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Big Ideas Math Book 7th Grade Answer Key Chapter 5 Ratios and Proportions

It is very difficult to get the best answer key for the math problems. A trustable guide will clear explanation and short answers are provided that help you to learn perfectly and to improve your math skills. One of such best online guide is Big Ideas Math Answers Grade 7 Chapter 5 Ratios and Proportions. Refer Big Ideas Math Book Grade 7 Chapter 5 Ratios and proportions within 10 to learning the easy way of maths practice. Get a clear idea on ratios and ratio tables, rates and unit rates, identifying proportional relationships, writing and solving proportions, graphs of proportional relationships, and scale drawings by practicing this chapter.

### Ratios and Proportions STEAM Video/ Performance Task

**STEAM Video**

Painting a Large Room

Shades of paint can be made by mixing other paints. What colors of paints can you mix to make green paint?

Watch the STEAM Video “Painting a Large Room.” Then answer the following questions.

1. Enid estimates that they need 2 gallons of paint to apply two coats to the wall shown. How many square feet does she expect \(\frac{1}{2}\) gallon of paint will cover?

Answer:

1/2 gallon of paint will cover 60 sq feet.

Explanation:

Enid estimates that they need 2 gallons of paint to apply two coats to the wall of 240 sq ft

The area of the shown figure = 24 x 10

= 240 square feet

To know how many sq feet does 1/2 gallon of paint will cover (240 . 1/2) . 1/2

= 240 . 1/4

= 60

So, 1/2 gallon of paint will cover 60 sq feet.

2. Describe a room that requires 5\(\frac{1}{2}\) gallons of paint to apply one coat of paint to each of the four walls.

Answer:

The room has 4 walls each wall is 330 sq feet.

Explanation:

A room requires 5(1/2) gallons of paint to apply one coat of paint to each of the four walls.

Enid estimates that they need 2 gallons of paint to apply two coats to the wall of 240 sq ft

If they coat only once, she would require 1 gallon of paint for a wall of 240 sq feet.

If she wants to use 5(1/2) = 11/2 gallons of paint, then the area of walls will be

= 240 . (11/2)

= 120 . 11 = 1320 Square feet

Each wall = 1320/4 =330 sq feet

The room has 4 walls each wall is 330 sq feet.

**Performance Task**

Mixing Paint

After completing this chapter, you will be able to use the concepts you learned to answer the questions in the STEAM Video Performance Task. You will be given the amounts of each tint used to make different colors of paint. For example:

You will be asked to solve various ratio problems about mixing paint. Given any color of paint, how can you make the paint slightly lighter in color?

### Ratios and Proportions Getting Ready for Chapter 5

**Chapter Exploration**

The Meaning of a Word Rate

When you rent snorkel gear at the beach, rate you should pay attention to the rental. The rental rate is in dollars per hour.

1. Work with a partner. Complete each step.

• Match each description with a rate.

• Match each rate with a fraction.

• Give a reasonable value for each fraction. Then give an unreasonable value.

2. Work with a partner Describe a situation to which the given fraction can apply. Show how to rewrite each expression as a division problem. Then simplify and interpret your result.

**Vocabulary**

The following vocabulary terms are defined in this chapter. Think about what each term might mean and record your thoughts.

proportional

constant of proportionality

scale drawing

### Lesson 5.1 Ratios and Ratio Tables

**EXPLORATION 1**

Describing Ratio Relationships

Work with a partner. Use the recipe shown.

a. Identify several ratios in the recipe.

b. You halve the recipe. Describe your ratio relationships in part (a) using the new quantities. Is the relationship between the ingredients the same as in part(a)? Explain.

Answer:

a. Stewed tomatoes : chicken broth = 9 ounces : 15 ounces

Stewed tomatoes : Chopped Spinach = 9 ounces : 9 ounces = 1 : 1

Chopped Spinach : chicken broth = 9 ounces : 15 ounces

Chopped Chicken : grated parmesan = 1 cup : 5 tablespoons

Chicken broth : Chopped Chicken = 15 ounces : 1 cup

Stewed tomatoes : chopped chicken = 9 ounces : 1 cup = Chopped Spinach : chopped chicken

Stewed tomatoes : grated parmesan = 9 ounces : 5 tablespoons = Chopped Spinach : grated parmesan

chicken broth : grated parmesan = 15 ounces : 5 tablespoons = 5 ounces : 1 tablespoon

b. The relationship between the ingredients the same as in part(a).

Explanation:

a. The several ratios of the recipe are Stewed tomatoes : chicken broth = 9 ounces : 15 ounces

Stewed tomatoes : Chopped Spinach = 9 ounces : 9 ounces = 1 : 1

Chopped Spinach : chicken broth = 9 ounces : 15 ounces

Chopped Chicken : grated parmesan = 1 cup : 5 tablespoons

Chicken broth : Chopped Chicken = 15 ounces : 1 cup

Stewed tomatoes : chopped chicken = 9 ounces : 1 cup = Chopped Spinach : chopped chicken

Stewed tomatoes : grated parmesan = 9 ounces : 5 tablespoons = Chopped Spinach : grated parmesan

chicken broth : grated parmesan = 15 ounces : 5 tablespoons = 5 ounces : 1 tablespoon

b. The relationship between the ingredients the same as in part(a).

**EXPLORATION 2**

Completing Ratio Tables

Work with a partner. Use the ratio tables shown.

a. Complete the first ratio table using multiple operations. Use the same operations to complete the second ratio table.

b. Are the ratios in the first table equivalent? the second table? Explain.

c. Do the strategies for completing ratio tables of whole numbers work for completing ratio tables of fractions? Explain your reasoning.

Answer:

a.

b. Yes

Explanation:

a. 5 x 2 = 10, 1 x 2 = 2, 1/4 x 2 = 1/2, 1/2 x 2 = 1

5 x 1.5 = 7.5, 1 x 1.5 = 1.5, 1/4 x 3/2 = 3/8, 1/2 x 3/2 = 3/4

5 x 4 = 20, 1 x 4 = 4, 1/4 x 4 = 1, 1/2 x 4= 2

b. Yes the ratios in the first table are equivalent to the second table.

**Try It**

Question 1.

You mix \(\frac{2}{3}\) teaspoon of baking soda with 3 teaspoons of salt. Find and interpret the value of the ratio of baking soda to salt.

Answer:

The ratio of baking soda to salt is 2 : 1.

Explanation:

Given that,

You mix \(\frac{2}{3}\) teaspoon of baking soda with 3 teaspoons of salt.

The ratio of baking soda to salt = 2/3 to 3 = 2/3 : 3

= 2/3 / 3 = 2/3 . 3/1 = 2/1 = 2 : 1

The ratio of baking soda to salt is 2 : 1.

**Find the missing values in the ratio table. Then write the equivalent ratios.**

Question 2.

Answer:

Explanation:

The missing values in the ratio table are

4 x 4 = 16. So, 5/2 x 4 = 10

5/2 . x = 5

x = 2

So, 4 . 2 = 8

The equivalent ratios are 5/2 : 4, 10 : 16, 5 : 8.

Question 3.

Answer:

Explanation:

0.4 . x = 1.2

x = 1.2/0.4 = 3

So, 0.75 . 3 = 2.25

0.4 . y = 1.6

y = 1.6/0.4 = 4

So, 0.75 . 4 = 3

The equivalent ratios are 0.4 : 0.75, 1.2 : 2.25, 1.6 : 3

**Self-Assessment for Concepts & Skills**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 4.

**WRITING AND INTERPRETING RATIOS**

You include \(\frac{1}{2}\) tablespoon of essential oils in a solution for every 12 tablespoons of jojoba oil. Find and interpret the value of the ratio of jojoba oil to essential oils.

Answer:

The amount of jojoba oil in the solution is the 24th amount of essential oil.

Explanation:

You include \(\frac{1}{2}\) tablespoon of essential oils in a solution for every 12 tablespoons of jojoba oil

The ratio of jojoba oil to essential oil = 12 : 1/2

= 12 ÷ 1/2

= 12 . 2/1

= 24

The amount of jojoba oil in the solution is the 24th amount of essential oil.

Question 5.

**NUMBER SENSE**

Find the missing values in the ratio table. Then write the equivalent ratios.

Answer:

Explanation:

1/12 . x = 2/3

x = 2/3 . 12

x = 8

So, Pounds = 3/2 . 8 = 12

3/2 . 7 = 21/2

1/12 . 7 = 7/12

The equivalent ratios are 3/2 : 1/12, 12 : 2/3, 21/2 : 7/12

**Self-Assessment for Problem Solving**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 6.

**DIG DEEPER!**

A satellite orbiting Earth travels 14\(\frac{1}{2}\) miles every 3 seconds. How far does the satellite travel in \(\frac{3}{4}\) minute?

Answer:

The satellite travels 217(1/2) miles in 3/4 minutes.

Explanation:

A satellite orbiting Earth travels 14\(\frac{1}{2}\) miles every 3 seconds.

So, it travels 14.5/3 miles = 29/6 miles in every 1 second

3/4 minutes = 3/4 . 60

= 45 seconds

Therefore, it travels 45 . 29/6 miles in 3/4 minutes

= 217(1/2) miles in 3/4 minutes.

Hence, the satellite travels 217(1/2) miles in 3/4 minutes.

Question 7.

An engine runs on a mixture of 0.1 quart of oil for every 3.5 quarts of gasoline. You make 3 quarts of the mixture. How much oil and how much gasoline do you use?

Answer:

We use 0.083 quarts of oil and 2.917 quarts of gasoline.

Explanation:

The ratio of oil to gasoline = 0.1 : 3.5 = 1 : 35

use the ratio table to find an equivalent ratio in which the total mixture is 3 quarts.

For 3 quarts mixture

Oil required is 1/36 * 3 = 3/36

= 1/12 = 0.083 quarts

Gasoline required is

35/36 * 3 = 35/12 = 2.917 quarts

We use 0.083 quarts of oil and 2.917 quarts of gasoline.

### Ratios and Ratio Tables Homework & Practice 5.1

**Review & Refresh**

**Solve the inequality. Graph the solution.**

Question 1.

4p + 7 ≥ 19

Answer:

p ≥ 3

Explanation:

Given that,

4p + 7 ≥ 19

Subtract 7 from both sides

4p + 7 – 7 ≥ 19 – 7

4p ≥ 12

Divide both sides by 4

4p/4 ≥ 12/4

p ≥ 3

Question 2.

14 < – 6n – 10

Answer:

-4 > n

Explanation:

Given that,

14 < – 6n – 10

Add 10 to both sides

14 + 10 < -6n – 10 + 10

24 < -6n

Divide both sides by -6

24/-6 > -6n/-6

-4 > n

Question 3.

– 3(2 + d) ≤ 15

Answer:

d ≥ -7

Explanation:

Given that,

– 3(2 + d) ≤ 15

Divide both sides by -3

– 3(2 + d)/-3 ≥ 15/-3

2 + d ≥ -5

d ≥ -5 – 2

d ≥ -7

**Find the quotient. Write fractions in simplest form.**

Question 4.

\(\frac{2}{9}\) ÷ \(\frac{4}{3}\)

Answer:

\(\frac{2}{9}\) ÷ \(\frac{4}{3}\) = 1/6 = 0.166

Explanation:

\(\frac{2}{9}\) ÷ \(\frac{4}{3}\) = 2/9 . 3/4

= (2 . 3) / (9 . 4)

= 1/6

Question 5.

