    \nQuestion 1.
    \nSally is going on vacation with her family. In 2 hours they travel 90.5 miles. If they travel at the same speed, write an equation that represents how far they will travel, d, in h hours.<\/p>\n

    Answer:
    \nThe equation that represents how far they will travel = 45.25 miles.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nSally is going on vacation with her family.
    \nIn 2 hours they travel 90.5 miles.
    \n90.5\/2 = 45.25 miles.
    \nso the equation that represents how far they will travel = 45.25 miles.<\/p>\n

    Question 2.
    \nThe table shows the weights of bunches of bananas and the price of each bunch. Identify the constant of proportionality. Write an equation to relate weight, w, to the price, p.
    \n\"Envision<\/p>\n

    Answer:
    \nThe constant of proportionality is 3.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nThe table shows the weights of bunches of bananas and the price of each bunch.
    \nthe price of the bananas is $1.35, $1.71, and $2.34.
    \nthe weight of bananas is 3 pounds, 3.8 pounds, and 5.2 pounds.
    \n1.35\/3 = 0.45.
    \n1.71\/3.8 = 0.45.
    \n2.34\/5.2 = 0.45.
    \nso the constant of proportionality = 3.<\/p>\n

    Lesson 2.5 Graph Proportional Relationships<\/strong><\/p>\n

    Quick Review<\/strong>
    \nThe graph of a proportional relationship is a straight line through the origin. You can identify the constant of proportionality, k, from the point (1, k) or by dividing for any point (x, y) except the origin.<\/p>\n

    Example<\/strong>
    \nIs the number of sunny days proportional to the number of rainy days? If so, find the constant of proportionality, and explain its meaning in this situation.
    \n\"Envision
    \nThe graph is a straight line through the origin, so it shows a proportional relationship. The constant of proportionality is which means for every1 rainy day, there were 2 sunny days<\/p>\n

    Practice<\/strong>
    \nQuestion 1.
    \nDoes the graph show a proportional relationship? Explain.
    \n\"Envision<\/p>\n

    Answer:
    \nThe proportional relationship is 1 : 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number of tickets is 4, 8, and 12.
    \nthe cost of the tickets = 4, 8, 12, and 14.
    \nthe points are (2,4), (4,8), (8, 12), and (12, 16).
    \n2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16.
    \nso the proportional relationship is 1:2.<\/p>\n

    Question 2.
    \nSketch a graph that represents a proportional relationship.<\/p>\n

    Answer:
    \nThe proportional relationship is 1 : 2.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe number of tickets is 4, 8, and 12.
    \nthe cost of the tickets = 4, 8, 12, and 14.
    \nthe points are (2,4), (4,8), (8, 12), and (12, 16).
    \n2 x 2 = 4, 4 x 2 = 8, 8 x 2 = 16.
    \nso the proportional relationship is 1:2.<\/p>\n

    Lesson 2.6 Apply Proportional Reasoning to solve Problems<\/strong><\/p>\n

    Quick Review<\/strong>
    \nThink about how two quantities are related before you decide to use proportional reasoning to solve a problem.<\/p>\n

    Example<\/strong>
    \nAnn-Marie makes gift baskets that each contain 3 pounds of gourmet cheese with every 2 boxes of crackers. She ordered 80 pounds of cheese and 50 boxes of crackers. How can she adjust her order to make 25 gift baskets, with no leftover items?
    \n\\(\\frac{3 \\times 25}{2 \\times 25}=\\frac{75}{50}\\)
    \nAnn-Marie can order 5 fewer pounds of cheese so that her order of 75 pounds of cheese and 50 boxes of crackers will make exactly 25 gift baskets.<\/p>\n

    Practice<\/strong>
    \nDecide which problems can be solved by using proportional reasoning. Select all that apply.<\/p>\n

    \u2610 Yani buys 4 dozen flyers for $7.25. What is the cost of 12 dozen?
    \n\u2610 First-class letters cost $0.49 for the first ounce and $0.22 for each additional ounce. What is the cost of a 5-oz letter?
    \n\"Envision
    \nWhat is the value of y when x = 8?
    \n\"Envision
    \nWhat is the value of y when x = 5?<\/p>\n

    Answer:
    \nThe value of y is 15 when x = 5.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe values of x and y are given.
    \nx = 1.5, 3, 5.2, 6.5, and 8.
    \ny = 4.5, 9, 15.6, 19.5, and 24.
    \n5 x 3 = 15.
    \n15\/3 = 5.
    \nso the value of y is 15 when x = 5.<\/p>\n

    Topic 2 Fluency Practice<\/h3>\n

    Pathfinder<\/strong>
    \nShade a path from START to FINISH. Follow the solutions from the least value to the greatest value. You can only move up, down, right, or left.
    \nI can… add, subtract, multiply, and divide integers.
    \n\"Envision<\/p>\n

    Answer:
    \n3 – 24 = -21, 3 x -13 = -39, 14 x -3 = -42, 44 – 49 = -5, -48\/16 = 3, -26 + 7 = -19, -7 – 27 = -34, 13 x -4 = -52, 8 – 19 = -11, -18 \/ -6 = 3, 3 x -5 = -15, 28\/-2 = 14, -35\/2 = 17.5 , 4 + – 25 = -21, -68 \/ 17 = -4, 23 – 46 = -23, -2 – 11 = 13, 15 + -(26) = -11, 17 – 23 = -6, 29 – 38 = -9, -12 \/ -3 = 4, 18 + (-32) = -14, 11 – 29 = -18, -60\/12 = -5, -3 x -2 = 6.<\/p>\n

    Explanation:
    \nIn the above-given question,
    \ngiven that,
    \nthe multiplication, division, addition, and subtraction of numbers.
    \n3 – 24 = -21, 3 x -13 = -39, 14 x -3 = -42, 44 – 49 = -5, -48\/16 = 3, -26 + 7 = -19, -7 – 27 = -34, 13 x -4 = -52, 8 – 19 = -11, -18 \/ -6 = 3, 3 x -5 = -15, 28\/-2 = 14, -35\/2 = 17.5 , 4 + – 25 = -21, -68 \/ 17 = -4, 23 – 46 = -23, -2 – 11 = 13, 15 + -(26) = -11, 17 – 23 = -6, 29 – 38 = -9, -12 \/ -3 = 4, 18 + (-32) = -14, 11 – 29 = -18, -60\/12 = -5, -3 x -2 = 6.
    \n\"Envision-Math-Common-Core-7th-Grade-Answers-Key-Topic-2-Analyze<\/p>\n","protected":false},"excerpt":{"rendered":"

    Go through the\u00a0enVision Math Common Core Grade 7 Answer Key\u00a0Topic 2 Analyze And Use Proportional Relationships\u00a0regularly and improve your accuracy in solving questions. enVision Math Common Core 7th Grade Answers Key Topic 2 Analyze And Use Proportional Relationships Topic 2 Essential Question How can you recognize and represent proportional relationships and use them to solve …<\/p>\n","protected":false},"author":1,"featured_media":25562,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/envisionmathanswerkey.com\/wp-content\/uploads\/2021\/09\/Envision-Math-Common-Core-Grade-7-Answer-Key-Topic-2-Analyze-And-Use-Proportional-Relationships.png","_links":{"self":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/14972"}],"collection":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/comments?post=14972"}],"version-history":[{"count":5,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/14972\/revisions"}],"predecessor-version":[{"id":28260,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/posts\/14972\/revisions\/28260"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media\/25562"}],"wp:attachment":[{"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/media?parent=14972"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/categories?post=14972"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/envisionmathanswerkey.com\/wp-json\/wp\/v2\/tags?post=14972"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}