10.08 ÷ 12

Answer:

10.08 ÷ 12 = 3024/25 = 120.96

Explanation:

10.08 ÷ 12 = 1008/100 ÷ 12

= 1008/100 . 12/1

= (1008 . 12)/100

= (100.8 . 3)/25

= 3024/25 = 120.96

Question 6.

– \(\frac{5}{6}\) ÷ \(\frac{3}{10}\) = -25/9 = -2.7777

Answer:

– \(\frac{5}{6}\) ÷ \(\frac{3}{10}\)

Explanation:

– \(\frac{5}{6}\) ÷ \(\frac{3}{10}\) = -5/6 . 10/3

= (-5 . 10)/(3 . 6)

= -25/9 = -2.7777

Question 7.

Which ratio can be represented by the tape diagram?

A. 3 : 4

B. 4 : 5

C. 4 : 9

D. 8 : 12

Answer:

D. 8 : 12

Explanation:

Quantity 1 has 2 boxes, quantity 2 has 3 boxes.

Out of all given ratios, 8 : 12 satisfy the condition

The ratio of given quantity = 2 : 3

2 * 4 : 3 * 4 = 8 : 12

**Concepts, Skills, & Problem Solving**

**OPEN-ENDED** Complete the ratio table using multiple operations. Are the ratios in the table equivalent? Explain. (See Exploration 2, p. 183.)

Question 8.

Answer:

Explanation:

4 x 2 = 8, 10 2 = 20

8 x 2 = 16, 20 x 2 = 40

16/8 = 2, 40/8 = 5

The equivalent ratios are 4:10, 8:20, 16:40, 2:5.

Question 9.

Answer:

Explanation:

4/5 x 10 = 8, 1/2 x 10 = 5

4/5 x 5 = 4, 1/2 x 5 = 5/2

4 x 1/25 = 4/25, 5/2 x 1/25 = 1/10

**WRITING AND INTERPRETING RATIOS** Find the ratio. Then find and interpret the value of the ratio.

Question 10.

club soda : white grape juice

Answer:

club soda : white grape juice = 2 : 1

Explanation:

Club soda = 4 cups

white grape juice = 2 cups

club soda : white grape juice = 4 : 2

= 2 : 1

Question 11.

mint leaves : chopped watermelon

Answer:

mint leaves : chopped watermelon = 1 : 6

Explanation:

mint leaves = 1/2 cup

chopped watermelon = 3 cups

mint leaves : chopped watermelon = 1/2 : 3

= 1 : 3 x 2

= 1 : 6

Question 12.

white grape juice to sugar

Answer:

white grape juice : sugar = 8 : 3

Explanation:

white grape juice = 2 cups

sugar=3/4 cup

white grape juice : sugar = 2 : 3/4

= 2 x 4: 3

= 8 : 3

Question 13.

lime juice to mint leaves

Answer:

lime juice to mint leaves is 3 : 2

Explanation:

lime juice = 3/4 cup

mint leaves = 1/2 cup

lime juice to mint leaves = 3/4 : 1/2

= 3/2 : 1

= 3 : 2

Question 14.

You have blue ribbon and red ribbon in the ratio \(\frac{1}{2}\) : \(\frac{1}{5}\) . Your friend finds the value of the ratio. Is your friend correct? Explain your reasoning.

Answer:

Wrong.

Explanation:

1/2 : 1/5 = 1/2 ÷ 1/5

= 1/2 . 5/1 = 5/2

**COMPLETING A RATIO TABLE** Find the missing values in the ratio table. Then write the equivalent ratios.

Question 15.

Answer:

Explanation:

To find the missing values,

1/6 . x = 2/3

x = 2/3 . 6

x = 4

So, calories = 20 . 4 = 80

20 . y = 10

y = 10/20 = 1/2

Miles = 1/6 . 1/2 = 1/12

10 . z = 90

z = 90/10

z = 9

Miles = 1/12 . 9 = 3/4

The equivalent ratios are 20 : 1/6, 80 : 2/3, 10 : 1/12, 90 : 3/4

Question 16.

Answer:

Explanation:

To find the missing values,

8/2 = 4

Minutes = 1/3/2 = 1/3 . 2 = 1/6

1/3 . x = 1/4

x = 1/4 . 3 = 3/4

Meters = 8 . 3/4 = 6

1/3 . y = 5/12

y = 5/12 . 3 = 5/4

Meters = 8 . 5/4 = 10

The equivalent ratios are 8 : 1/3, 4 : 2/3, 6 : 1/4, 10 : 5/12

Question 17.

Answer:

Explanation:

To find the missing values,

1/2 . x = 1

x = 2

1/24 . 2 = 1/12 = Feet

1/24 . y = 1/8

y = 1/8 . 24 = 3

Inches = 1/2 . 3 = 3/2

1/2 . z = 1/4

z = 1/4 . 2 = 1/2

Feet = 1/24 . 1/2 = 1/48

The equivalent ratios are 1/24 : 1/2, 1/12 : 1, 1/8 : 3, 1/48 : 1/4

Question 18.

Answer:

Explanation:

To find missing values,

1.5 . x = 1

x = 1/1.5 = 2/3

Tea = 3.75 . 2/3 = 2.5

1.5 . y = 3.5

y = 35/15 = 7/3

Tea = 3.75 . 7/3 = 8.75

1.5 . z = 2.5

z = 5/3

3.75 . 5/3 = 6.25

Question 19.

**CRITICAL THINKING**

Are the two statements equivalent? Explain your reasoning.

The ratio of boys to girls is 2 to 3.

The ratio of girls to boys is 3 to 2.

Answer:

Yes, both the statements are equivalent.

Explanation:

The ratio of boys to girls is 2 to 3.

boys : girls = 2 : 3

The ratio of girls to boys is 3 to 2.

girls : boys = 3 : 2

Question 20.

**MODELING REAL LIFE**

A city dumps plastic shade balls into a reservoir to prevent water from evaporating during a drought. It costs $5760 for 16,000 shade balls. How much does it cost for 12,000 shade balls?

Answer:

The cost for 12,000 shade balls is $4320

Explanation:

It costs $5760 for 16,000 shade balls.

The cost for 12,000 shade balls = (5760 x 12,000)/16000

= 69120/16

= 4320

So, the The cost for 12,000 shade balls is $4320

Question 21.

**MODELING REAL LIFE**

An oil spill spreads 25 square meters every \(\frac{1}{6}\) hour. What is the area of the oil spill after 2 hours?

Answer:

The area that the oil spill covers after 2 hours is 300 sq meters.

Explanation:

An oil spill spreads 25 square meters every 1/6 hour

The unit rate = 25 ÷ 1/6

= 25 x 6 = 150 square meters per hour

Every 1 hour spreads 150 square meters

The area that the oil spill covers after 2 hours = 2 x 150 = 300 sq meters

Question 22.

**MODELING REAL LIFE**

You mix 0.25 cup of juice concentrate for every 2 cups of water to make 18 cups of juice. How much juice concentrate do you use? How much water do you use?

Answer:

To make 18 cups of juice we use 16 cups of water and 2 cups of juice concentrate

Explanation:

The ratio between cups of water and cups of juice concentration is 2 to 1/4

The unit rate = 2 / 1/4

= 2 x 4 = 8 cups of water per cup of juice concentrate

So, to make 9 cups of juice we use 8 cups of water and 1 cup of juice concentrate

To make 18 cups of juice we use 16 cups of water and 2 cups of juice concentrate

Question 23.

A store sells 2\(\frac{1}{4}\) pounds of mulch for every 1\(\frac{1}{2}\) pounds of gravel sold. The store sells 180 pounds of mulch and gravel combined. How many pounds of each item does the store sell?

Answer:

Mulch = 108 pounds, gravel = 72 pounds

Explanation:

Let 2(1/4)x = 9/4 x be the amount of mulch so that 1(1/2)x = 3x/2 is the amount of gravel the store sell, both in pounds.

Given that thestore sells a total of 180 pounds of mulch and gravel, we can write

9x/4 + 3x/2 = 180

9x + 6x = 180 x 4

15x = 720

x = 720/15

x = 48

So, the store sells 9/4(48) = 108 pounds of mulch and 3/2(48) = 72 pounds of gravel.

Question 24.

**DIG DEEPER!**

You mix \(\frac{1}{4}\) cup of red paint for every \(\frac{1}{2}\) cup of blue paint to make 3 gallons of purple paint.

a. How much red paint do you use? How much blue paint do you use?

b. You decide that you want to make a lighter purple paint. You make the new mixture by adding \(\frac{1}{4}\) cup of white paint for every \(\frac{1}{4}\) cup of red paint and \(\frac{1}{2}\) cup of blue paint. How much red paint, blue paint, and white paint do you use to make 1\(\frac{1}{2}\) gallons of the lighter purple paint?

Answer:

a. We use 12 cups of red paint, 24 cups of blue paint

b. white paint = 6 cups, red paint = 6 cups, blue paint = 12 cups.

Explanation:

You mix \(\frac{1}{4}\) cup of red paint for every \(\frac{1}{2}\) cup of blue paint to make 3 gallons of purple paint.

a. The ratio of red paint to blue paint = 1/4 : 1/2

1/4x + 1/2x = (x + 2x)/4 = 3x/4

3x/4 = 3

x = 3 . 4/3

x = 4 cups

1 cup = 1/16 gallons

So, 4 cups = 4/16 = 1/4 gallons

12 . 4 = 48 cups = 1/4 . 12 = 3 gallons

So, 1/4(48) = 12 cups of red paint and 1/2(48) = 24 cups of blue paint is required to make 3 gallons

b. You make the new mixture by adding \(\frac{1}{4}\) cup of white paint for every \(\frac{1}{4}\) cup of red paint and \(\frac{1}{2}\) cup of blue paint.

(1/4 + 1/4 +1/2) = 1

1 cup = 1/16 gallons

The total lighter purple paint = 1(1/2) = 3/2 gallons

Total number of cups = (3/2) 16

= 24

white paint = 1/4 (24) = 6 cups

red paint = 1/4 (24) = 6 cups

blue paint = 1/2 (24) = 12 cups

### Lesson 5.2 Rates and Unit Rates

**EXPLORATION 1**

Writing Rates

Work with a partner.

a. How many degrees does the minute hand on a clock move every 15 minutes? Write a rate that compares the number of degrees moved by the minute hand to the number of hours elapsed.

b. Can you use the rate in part(a) to determine how many degrees the minute hand moves in \(\frac{1}{2}\) hour? Explain your reasoning.

c. Write a rate that represents the number of degrees moved by the minute hand every hour. How can you use this rate to find the number of degrees moved by the minute hand in 2\(\frac{1}{2}\) hours?

d. Draw a clock with hour and minute hands. Draw another clock that shows the time after the minute hand moves 900°. How many degrees does the hour hand move in this time? in one hour? Explain your reasoning.

**Try It**

Question 1.

There is \(\frac{1}{4}\) gram of fat for every \(\frac{1}{3}\) tablespoon of powdered peanut butter. How many grams of fat are there for every tablespoon of the powder?

Answer:

We have 3/4 grams of fat for every tablespoon of the powder.

Explanation:

There is \(\frac{1}{4}\) gram of fat for every \(\frac{1}{3}\) tablespoon of powdered peanut butter.

There are 1/4 / 1/3 = 1/4. 3 = 3/4 grams of fat for every tablespoon of the powder.

Question 2.

**WHAT IF?**

The scientist later states that the iron travels 3 feet every 10 minutes. Does this change your answer in Example 2? Explain.

Answer:

The liquid iron travels about 432 feet in 1 day.

Explanation:

The iron travels 3 feet for every 10 minutes

The ratio of feet to minutes is 3 : 10

Divide the quantity by 10 to get the unit rate per minute. Then multiply each quantity by 1440 to find the distance traveled in 24 hours.

So, the liquid iron travels about 432 feet in 1 day.

**Self-Assessment for Concepts & Skills**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.

**VOCABULARY**

How can you tell when a rate is a unit rate?

Answer:

A unit rate is described as hoe many units of the first type of quantity corresponds to one unit of the second type of quantity. When the denominator is 1, then you can say that that rate is a unit rate. Few examples of unit rate are m/sec, feet/hour.

Question 4.

**WRITING**

Explain why rates are usually written as unit rates.

Answer:

Generally, a rate is a ratio of two quantities. Rates are usually written as unit rates because they are easier to read, easier to understand and easier to compare.

**Find the unit rate.**

Question 5.

$1.32 for 12 ounces

Answer:

$0.11 for 1 ounce

Explanation:

$1.32 for 12 ounces = 1.32 : 12

= 1.32/12 : 12/12

= 0.11 : 1

Question 6.

\(\frac{1}{4}\) gallon for every \(\frac{3}{10}\) mile

Answer:

5/6 gallon for every 1 mile

Explanation:

\(\frac{1}{4}\) gallon for every \(\frac{3}{10}\) mile = 1/4 : 3/10

= 1/4 x 10/3 : 3/10 x 10/3

= 5/6 : 1

Question 7.

**USING TOOLS**

Find the missing values in the ratio table. Then write the unit rate of grams per cup and the unit rate of cups per gram.

Answer:

The unit rate of grams per cup = 15/4 : 1

The unit rate of cups per gram = 1 : 4/15

Explanation:

2/3 . x = 1/6

x = 1/6 . 3/2 = 1/4

1/4 . 5/2 = 5/8 = grams

5/2 . y = 1

y = 1 . 2/5 = 2/5

2/3 . 2/5 = 4/15 = cups

5/2 . 3/2 = 15/4

So, 2/3 . 3/2 = 1 = cups

2/3 . z = 4

z = 4 . 3/2 = 6

So, 5/2 . 6 = 15

grams: cups = 5/2 : 2/3

The unit rate of grams per cup = 5/2 . 3/2 : 1 = 15/4 : 1

cups : grams = 4/15 : 1

**Self-Assessment for Problem Solving**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 8.

Two people compete in a five-mile go-kart race. Person A travels \(\frac{1}{10}\) mile every 15 seconds. Person B travels \(\frac{3}{8}\) mile every 48 seconds. Who wins the race? What is the difference of the finish times of the competitors?

Answer:

Person B wins the race.

The difference of the finish times of the competitors = 1.833 minutes

Explanation:

Two people compete in a five-mile go-kart race.

Person A travels 1/10 miles for 15 seconds = 1/10 : 15

Person A unit rate = 1/150 : 1

It means, A travels 1/150 miles per1 second

So, person A completes the race in 5 x 150 = 750 seconds = 750/60 = 12.5 minutes

Person B travels 3/8 mile for 48 seconds = 3/8 : 48

Person B unit rate = 3/384 : 1

It means B travels 3/384 miles in 1 second.

So, person B completes the race in 5 . (384/3) = 1920/3 = 640

= 640/60 = 10.666 minutes

Hence, person B wins the race.

The difference of the finish times of the competitors = 12.5 – 10.66 = 1.8333 minutes

Question 9.

**DIG DEEPER!**

A bus travels 0.8 mile east every 45 seconds. A second bus travels 0.55 mile west every 30 seconds. The buses start at the same location. Use two methods to determine how far apart the buses are after 15 minutes. Explain your reasoning.

Answer:

Two buses are 1/2 mile apart after 15 minutes.

Explanation:

A bus travels 0.8 mile east every 45 seconds = 0.8 : 45

The unit rate = 0.8/45 : 1

A second bus travels 0.55 mile west every 30 seconds = 0.55 : 30

The unit rate = 0.55/30 : 1

Find how much distance, busses travelled after 15 minutes = 15 x 60 = 900 seconds

First bus travels (0.8/45) x 900 = 16 miles

Second bus travels (0.55/30) x 900 = 16.5 miles

The difference in their distances = 16.5 – 16 = 1/2 mile

### Rates and Unit Rates Homework & Practice 5.2

**Review & Refresh**

**Find the missing values in the ratio table. Then write the equivalent ratios.**

Question 1.

Answer:

The equivalent ratios are 3/4 : 1/3, 3/2 : 2/3, 3 : 3/4, 1 : 4/9

Explanation:

To find the missing values,

1/3 . x = 2/3

x = 2

3/4 . 2 = 3/2 = flour

3/4 . y = 3

y = 4

1/3 . 4 = 4/3 = oats1

3/4 . z = 1

z = 4/3

4/3 . 1/3 = 4/9 = oats2

The equivalent ratios are 3/4 : 1/3, 3/2 : 2/3, 3 : 3/4, 1 : 4/9

Question 2.

Answer:

The equivalent ratios are 1/4 : 1/2, 3/4 : 3/2, 3/2 : 3, 5 : 10

Explanation:

To find the missing values,

1/4 . x = 3/4

x = 3

1/2 . 3 = 3/2 = minutes1

1/2 . y = 3

y = 6

1/4 . 6 = 3/2 = pages

1/4 . z = 5

z = 20

1/2 . 20 = 10 = minutes 2

The equivalent ratios are 1/4 : 1/2, 3/4 : 3/2, 3/2 : 3, 5 : 10

**Copy and complete the statement using <, >, or =.**

Question 3.

Answer:

9/2 > 8/3

Explanation:

9/2 = 4.5

8/3 = 2.6666

So, 9/2 > 8/3

Question 4.

Answer:

-8/15 < 10/18

Explanation:

-8/15 = -0.5333

10/18 = 0.555

So, -8/15 < 10/18

Question 5.

Answer:

-6/24 = -2/8

Explanation:

-6/24 = -0.25

-2/8 = -0.25

So, -6/24 = -2/8

**Concepts, Skills, & Problem Solving**

**WRITING RATES** Find the number of degrees moved by the minute hand of a clock in the given amount of time. Explain your reasoning. (See Exploration 1, p. 189.)

Question 6.

\(\frac{2}{3}\) hour

Answer:

240 degrees.

Explanation:

The minute hand passes through 60 minutes ( = 360 degrees) in 1 hour.

In 2/3 hours it passes through 2/3 * 360 = 240 degrees.

Question 7.

\(\frac{7}{12}\) hour

Answer:

210 degrees

Explanation:

The minute hand passes through 60 minutes ( = 360 degrees) in 1 hour.

In 7/12 hours it passes through 7/12 * 360 = 210 degrees

Question 8.

1\(\frac{1}{4}\) hours

Answer:

450 degrees = (360 + 90) degrees

Explanation:

The minute hand passes through 60 minutes ( = 360 degrees) in 1 hour.

In 1(1/4) = 5/4 hours it passes through 5/4 * 360 = 450 degrees

**FINDING UNIT RATES** Find the unit rate.

Question 9.

180 miles in 3 hours

Answer:

6 miles per 1 hour

Explanation:

180 miles in 3 hours = 18 : 3

The unit rate is 18/3 : 1 = 6 : 1

Question 10.

256 miles per 8 gallons

Answer:

32 miles per 1 gallon

Explanation:

256 miles per 8 gallons = 256 : 8

The unit rate is 256/8 : 1 = 32 : 1

Question 11.

\(\frac{1}{2}\) pound : 5 days

Answer:

1/10 pounds for 1 day

Explanation:

\(\frac{1}{2}\) pound : 5 days = 1/2 : 5

The unit rate is 1/2/5 : 1

= 1/10 : 1

Question 12.

4 grams for every \(\frac{3}{4}\) serving

Answer:

16/3 grams per serving

Explanation:

4 grams for every \(\frac{3}{4}\) serving = 4 : 3/4

The unit rate is = 4 x (4/3) : 1

= 16/3 : 1

Question 13.

$9.60 for 4 pounds

Answer:

$2.4 per 1 pound

Explanation:

$9.60 for 4 pounds = 9.60 : 4

The unit rate is 9.60/4 : 1

= 2.4 : 1

Question 14.

$4.80 for 6 cans

Answer:

$0.8 per 1 can

Explanation:

$4.80 for 6 cans = 4.80 : 6

The unit rate is 4.80/6 : 1

= 0.8 : 1

Question 15.

297 words in 5.5 minutes

Answer:

54 words in 1 minute

Explanation:

297 words in 5.5 minutes = 297 : 5.5

The unit rate is 297/5.5 : 1

= 54 : 1

Question 16.

\(\frac{1}{3}\) kilogram: \(\frac{2}{3}\) foot

Answer:

1/2 kilogram per 1 foot

Explanation:

\(\frac{1}{3}\) kilogram: \(\frac{2}{3}\) foot = 1/3 : 2/3

The unit rate is 1/3 x (3/2) : 1

= 1/2 : 1

Question 17.

\(\frac{5}{8}\) ounce per \(\frac{1}{4}\) pint

Answer:

5/2 ounce per 1 pint

Explanation:

\(\frac{5}{8}\) ounce per \(\frac{1}{4}\) pint = 5/8 : 1/4

The unit rate is 5/8 x 4 : 1

= 5/2 : 1

Question 18.

21\(\frac{3}{4}\) meters in 2\(\frac{1}{2}\) hours

Answer:

87 meters in 1 hour

Explanation:

21\(\frac{3}{4}\) meters in 2\(\frac{1}{2}\) hours = 21(3/4) : 2(1/2)

= 87/4 : 5/2

The unit rate is 87/4 x (2/5) : 1

= 87/10 : 1

**USING TOOLS** Find the missing values in the ratio table. Then write the equivalent ratios.

Question 19.

Answer:

The equivalent ratios are 25 : 1/3, 50 : 2/3, 75 : 1, 100 : 4/3

Explanation:

To find the missing values,

25 . x = 50

x = 50/25 = 2

1/3 . 2 = 2/3 = servings

1/3 . y = 1

y = 3

25 . 3 = 75 = calories1

1/3 . z = 4/3

z = 4

25 . 4 = 100 = calories2

The equivalent ratios are 25 : 1/3, 50 : 2/3, 75 : 1, 100 : 4/3

Question 20.

Answer:

The equivalent ratios are 4 : 3/4, 4/3 : 1/4, 16/3 : 1, 16 : 3

Explanation:

To find the missing values,

4x = 4/3

x = 1/3

3/4 . 1/3 = 1/4 = time1

3/4 y = 1

y = 4/3

4 . 4/3 = 16/3 = oxygen

4z = 16

z = 16/4 = 4

3/4 . 4 = 3 = time2

The equivalent ratios are 4 : 3/4, 4/3 : 1/4, 16/3 : 1, 16 : 3

Question 21.

**PROBLEM SOLVING**

In January 2012, the U.S.population was about 313 million people. In January 2017, it was about 324 million. What was the average rate of population change per year?

Answer:

The average rate of population change per year = 2.2 million

Explanation:

In January 2012, the U.S.population was about 313 million people

In January 2017, it was about 324 million

After 5 years, the population change = 324 – 313 = 11 million

The average rate of population change per year = 11/5 = 2.2 million

Question 22.

**MODELING REAL LIFE**

You can sand \(\frac{4}{9}\) square yard of wood in \(\frac{1}{2}\) hour. How many square yards can you sand in 3.2 hours? Justify your answer.

Answer:

128/45 square yards can you sand in 3.2 hours.

Explanation:

You can sand \(\frac{4}{9}\) square yard of wood in \(\frac{1}{2}\) hour

In 3.2 hours, you can sand 4/9 x 3.2 x 2 sq yads

= 25.6/9 = 128/45

So, 128/45 square yards can you sand in 3.2 hours.

Question 23.

**REASONING**

Tell whether the rates are equivalent. Justify your answer.

Question 23.

75 pounds per 1.5 years

38.4 ounces per 0.75 year

Answer:

Rates are not equivalent.

Explanation:

75 pounds per 1.5 years = 75 : 1.5

= 750/15 = 50

38.4 ounces per 0.75 year = 38.4 : 0.75

= 384 : 7.5 = 51.2

So, rates are not equivalent

Question 24.

7\(\frac{1}{2}\) miles for every \(\frac{3}{4}\) hour

\(\frac{1}{2}\) mile for every 3 minutes

Answer:

Rates are not equivalent.

Explanation:

7\(\frac{1}{2}\) miles for every \(\frac{3}{4}\) hour = 7(1/2) : 3/4

= 15/2 : 3/4

= 15/2 . 4/3 = 10

\(\frac{1}{2}\) mile for every 3 minutes = 1/2 : 3

= 1 : 6

So, rates are not equivalent

Question 25.

**PROBLEM SOLVING**

The table shows nutritional information for three beverages.

a. Which has the most calories per fluid ounce?

b. Which has the least sodium per fluid ounce?

Answer:

a. whole milk has the most calories per fluid ounce

b. Oange juice has the least sodium per fluid ounce.

Explanation:

a.

Convert serving size to fluid ounce

Whole Milk = 1 c = 8 fl oz

Orange juice = 1 pt = 16 fl oz

Whole milk = 146 : 8 = 146/8 : 1

= 18.25

There are 18.25 calories in 1 fluid ounce of Whole Milk.

Orange juice = 210 : 16 = 210/16 : 1

= 13.125

There are 13.125 calories in 1 fluid ounce of Orange Juice.

Apple juice = 351 : 24

= 351/24 : 1

There are 14.625 calories in 1 fluid ounce of Apple Juice.

So, whole milk has the most calories per fluid ounce

b. Whole milk = 98 : 8

= 98/8 : 1

= 12.25

Orange juice = 10 : 16

= 10/16 : 1

= 0.625

Apple juice = 21 : 24

= 21/24 : 1

= 0.875

So, orange juice has the least sodium per fluid ounce.

Question 26.

**MODELING REAL LIFE**

A shuttle leaving Earth’s atmosphere travels 15 miles every 2 seconds. When entering the Earth’s atmosphere, the shuttle travels 2\(\frac{3}{8}\) miles per \(\frac{1}{2}\) second. Find the difference in the distances traveled after 15 seconds when leaving and entering the atmosphere.

Answer:

The difference in the distances traveled is 2503 miles.

Explanation:

A shuttle leaving Earth’s atmosphere travels 15 miles every 2 seconds. = 15 : 2

= 15/2 : 1

= 7.5 miles for every second

When entering the Earth’s atmosphere, the shuttle travels 2\(\frac{3}{8}\) miles per \(\frac{1}{2}\) second = 2(3/8) : /2

= 19/8 : 1/2

= 19/8 . 2 = 19/4

= 4.75 miles per second

After 15 minutes = 15 x 60 = 900 seconds,

A shuttle leaving Earth’s atmosphere travels = 7.5 x 900

= 6750 miles

When entering the Earth’s atmosphere, the shuttle travels = 4.75 x 900

= 4247 miles

The difference in the distances traveled = 6750 – 4247

= 2503 miles

Question 27.

**RESEARCH**

Fire hydrants are one of four different colors to indicate the rate at which water comes from the hydrant.

a. Use the Internet to find the ranges of rates indicated by each color.

b. Research why a fire fighter needs to know the rate at which water comes out of a hydrant.

Answer:

a. Blue – 1500 or more gallons per minute

Green – 1000 to 1499 gallons per minute

Red – Less than 500 gallons per minute

Orange – 500 to 999 gallons per minute

b. Fire hydrants are flow-tested at a residual pressure of 20 psi. Therefore, firefighters should understand the typical flow rates of fire hydrants at a pressure. They should not expect that the fire hydrant will be supplied at an increased pressure, resulting in a greater flow rate.

Explanation:

a. Blue – 1500 or more gallons per minute

Green – 1000 to 1499 gallons per minute

Red – Less than 500 gallons per minute

Orange – 500 to 999 gallons per minute

Question 28.

**DIG DEEPER!**

You and a friend start riding bikes toward each other from opposite ends of a 24-mile biking route. You ride 2\(\frac{1}{6}\) miles every \(\frac{1}{4}\) hour. Your friend rides 7\(\frac{1}{3}\) miles per hour.

a. After how many hours do you meet?

b. When you meet, who has traveled farther? How much farther?

Answer:

a. They meet after 1(1/2) hour.

b. You have traveled farther 2 miles.

Explanation:

You ride 2\(\frac{1}{6}\) miles every \(\frac{1}{4}\) hour.

= 2(1/6) : 1/4

= 13/6 : 1/4

= 13/6 . 4 = (13 . 2)/3 = 26/3 : 1

Your friend rides 7\(\frac{1}{3}\) miles per hour.

= 7(1/3) : 1

= 22/3 : 1

After 1 hour, you traveled = 26/3 miles, your friend traveled 22/3 miles

= 26/3 + 22/3 = 48/3

= 16 miles

Since you and your friend moving towards each other at 16 miles per hour

we can write 24/16 = 3/2 = 1.5 hours.

So, they meet after 1(1/2) hour.

b. You have traveled 26/3 (1.5) = 13 miles at the time of the meeting

Your friend has traveled 22/3 (1.5) = 11 miles at the time of the meeting.

You have traveled farther 2 miles.

### Lesson 5.3 Identifying Proportional Relationships

**EXPLORATION 1**

Determining Proportional Relationships

Work with a partner.

a. You can paint 50 square feet of a surface every 40 minutes. How long does it take you to paint the mural shown? Explain how you found your answer.

b. The number of square feet you paint is proportional to the number of minutes it takes you. What do you think it means for a quantity to be proportional to another quantity?

c. Assume your friends paint at the same rate as you. The table shows how long it takes you and different numbers of friends to paint a fence. Is x proportional to y in the table? Explain.

d. How long will it take you and four friends to paint the fence? Explain how you found your answer.

Answer:

a. It takes 3 hours to paint the mural.

b. It means if the number of squares feet increases or decreases then the number of minutes to paint also increases or decreases respectively.

c. x is not proportional to y.

Explanation:

a. The area of mural = 25 x 9 = 225 sq ft

You can paint 50 square feet of a surface every 40 minutes

To paint the given mural it takes (40 x 225)/50 = 180 minutes = 3 hours

b. The number of square feet you paint is proportional to the number of minutes it takes you.

It means if the number of squares feet increases or decreases then the number of minutes to paint also increases or decreases respectively.

c. x : y = 1 : 4, 2 : 2 = 1 : 1

3 : 4/3 = 9 : 4, 4 : 1

All those are not equa so, x is not proportional to y.

**Try It**

**Tell whether the ratios form a proportion.**

Question 1.

1 : 2 and 5 : 10

Answer:

1 : 2 and 5 : 10 form a proportion.

Explanation:

Compare the values of the ratios

1/2

5/10 = (5 ÷ 5) / (10 ÷ 5) = 1/2

The values of the ratios are equivalent

So, 1 : 2 and 5 : 10 form a proportion.

Question 2.

4 : 6 and 18 : 24

Answer:

4 : 6 and 18 : 24 do not form a proportion.

Explanation:

Compare the values of the ratios

4/6 = (4 ÷ 2)/(6 ÷ 2) = 2/3

18/24 = (18 ÷ 6)/(24 ÷ 6) = 3/4

The values of the ratios are not equivalent

So, 4 : 6 and 18 : 24 do not form a proportion.

Question 3.

4.5 to 3 and 6 to 9

Answer:

4.5 to 3 and 6 to 9 do not form a proportion.

Explanation:

Compare the values of the ratios

4.5/3 = 45/30

= (45 ÷ 15) / (30 ÷ 15) =3/2

6/9 = (6 ÷ 3) / (9 ÷ 3)

= 2/3

The values of the ratios are not equivalent

So, 4.5 to 3 and 6 to 9 do not form a proportion.

Question 4.

\(\frac{1}{2}\) to \(\frac{1}{4}\) and 8 to 4

Answer:

\(\frac{1}{2}\) to \(\frac{1}{4}\) and 8 to 4 form a proportion

Explanation:

Compare the values of the ratios

1/2 : 1/4 = (1/2) / (1/4)

= 1/2 . 4 = 2 : 1

8 : 4 = (8 ÷ 4) / (4 ÷ 4) = 2 : 1

The values of the ratios are equivalent

So, \(\frac{1}{2}\) to \(\frac{1}{4}\) and 8 to 4 form a proportion

**Tell whether the ratios form a proportion.**

Question 5.

6 : 2 and 12 : 1

Answer:

6 : 2 and 12 : 1 do not form a proportion.

Explanation:

Compare the values of the ratios

6 : 2 = (6 ÷ 2) : (2 ÷ 2) = 3 : 1

12 : 1 = (12 ÷ 1) : (1 ÷ 1) = 12 : 1

The values of the ratios are not equivalent

So, 6 : 2 and 12 : 1 do not form a proportion.

Question 6.

8 : 12 and \(\frac{2}{3}\) : 1

Answer:

8 : 12 and \(\frac{2}{3}\) : 1 form a proportion

Explanation:

Use the cross product property to determine whether the ratios form a proportion.

8/12 = 2/3 : 1

8/12 = 2/3

8 . 3 = 12 . 2

24 = 24

The cross productsare equal.

So, 8 : 12 and \(\frac{2}{3}\) : 1 form a proportion

**Tell whether x and y are proportional.**

Question 7.

Answer:

x and y are proportional.

Explanation:

Compare the values of ratios x to y

1/2, 2/4 = 1/2, 3/6 = 1/2, 4/8 = 1/2

The values of the ratios are equivalent.

So, x and y are proportional.

Question 8.

Answer:

x and y are not proportional.

Explanation:

Compare the values of ratios x to y

2/4 = 1/2, 4/2 = 2, 6/1 = 6, 8/1/2 = 16, 10/1/4 = 40

The values of the ratios are not equivalent.

So, x and y are not proportional.

**Self-Assessment for Concepts & Skills**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

**PROPORTIONS** Tell whether the ratios form a proportion.

Question 9.

4 : 14 and 12 : 40

Answer:

4 : 14 and 12 : 40 do not form a proportion.

Explanation:

Use the cross product property to determine whether the ratios form a proportion.

4/14 = 12/40

Cross multiply

4 . 40 = 12 . 14

160 ≠ 168

The cross products are not equal

So, 4 : 14 and 12 : 40 do not form a proportion

Question 10.

9 : 3 and 45 : 15

Answer:

9 : 3 and 45 : 15 form a proportion.

Explanation:

Compare the values of the ratios

9/3 = (9 ÷ 3) / (3 ÷ 3) = 3/1

45/15 = (45 ÷ 15) / (15 ÷ 15) = 3/1

The ratios are equivalent

So, 9 : 3 and 45 : 15 form a proportion

Question 11.

**VOCABULARY**

Explain how to determine whether two quantities are proportional.

Answer:

The two quantities are proportional means they are having equivalent ratios.

Question 12.

**WHICH ONE DOESN’T BELONG?**

Which ratio does not belong with the other three? Explain your reasoning.

Answer:

3/5 does not belong with the other three.

Explanation:

4/10 = 2/5

2/5

3/5

6/15 = 2/5

So, 3/5 does not belong with the other three.

**Self-Assessment for Problem Solving**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 13.

After making 20 servings of pasta, a chef has used 30 cloves of garlic. The chef used 6 cloves to make the first 4 servings. How many cloves of garlic are used to make 10 servings? Justify your answer.

Answer:

15 cloves of garlic are used to make 10 servings?

Explanation:

After making 20 servings of pasta, a chef has used 30 cloves of garlic.

The ratio of garlic cloves to serving of pasta = 30 : 20

= 30/20 : 1

= 3/2 : 1

The chef used 6 cloves to make the first 4 servings

The ratio of garlic cloves to serving of pasta = 6 : 4

= 6/4 : 1

= 3/2 : 1

It means 1.5 garlic cloves are used to make 1 serving.

The ratios are proportional

So, to make 10 servings, multiply 1.5 by 10

= 1.5 x 10 = 15 cloves

Therefore, 15 cloves of garlic are used to make 10 servings.

Question 14.

**DIG DEEPER!**

A runner completes a 25-mile race in 5 hours. The runner completes the first 7.5 miles in 1.5 hours.

a. Do these rates form a proportion? Justify your answer.

b. Can you determine, with certainty, the time it took the runner to complete 10 miles? Explain your reasoning.

Answer:

a. Yes, these form a proportion

b. It takes 2 hours for the runner to complete 10 miles.

Explanation:

a. A runner completes a 25-mile race in 5 hours.

The ratio of miles to hours = 25 : 5

= 25/5 : 1

= 5 : 1

The runner completes the first 7.5 miles in 1.5 hours.

The ratio of miles to hours = 7.5/1.5

= 75/15 : 1

= 5 : 1

It means the runner completes 5 miles in 1 hour

The ratios are proportional

b. So, it takes 10/5 = 2 hours for the runner to complete 10 miles.

### Identifying Proportional Relationships Homework & Practice 5.3

**Review & Refresh**

**Find the unit rate.**

Question 1.

30 inches per 5 years

Answer:

6 inches per 1 year

Explanation:

30 inches per 5 years = 30 : 5

= 30/5 : 1

= 6 : 1

6 inches per 1 year

Question 2.

486 games every 3 seasons

Answer:

162 games every season

Explanation:

486 games every 3 seasons = 486 : 3

= 486/3 : 1

= 162 : 1

162 games every season

Question 3.

8750 steps every 1.25 hours

Answer:

7000 steps every hour

Explanation:

8750 steps every 1.25 hours = 8750 : 1.25

= 8750/1.25 : 1

= 7000 : 1

7000 steps every hour

Question 4.

3.75 pints out of every 5 gallons

Answer:

0.75 prints out of every gallon

Explanation:

3.75 pints out of every 5 gallons = 3.75 : 5

= 3.75/5 : 1

= 3/4 : 1

0.75 prints out of every gallon

**Add or subtract.**

Question 5.

– 28 + 15

Answer:

– 28 + 15 = -13

Explanation:

– 28 + 15 = -13

Question 6.

– 6 + (- 11)

Answer:

– 6 + (- 11) = -17

Explanation:

– 6 + (- 11) = -6 – 11

= -17

Question 7.

– 10 – 8

Answer:

– 10 – 8 = -18

Explanation:

– 10 – 8 = -18

Question 8.

– 17 – (- 14)

Answer:

– 17 – (- 14) = -3

Explanation:

– 17 – (- 14) = -17 + 14

= -3

**Solve the equation.**

Question 9.

\(\frac{x}{6}\) = 25

Answer:

x = 150

Explanation:

\(\frac{x}{6}\) = 25

Multiply both sides by 6

x/6 • 6 = 25 • 6

x = 150

Question 10.

8x = 72

Answer:

x = 9

Explanation:

8x = 72

Divide both sides by 8

8x/8 = 72/8

x = 9

Question 11.

150 = 2x

Answer:

x = 75

Explanation:

150 = 2x

Divide both sides by 2

150/2 = 2x/2

75 = x

Question 12.

35 = \(\frac{x}{4}\)

Answer:

x = 140

Explanation:

35 = \(\frac{x}{4}\)

Multiply both sides by 4

35 • 4 = x/4 • 4

140 = x

**Concepts, Skills, & Problem Solving**

**REASONING** You can paint 75 square feet of a surface every 45 minutes. Determine how long it takes you to paint a wall with the given dimensions. (See Exploration 1, p. 195.)

Question 13.

8 ft × 5 ft

Answer:

24 minutes

Explanation:

You can paint 75 square feet of a surface every 45 minutes

= 75 : 45

= 75/45 : 1

= 5/3 : 1

So, you can paint 5/3 sq feet of a surface in 1 minute.

The given area is 8 ft × 5 ft = 40 sq ft

Divide 40 by 5/3

= 40 / 5/3

= 40 . 3/5

= 24 minutes

Question 14.

7 ft × 6 ft

Answer:

25 minutes 12 seconds

Explanation:

The given area is 7 ft × 6 ft = 42 sq ft

Divide 42 by 5/3

= 42/5/3

= 42 . 3/5

= 126/5

= 25(1/5)

= 25 minutes 12 seconds

Question 15.

9 ft × 9 ft

Answer:

48 minutes 36 seconds

Explanation:

The given area is 9 ft × 9 ft = 81 sq ft

Divide 81 by 5/3

= 81/5/3

= 81 . 3/5

= 243/5

= 48(3/5)

= 48 minutes 36 seconds

**PROPORTIONS** Tell whether the ratios form a proportion.

Question 16.

1 to 3 and 7 to 21

Answer:

1 to 3 and 7 to 21 form a proportion.

Explanation:

Compare the values of the ratios

1 to 3 = 1 : 3

7 to 21 = 7 : 21

= (7 ÷ 7) : (21 ÷ 7)

= 1 : 3

The ratios are equivalent

So, 1 to 3 and 7 to 21 form a proportion.

Question 17.

1 : 5 and 6 : 30

Answer:

1 : 5 and 6 : 30 form a proportion

Explanation:

Compare the values of the ratios

1 : 5 = 1/5

6 : 30 = (6 ÷ 6) : (30 ÷ 6)

= 1/5

The ratios are equivalent

So, 1 : 5 and 6 : 30 form a proportion

Question 18.

3 to 4 and 24 to 18

Answer:

3 to 4 and 24 to 18 do not form a proportion

Explanation:

Compare the values of the ratios

3 to 24 = 3 : 24

= (3 ÷ 3) : (24 ÷ 3)

= 1 : 8

24 to 18 = 24 : 18

= (24 ÷ 6) : (18 ÷ 6)

= 4 : 3

The ratios are not equivalent

So, 3 to 4 and 24 to 18 do not form a proportion

Question 19.

3.5 : 2 and 14 : 8

Answer:

3.5 : 2 and 14 : 8 do not form a proportion

Explanation:

Compare the values of the ratios

35 : 2 = 35/2

14 : 8 = (14 ÷ 2) : (8 ÷ 2)

= 7 : 4 = 7/4

The ratios are not equivalent

So, 3.5 : 2 and 14 : 8 do not form a proportion

Question 20.

24: 30 and 3 : \(\frac{7}{2}\)

Answer:

24: 30 and 3 : \(\frac{7}{2}\) do not form a proportion.

Explanation:

Compare the values of the ratios

24 : 30 = 24/30

= (24 ÷ 2)/(30 ÷ 2)

= 12/15

3 : 7/2 = 6 : 7

= 6/7

The ratios are not equivalent

So, 24: 30 and 3 : \(\frac{7}{2}\) do not form a proportion.

Question 21.

\(\frac{21}{2}\) : 3 and 16 : 6

Answer:

\(\frac{21}{2}\) : 3 and 16 : 6 do not form a proportion.

Explanation:

21/2 : 3 = 21 : 6

= (21 ÷ 3) : (6 ÷ 3)

= 7 : 2

16 : 6 = (16 ÷ 2) : (6 ÷ 2)

= 8 : 3

The ratios are not equivalent

So, \(\frac{21}{2}\) : 3 and 16 : 6 do not form a proportion.

Question 22.

0.6 : 0.5 and 12 : 10

Answer:

0.6 : 0.5 and 12 : 10 form a proportion

Explanation:

Compare the values of the ratios

0.6 : 0.5 = 6/5

12 : 10 = (12 ÷ 2) : (10 ÷ 2)

= 6 : 5

The ratios are equivalent

So, 0.6 : 0.5 and 12 : 10 form a proportion

Question 23.

2 to 4 and 11 to \(\frac{11}{2}\)

Answer:

2 to 4 and 11 to \(\frac{11}{2}\) do not form a proportion.

Explanation:

Compare the values of the ratios

2 to 4 = 2 : 4

= (2 ÷ 2) : (4 ÷ 2)

= 1 : 2

11 to \(\frac{11}{2}\) = 11 : 11/2

= 22 : 11

= (22 ÷ 11) : (11 ÷ 11)

= 2 : 1

The ratios are not equivalent

So, 2 to 4 and 11 to \(\frac{11}{2}\) do not form a proportion.

Question 24

\(\frac{5}{8}\) : \(\frac{2}{3}\) and \(\frac{1}{4}\) : \(\frac{1}{3}\)

Answer:

\(\frac{5}{8}\) : \(\frac{2}{3}\) and \(\frac{1}{4}\) : \(\frac{1}{3}\) do not form a proportion.

Explanation:

Compare the values of the ratios

\(\frac{5}{8}\) : \(\frac{2}{3}\) = 5/8 / 2/3

= 5/8 . 3/2

= 15/16

\(\frac{1}{4}\) : \(\frac{1}{3}\) = 1/4 / 1/3

= 1/4 . 3 = 3/4

The ratios are not equivalent

So, \(\frac{5}{8}\) : \(\frac{2}{3}\) and \(\frac{1}{4}\) : \(\frac{1}{3}\) do not form proportion

**IDENTIFYING PROPORTIONAL RELATIONSHIPS** Tell whether x and y are proportional.

Question 25.

Answer:

x and y are not proportional.

Explanation:

Compare the values of the ratios x to y.

1/7, 2/8 = 1/4, 3/9 = 1/3

The values of the ratios are not equivalent.

So, x and y are not proportional.

Question 26.

Answer:

x and y are proportional.

Explanation:

Compare the values of the ratios x to y.

2/5, 4/10 = 2/5, 6/15 = 2/5

The values of the ratios are equivalent

So, x and y are proportional.

Question 27.

Answer:

x and y are proportional.

Explanation:

Compare the values of the ratios x to y.

0.25/4 = 1/16, 0.5/8 = 1/6, 0.75/12 = 1/16

The values of the ratios are equivalent

So, x and y are proportional.

Question 28.

Answer:

x and y are not proportional.

Explanation:

Compare the values of the ratios x to y.

2/3 / 7/10 = 2/3 . 10/7 = 20/21, 1/3/5 = 1/3 . 5 = 5/3

4/3 / 1/2 = 4/3 . 2 = 8/3

The values of the ratios are not equivalent

So, x and y are not proportional.

**YOU BE THE TEACHER** Your friend determines whether x and y are proportional. Is your friend correct? Explain your reasoning.

Question 29.

Answer:

Correct.

Explanation:

Compare the values of the ratios x to y.

8 + 1 = 9

3 + 1 = 4

The values of the ratios x to y are equal.

So, x and y are proportional.

Question 30.

Answer:

Wrong

Explanation:

Compare the values of the ratios x to y.

2/6 = 1/3

4/12 = 1/3

8/18 = 4/9

The values of the ratios x to y are not equal.

So, x and y are not proportional.

**PROPORTIONS** Tell whether the rates form a proportion.

Question 31.

7 inches in 9 hours;

42 inches in 54 hours

Answer:

Ratios form a proportion.

Explanation:

7 inches in 9 hours = 7 : 9

42 inches in 54 hours = 42 : 54

= (42 ÷ 6) : (54 ÷ 6)

= 7 : 9

The ratios are equivalent

So, ratios form a proportion.

Question 32.

12 players from 21 teams;

15 players from 24 teams

Answer:

The ratios do not form a proportion.

Explanation:

12 players from 21 teams; = 12 : 21

= (12 ÷ 3) : (21 ÷ 3)

= 4 : 7

15 players from 24 teams = 15 : 24

= (15 ÷ 3) : (24 ÷ 3)

= 5 : 8

The ratios are not equivalent

So, ratios do not form a proportion.

Question 33.

385 calories in 3.5 servings; 300 calories in 3 servings

Answer:

385 calories in 3.5 servings; 300 calories in 3 servings do not form a proportion.

Explanation:

385 calories in 3.5 servings = 385 : 3.5

= 110

300 calories in 3 servings = 300 : 3

= 100

The ratios are not equivalent

So, 385 calories in 3.5 servings; 300 calories in 3 servings do not form a proportion.

Question 34.

4.8 laps every 8 minutes; 3.6 laps every 6 minutes

Answer:

4.8 laps every 8 minutes; 3.6 laps every 6 minutes form a proportion.

Explanation:

4.8 laps every 8 minutes = 4.8 : 8

= (4.8 ÷ 8) : (8 ÷ 8)

= 3/5 : 1

3.6 laps every 6 minutes = 3.6 : 6

= (3.6 ÷ 6) : (6 ÷ 6)

= 3/5 : 1

The ratios are equivalent

So, 4.8 laps every 8 minutes; 3.6 laps every 6 minutes form a proportion.

Question 35.

\(\frac{3}{4}\) pound for every 5 gallons; \(\frac{4}{5}\) pound for every 5\(\frac{1}{3}\) gallons

Answer:

\(\frac{3}{4}\) pound for every 5 gallons; \(\frac{4}{5}\) pound for every 5\(\frac{1}{3}\) gallons do not form a proportion.

Explanation:

\(\frac{3}{4}\) pound for every 5 gallons = 3/4 : 5

= 3 : 20

\(\frac{4}{5}\) pound for every 5\(\frac{1}{3}\) gallons = 4/5 : 5(1/3)

= 4/5 : 16/3

= 4/5 . 3/16

= 15/16

The ratios are not equivalent

So, \(\frac{3}{4}\) pound for every 5 gallons; \(\frac{4}{5}\) pound for every 5\(\frac{1}{3}\) gallons do not form a proportion.

Question 36.

**MODELING REAL LIFE**

You do 90 sit-ups in 2 minutes. Your friend does 126 sit-ups in 2.8 minutes. Do these rates form a proportion? Explain.

Answer:

The rates form a proportion

Explanation:

You do 90 sit-ups in 2 minutes = 90/2

= 45 sit-ups in 1 minute

Your friend does 126 sit-ups in 2.8 minutes = 126/2.8

= 45 sit-ups in 1 minute

They are equivalent.

So, the rates form a proportion

Question 37.

**MODELING REAL LIFE**

Find the heart rates of you and your friend. Do these rates form a proportion? Explain.

Answer:

The rates do not form a proportion

Explanation:

Your heart rate = 22/20

= 11/10

Your friend heat rate = 18/15 = 6/5

The ratios are not equivalent

So, the rates do not form a proportion

Question 38.

**PROBLEM SOLVING**

You earn $56 walking your neighbor’s dog for 8 hours. Your friend earns $36 painting your neighbor’s fence for 4 hours. Are the pay rates equivalent? Explain.

Answer:

The pay rates are not equivalent.

Explanation:

You earn $56 walking your neighbor’s dog for 8 hours.

= 56 : 8

= 7

Your friend earns $36 painting your neighbor’s fence for 4 hours

= 36 : 4

= 9

So, the pay rates are not equivalent.

Question 39.

**GEOMETRY**

Are the heights and bases of the two triangles proportional? Explain.

Answer:

The heights and bases of the two triangles proportional

Explanation:

Height to base of first triangle = 12 : 15

= (12 ÷ 3) : (15 ÷ 3)

= 4 : 5

Height to base of second triangle = 8 : 10

= (8 ÷ 2) : (10 ÷ 5)

= 4 : 5

The ratios are equivalent

So, the heights and bases of the two triangles proportional

Question 40.

**REASONING**

A pitcher coming back from an injury limits the number of pitches thrown in bullpen sessions as shown.

a. Which quantities are proportional?

b. How many pitches that are not curveballs will the pitcher likely throw in Session 5?

Answer:

a. None of the quantities are proportional.

b. Curveballs are 20, pitches are 50.

Explanation:

a. By observing the table we can say that,

None of the quantities are proportional.

b. In session 5, the pitches are 5 . 10 = 50

curveballs are 5 . 4 = 20

Question 41.

**STRUCTURE**

You add the same numbers of pennies and dimes to the coins shown. Is the new ratio of pennies to dimes proportional to the original ratio of pennies to dimes? If so, illustrate your answer with an example. If not, show why with a counterexample.

Answer:

The new ratio of pennies to dimes is not proportional to the original ratio of pennies to dimes.

Explanation:

Number of pennies to dimes in image a = 3 : 2

Number of pennies to dimes in image b = 4 : 4

= 1 : 1

The ratios are not equivalent

So, the new ratio of pennies to dimes is not proportional to the original ratio of pennies to dimes.

Question 42.

**REASONING**

You are 13 years old, and your cousin is 19 years old. As you grow older, is your age proportional to your cousin’s age? Explain your reasoning.

Answer:

Your age is not proportional to your cousin’s age

Explanation:

You are 13 years old, and your cousin is 19 years old.

= 13 : 19

As you grow older

Suppose after 3 years the ratio is (13 + 3) / (19 + 3) = 16/21

13/19≠ 16/21

So, your age is not proportional to your cousin’s age

Question 43.

**MODELING REAL LIFE**

The shadow of the moon during a solar eclipse travels 2300 miles in 1 hour. In the first 20 minutes, the shadow traveled 766\(\frac{2}{3}\) miles. How long does it take for the shadow to travel 1150 miles? Justify your answer.

Answer:

It takes 30 minutes for the shadow to travel 1150 miles

Explanation:

The shadow of the moon during a solar eclipse travels 2300 miles in 1 hour

= 2300 : 1

In the first 20 minutes, the shadow traveled 766\(\frac{2}{3}\) miles.

= 766(2/3) : 20

= 2300/3 : 20

So, it travels 2300/3 miles in 20 minutes

The time taken for the shadow to travel 1150 miles is 1150 . 20 x (3/2300)

= 69000/2300

= 30

Therefore, The time taken for the shadow to travel 1150 miles is 30 minutes

Question 44.

**MODELING REAL LIFE**

In 60 seconds, a car in a parade travels 0.2 mile. The car traveled the last 0.05 mile in 12 seconds. How long did it take for the car to travel 0.1 mile? Justify your answer.

Answer:

It takes 32 seconds for the car to travel 0.1 mile.

Explanation:

In 60 seconds, a car in a parade travels 0.2 mile.

Car travels to time = 0.2 : 60

The car traveled the last 0.05 mile in 12 seconds

= 0.05 : 12

So, the car travels (0.2 – 0.05 = 0.15 miles) in (60 – 12 = 48 seconds)

The time taken to travel 0.1 mile = (48 x 0.1)/0.15 = 32 seconds.

Question 45.

**OPEN-ENDED**

Describe (a) a real-life situation where you expect two quantities to be proportional and (b) a real-life situation where you do not expect two quantities to be proportional. Explain your reasoning.

Answer:

(a) We will the real-life situation as the relationship between the number of gallons of fuel that we put in the tank and the amount of money we have to pay.

(b) The real-life situation is 2 persons buy different items at a store and pay amount.

Explanation:

(a) Let us take two people to put fuel their cars. Person 1 puts 5 gallons of fuel and pays 20 dollars.

Person 2 puts 8 gallons of fuel pays 32 dollars.

The ratio to the number of gallons to dollars is 5 : 20, 8 : 32

= 1 : 4, 1 : 4

So, the ratios are proportional.

(b) Let us assume 2 persons buy different products at the store and pays the bill.

Person 1 buys 2 items per 15 dollars and person 2 buys 4 items for 32 dollars.

Question 46.

**PROBLEM SOLVING**

A specific shade of red nail polish requires 7 parts red to 2 parts yellow. A mixture contains 35 quarts of red and 8 quarts of yellow. Is the mixture the correct shade? If so, justify your answer. If not, explain how you can fix the mixture to make the correct shade of red.

Answer:

To fix the mixture add 2 more quarts of yellow.

Explanation:

The ratio of nail polish = 7/2 = (7 . 5) / (2 . 5) = 35/10 red/yellow

The ratio of mixture = 35/8 red/yellow

To fix the mixture make the two ratios 35/10 and 35/8 equivalent

Add 2 more quarts of yellow to the mixture

35/10 = 35/(8 + 2)

To fix the mixture add 2 more quarts of yellow.

Question 47.

LOGIC

The quantities x and y are proportional. Use each of the integers 1–5 to complete the table. Justify your answer.

Answer:

Explanation:

The quantities x and y are proportional.

Let us take x/2 = y

So, 10/2 = 5

6/2 = 3

0.5 x 2 = 1

4/2 = 2

Question 48.

**CRITICAL THINKING**

Ratio A and Ratio B form a proportion. Ratio B and Ratio C also form a proportion. Do Ratio A and Ratio C form a proportion? Justify your answer.

Answer:

Ratio A and Ratio C form a proportion.

Explanation:

Ratio A and Ratio B form a proportion

Ratio B and Ratio C also form a proportion

So, Ratio A : Ratio B = Ratio B : Ratio C

Ratio A = (Ratio B)/Ratio C x Ratio B

Ratio C = (Ratio B x Ratio B)/Ratio A

Ratio A : Ratio C = [(Ratio B)/Ratio C x Ratio B] : [(Ratio B x Ratio B)/Ratio A]

= 1/Ratio C : 1/Ratio A

= Ratio A : Ratio C

So, Ratio A and Ratio C form a proportion

### Lesson 5.4 Writing and Solving Proportions

**EXPLORATION 1**

Solving a Ratio Problem

Work with a partner. A train travels 50 miles every 40 minutes. To determine the number of miles the train travels in 90 minutes, your friend creates the following table.

a. Explain how you can find the value of x.

b. Can you use the information in the table to write a proportion? If so, explain how you can use the proportion to find the value of x. If not, explain why not.

c. How far does the train below travel in 2 hours?

d. Share your results in part(c) with other groups. Compare and contrast methods used to solve the problem.

Answer:

a. x = 112.5

b. yes

c. 120 miles

Explanation:

a. 50/40 = x/90

(50/40) . 90 = x

4500/40 = x

x = 112.5

b. We can use information in the table to write a proportion.

c. The train travels 30 miles in 1/2 hour.

The distance traveled in 2 hours is 30. 2 . 2 = 120 miles

**Try It**

**Solve the proportion.**

Question 1.

\(\frac{5}{8}=\frac{20}{d}\)

Answer:

d = 32

Explanation:

\(\frac{5}{8}=\frac{20}{d}\)

cross multiply the fractions

5d = 20 . 8

5d = 160

d = 160/5

d = 32

Question 2.

\(\frac{7}{z}=\frac{14}{10}\)

Answer:

z = 5

Explanation:

\(\frac{7}{z}=\frac{14}{10}\)

cross multiply the fractions

7 . 10 = 14 . z

70 = 14z

70/14 = z

5 = z

Question 3.

\(\frac{21}{24}=\frac{x}{8}\)

Answer:

x = 7

Explanation:

\(\frac{21}{24}=\frac{x}{8}\)

cross multiply the fractions

21 . 8 = 24x

168 = 24x

168/24 = x

x = 7

Solve the proportion.

Question 4.

\(\frac{w}{6}=\frac{6}{9}\)

Answer:

w = 4

Explanation:

\(\frac{w}{6}=\frac{6}{9}\)

cross multiply the fractions

9w = 6 . 6

9w = 36

w = 36/9

w = 4

Question 5.

\(\frac{12}{10}=\frac{a}{15}\)

Answer:

a = 18

Explanation:

\(\frac{12}{10}=\frac{a}{15}\)

cross multiply the fractions

12 . 15 = 10a

180 = 10a

a = 180/10

a = 18

Question 6.

\(\frac{y}{10}=\frac{3}{5}\)

Answer:

y = 6

Explanation:

\(\frac{y}{10}=\frac{3}{5}\)

cross multiply the fractions

5y = 3 . 10

5y = 30

y = 30/5

y = 6

**Solve the proportion.
**Question 7.

\(\frac{2}{7}=\frac{x}{28}\)

Answer:

x = 8

Explanation:

\(\frac{2}{7}=\frac{x}{28}\)

cross multiply the fractions

2 . 28 = 7x

56 = 7x

x = 56/7

x = 8

Question 8.

\(\frac{12}{5}=\frac{6}{y}\)

Answer:

y = 5/2

Explanation:

\(\frac{12}{5}=\frac{6}{y}\)

cross multiply the fractions

12y = 6 . 5

12y = 30

y = 30/12

y = 5/2

Question 9.

\(\frac{40}{z+1}=\frac{15}{6}\)

Answer:

z = 15

Explanation:

\(\frac{40}{z+1}=\frac{15}{6}\)

cross multiply the fractions

40 . 6 = 15(z + 1)

240 = 15z + 15

240 – 15 = 15z

225 = 15z

z = 225/15

z = 15

**Find the value of x so that the ratios are equivalent.**

Question 10.

2 : 4 and x : 6

Answer:

x = 3

Explanation:

2/4 = x/6

cross multiply the fractions

2 . 6 = 4x

12 = 4x

x = 12/4

x = 3

Question 11.

x : 5 and 8 : 2

Answer:

x = 20

Explanation:

x/5 = 8/2

x/5 = 4

x = 4 . 5

x = 20

Question 12.

4 to 3 and 10 to x

Answer:

x = 15/2

Explanation:

4/3 = 10/x

cross multiply the fractions

4x = 3 . 10

4x = 30

x = 30/4

x = 15/2

Question 13.

Write a proportion that can be used to find the number of tomatoes in the new recipe.

Answer:

The number of tomatoes in the new recipe are 4.

Explanation:

In the original recipe, the ratio of the black beans to the number of tomatoes is 1.5 : 1

For the new recipe to be proportional to the original recipe, these ratios must be equivalent.

So the vales of the ratios must be equal

1.5/1 = 6/x

1.5 = 6/x

1.5x = 6

x = 6/1.5

x = 4

**Self-Assessment for Concepts & Skills**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

**SOLVING A PROPORTION** Solve the proportion.

Question 14.

\(\frac{5}{12}=\frac{b}{36}\)

Answer:

b = 15

Explanation:

\(\frac{5}{12}=\frac{b}{36}\)

Cross multiply the fractions

(5 x 36) = 12b

180 = 12b

b = 180/12

b = 15

Question 15.

\(\frac{6}{p}=\frac{42}{35}\)

Answer:

p = 5

Explanation:

\(\frac{6}{p}=\frac{42}{35}\)

Cross multiply the fractions

6 x 35 = 42p

210 = 42p

p = 210/42

p = 5

Question 16.

**WRITING AND SOLVING A PROPORTION**

Find the value of x so that the ratios x : 9 and 5 : 6 are equivalent.

Answer:

x = 15/2

Explanation:

The ratios x : 9 and 5 : 6 are equivalent.

x/9 = 5/6

Cross multiply the fractionsx

6x = 9 . 5

6x = 45

x = 45/6

x = 15/2

Question 17.

**DIFFERENT WORDS, SAME QUESTION**

Which is different? Find “both” answers.

Answer:

Solve 12/x = 3/8 is different, x = 32

For remaining x = 2

Explanation:

3/x = 12/8

3 • 8 = 12x

24 = 12x

x = 2

3 : x and 12 : 8 are equivalent

3/x = 12/8

So, x = 2

3 : 12 and x : 8 are equivalent

3/12 = x/8

3 • 8 = 12x

x = 2

12/x = 3/8

12 • 8 = 3x

3x= 96

x = 96/3 = 32

**Self-Assessment for Problem Solving**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 18.

You burn 35 calories every 3 minutes running on a treadmill. You want to run for at least 15 minutes, but no more than 30 minutes. What are the possible numbers of calories that you will burn? Justify your answer.

Answer:

The possible number of calories burned in between 560/3 and 350.

Explanation:

You burn 35 calories every 3 minutes running on a treadmill.

35 : 3

= 35/3 : 1

You want to run for at least 15 minutes, but no more than 30 minutes means you want to run in between 16 and 30 minutes

If you run for 16 minutes then, the number of calories burn = (35/3) x 16

= 560/3

If you run for 30 minutes, then the number of calories burn = (35/3) x 30

= 1050/3 = 350

So, the possible number of calories burned in between 560/3 and 350.

Question 19.

**DIG DEEPER!**

Two boats travel at the same speed to different destinations. Boat A reaches its destination in 12 minutes. Boat B reaches its destination in 18 minutes. Boat B travels 3 miles farther than Boat A. How fast do the boats travel? Justify your answer.

Answer:

The boat travels with 1/2 mile/hr.

Explanation:

Boat A reaches its destination in 12 minutes

Boat B reaches its destination in 18 minutes

Let us take the speed as x miles/minute

So, Boat A travels 12x miles

Boat B travels 18x miles

Boat B travels 3 miles farther than Boat A

18x + 3 = 12x

18x – 12x = -3

6x = -3

x = 3/6

x = 1/2 mile/hr

So, the boat travels with 1/2 mile/hr

### Writing and Solving Proportions Homework & Practice 5.4

**Review & Refresh**

**Tell whether and are proportional.**

Question 1.

Answer:

x and y are not proportional.

Explanation:

Compare the values of the ratios x to y

4/6 = 2/3

6/8 = 3/4

8/10 = 4/5

The values of the ratios are not equivalent

So, x and y are not proportional.

Question 2.

Answer:

x and y are proportional.

Explanation:

Compare the values of the ratios x to y

2/5/3 = 2/15

4/5/6 = 2/15

4/30 = 2/15

The values of the ratios are equivalent

So, x and y are proportional.

**Plot the ordered pair in a coordinate plane.**

Question 3.

A (- 5, – 2)

Answer:

Explanation:

In point A (- 5, – 2)

The x coordinate and y coordinate values are negative.

so, the point lies in 3rd quadrant.

Take 5 units on the x-axis, 2 units on the y-axis, and locate the point.

Question 4.

B (- 3, 0)

Answer:

Explanation:

In the point B (- 3, 0)

The x coordinate, y coordinates are negative and zero.

So, the point lies in the x-axis.

Question 5.

C (- 1, 2)

Answer:

Explanation:

In point C (- 1, 2) has x coordinate negative value and y coordinate positive value.

so, the point lies in the 2nd quadrant.

Take -1 units on the x-axis, 2 units on the y-axis, and mark the point.

Question 6.

D (1, 4)

Answer:

Explanation:

In point D (1, 4) the x, y coordinates are positive.

So, the point lies in quadrant 1.

Take 1 unit on the x-axis, 4 units on the y-axis, and mark the point.

Question 7.

Which expression is equivalent to (3w – 8) – 4(2w + 3)?

A. 11w + 4

B. – 5w – 5

C. – 5w + 4

D. – 5w – 20

Answer:

D. – 5w – 20

Explanation:

(3w – 8) – 4(2w + 3) = 3w – 8 – 8w – 12

= -5w -20

**Concepts, Skills, & Problem Solving**

**SOLVING A RATIO PROBLEM** Determine how far the vehicle travels in 3 hours. (See Exploration 1, p. 203.)

Question 8.

A helicopter travels 240 miles every 2 hours.

Answer:

The helicopter travels 360 miles in 3 hours.

Explanation:

A helicopter travels 240 miles every 2 hours = 240 : 2

= 240/2 : 1

= 120 : 1

The helicopter travels 120 miles in 1 hour

The distance traveled by the vehicle in 3 hours = 120 * 3

= 360 miles

Question 9.

A motorcycle travels 25 miles every 0.5 hour.

Answer:

A motorcycle travels 150 miles in 3 hours.

Explanation:

A motorcycle travels 25 miles every 0.5 hour = 25 : 0.5

= 25/0.5 : 1

= 50 : 1

The motorcycle travels 50 miles in 1 hour

So, the distance traveled by the vehicle in 3 hours = 50 * 3 = 150 miles

Question 10.

A train travels 10 miles every \(\frac{1}{4}\) hour.

Answer:

A train travels 120 miles in 3 hours.

Explanation:

A train travels 10 miles every \(\frac{1}{4}\) hour = 10 : 1/4

= 4 * 10 : 1

= 40 : 1

A train travels 40 miles in 1 hour

So, the distance traveled by the vehicle in 3 hours = 40 * 3 = 120 miles

Question 11.

A ferry travels 45 miles every 1\(\frac{1}{2}\) hours.

Answer:

A ferry travels 90 miles in 3 hours

Explanation:

A ferry travels 45 miles every 1\(\frac{1}{2}\) hours = 45 : 1(1/2)

= 45 : 3/2

= 45 * 2 : 3

= 90 : 3

= 30 : 1

A ferry travels 30 miles in 1 hour

So, the distance traveled by the vehicle in 3 hours = 30 * 3 = 90

**SOLVING A PROPORTION** Solve the proportion. Explain your choice of method.

Question 12.

\(\frac{1}{4}=\frac{z}{20}\)

Answer:

z = 5

Explanation:

\(\frac{1}{4}=\frac{z}{20}\)

Cross multiply the fractions

20 = 4z

z = 20/4

z = 5

Question 13.

\(\frac{3}{4}=\frac{12}{y}\)

Answer:

y = 16

Explanation:

\(\frac{3}{4}=\frac{12}{y}\)

The product of 3 and waht number is 12?

Because the product pf 3 and 4 is 12, multiply the denominator by 4 to find y

4 * 4 = 16

So, y = 16

Question 14.

\(\frac{35}{k}=\frac{7}{3}\)

Answer:

k = 15

Explanation:

\(\frac{35}{k}=\frac{7}{3}\)

Divide both sides by 35

(35/k)/35 = (7/3)/35

1/k = 1/15

k = 15

Question 15.

\(\frac{b}{36}=\frac{5}{9}\)

Answer:

b = 20

Explanation:

\(\frac{b}{36}=\frac{5}{9}\)

Cross multiply the proportions

9b = 5 * 36

9b = 180

b = 180/9

b = 20

Question 16.

\(\frac{x}{8}=\frac{3}{12}\)

Answer:

x = 2

Explanation:

\(\frac{x}{8}=\frac{3}{12}\)

Cross multiply the fractions

12x = 3 * 8

12x = 24

x = 24/12

x = 2

Question 17.

\(\frac{3}{4}=\frac{v}{14}\)

Answer:

v = 21/2

Explanation:

\(\frac{3}{4}=\frac{v}{14}\)

Cross multiply the fractions

3 * 14 = 4v

42 = 4v

v = 42/4

v = 21/2

Question 18.

\(\frac{15}{8}=\frac{45}{c}\)

Answer:

c = 24

Explanation:

\(\frac{15}{8}=\frac{45}{c}\)

Cross multiply the fractions

15c = 45 * 8

15c = 360

c = 360/15

c = 24

Question 19.

\(\frac{35}{28}=\frac{n}{12}\)

Answer:

n = 15

Explanation:

\(\frac{35}{28}=\frac{n}{12}\)

Cross multiply the fractions

35 * 12 = 28n

420 = 28n

n = 420/28

n = 15

Question 20.

\(\frac{a}{6}=\frac{15}{2}\)

Answer:

a = 45

Explanation:

\(\frac{a}{6}=\frac{15}{2}\)

Cross multiply the fractions

2a = 15 * 6

2a = 90

a = 90/2

a = 45

Question 21.

\(\frac{y}{9}=\frac{44}{54}\)

Answer:

y = 22/3

Explanation:

\(\frac{y}{9}=\frac{44}{54}\)

Cross multiply the fractions

54y = 44 * 9

54y = 396

y = 396/54

y = 22/3

Question 22.

\(\frac{4}{24}=\frac{c}{36}\)

Answer:

c = 6

Explanation:

\(\frac{4}{24}=\frac{c}{36}\)

Cross multiply the fractions

4 * 36 = 24c

144 = 24c

c = 144/24

c = 6

Question 23.

\(\frac{20}{16}=\frac{d}{12}\)

Answer:

d = 15

Explanation:

\(\frac{20}{16}=\frac{d}{12}\)

Cross multiply the fractions

20 * 12 = 16d

240 = 16d

d = 240/16

d = 15

Question 24.

\(\frac{10}{7}=\frac{8}{k}\)

Answer:

k = 5.6

Explanation:

Cross multiply the fractions

10k = 8 * 7

10k = 56

k = 56/10

k = 5.6

Question 25.

\(\frac{5}{n}=\frac{16}{32}\)

Answer:

n = 10

Explanation:

\(\frac{5}{n}=\frac{16}{32}\)

Cross multiply the fractions

5 * 32 = 16n

160 = 16n

n = 160/16

n = 10

Question 26.

\(\frac{9}{10}=\frac{d}{6.4}\)

Answer:

d = 5.76

Explanation:

\(\frac{9}{10}=\frac{d}{6.4}\)

Cross multiply the fractions

10d = 6.4 * 9

10d = 57.6

d = 57.6/10

d = 5.76

Question 27.

\(\frac{2.4}{1.8}=\frac{7.2}{k}\)

Answer:

k = 5.4

Explanation:

\(\frac{2.4}{1.8}=\frac{7.2}{k}\)

Cross multiply the fractions

2.4k = 7.2 * 1.8

2.4k = 12.96

k = 12.96/2.4

k = 5.4

Question 28.

**YOU BE THE TEACHER**

Your friend solves the proportion \(\frac{m}{8}=\frac{15}{24}\). Is your friend correct? Explain your reasoning.

Answer:

Correct

Explanation:

m/8 = 15/24

Cross multiply the fractions

24m = 15 * 8

24m = 120

m = 120/24

m = 5

Question 29.

**NUMBER SENSE**

Without solving, determine whether \(\frac{x}{4}=\frac{15}{3}\) and \(\frac{x}{15}=\frac{4}{3}\) have the same solution. Explain your reasoning.

Answer:

Both have same solution.

Explanation:

\(\frac{x}{4}=\frac{15}{3}[/latex

Cross multiply the fractions

3x = 15 * 4

[latex]\frac{x}{15}=\frac{4}{3}\)

Cross multiply the fractions

3x = 4 * 15

We got same equation

So, both have same solution.

**WRITING A PROPORTION** Use the table to write a proportion.

Question 30.

Answer:

6/7 = 18/w, w = 6

Explanation:

12/14 = 18/w

Cross multiply the fractions

12w = 18 * 4

w = 72/12

w = 6

12/14 = 18/w

6/7 = 18/w

Question 31.

Answer:

n/85 = 1/5, n = 17

Explanation:

n/85 = 34/170

n/85 = 1/5

Cross multiply the fractions

5n = 85

n = 85/5

n = 17

Question 32.

Answer:

15/2.5 = m/4, m = 24

Explanation:

15/2.5 = m/4

Cross multiply the fractions

15 * 4 = 2.5m

60 = 2.5m

m = 60/2.5

m = 24

Question 33.

Answer:

x = 11.2

Explanation:

100/x = 200/22.4

Cross multiply the fractions

100 * 22.4 = 200x

2240 = 200x

x = 2240/200

x = 11.2

**WRITING AND SOLVING A PROPORTION** Find the value of x so that the ratios are equivalent.

Question 34.

1 : 8 and 4 : x

Answer:

x = 32

Explanation:

1/8 = 4/x

Cross multiply the fractions

x = 8 * 4

x = 32

Question 35.

4 to 5 and x to 20

Answer:

x = 16

Explanation:

4/5 = x/20

Cross multiply the fractions

20 * 4 = 5x

80 = 5x

x = 80/5

x = 16

Question 36.

3 : x and 12 : 40

Answer:

x = 10

Explanation:

3/x = 12/40

Cross multiply the fractions

3 * 40 = 12x

120 = 12x

x = 120/12

x = 10

Question 37.

x to 0.25 and 6 to 1.5

Answer:

x = 1

Explanation:

x/0.25 = 6/1.5

Cross multiply the fractions

1.5x = 6 * 0.25

1.5x = 1.5

x = 1.5/1.5

x = 1

Question 38.

x : \(\frac{5}{2}\) and 8 : 10

Answer:

x = 2

Explanation:

x : 5/2 = 8 : 10

Cross multiply the fractions

10x = 8 * 5/2

10x = 20

x = 20/10

x = 2

Question 39.

\(\frac{7}{4}\) to 14 and x to 32

Answer:

x = 4

Explanation:

7/4 : 14 = x : 32

Cross multiply the fractions

14x = 32 * (7/4)

14x = 56

x = 56/14

x = 4

Question 40.

**WRITING A PROPORTION**

Your science teacher has a photograph of the space shuttle Atlantis. Every 1 centimeter in the photograph represents 200 centimeters on the actual shuttle. Which of the proportions can you use to find the actual length x of Atlantis? Explain.

Answer:

1/200 = x/19.5

Explanation:

Every 1 centimeter in the photograph represents 200 centimeters on the actual shuttle

= 1 : 200

Length of atlantis on model = 19.5

The actual length on Atlantis = x

Scale 1:200

When length on model = 1

Actual length = 200

When actual length = x

Length on model = 19.5

Scale = x : 19.5

Equating both scales

1/200 = x/19.5

Question 41.

**MODELING REAL LIFE**

In an orchestra, the ratio of trombones to violas is 1 to 3. There are 9 violas. How many trombones are in the orchestra?

Answer:

There are 3 trombones in the orchestra.

Explanation:

t/9 = 1/3

3t = 9

t = 9/3

t = 3

There are 3 trombones in the orchestra.

Question 42.

**MODELING REAL LIFE**

A dance team has 80 dancers. The ratio of seventh-grade dancers to all dancers is 5:16. Find the number of seventh-grade dancers on the team.

Answer:

The number of seventh-grade dancers on the team is 25.

Explanation:

A dance team has 80 dancers

Let us take the seventh-grade dancers as x

The ratio of seventh-grade dancers to all dancers is 5:16

x : 80 = 5 : 16

x/80 = 5/16

Cross multiply the fractions

16x = 5 *80

16x = 400

x = 400/16

x = 25

The number of seventh-grade dancers on the team is 25.

Question 43.

**MODELING REAL LIFE**

There are 144 people in an audience. The ratio of adults to children is 5 to 3. How many are adults?

Answer:

The number of adults in the audience are 18.

Explanation:

The ratio of adults to children is 5 to 3

adults : children = 5 : 3

There are 144 people in an audience.

5x + 3x = 144

8x = 144

x = 144/8

x = 18

So, the number of adults in the audience are 18.

Question 44.

**PROBLEM SOLVING**

You have $50 to buy T-shirts. You can buy 3 T-shirts for $24. Do you have enough money to buy 7 T-shirts? Justify your answer.

Answer:

No.

Explanation:

You can buy 3 T-shirts for $24

= 24 : 3

= 24/3 : 1

= 8 : 1

So, you can buy 1 T-shirt per $8.

The money required to buy 7 T-shirts = 7 * 8 = $56

The remaining money after buying 3 T-shirts is 50 – 24 = $26

$26 is not enough to buy 7 T-shirts.

Question 45.

**PROBLEM SOLVING**

You buy 10 vegetarian pizzas and pay with $100. How much change do you receive?

Answer:

The change you receive = $15

Explanation:

3 vegetarian pizzas for $25.50 = 25.50 : 3

= 25.50/3 : 1

= 8.5 : 1

So, you have to pay $8.5 for 1 vegetarian pizza

If you buy 10 vegetarian pizzas, then amount is 10 * 8.5 = $85

The change you receive = 100 – 85 = 15

Question 46.

**MODELING REAL LIFE**

A person who weighs 120 pounds on Earth weighs 20 pounds on the Moon. How much does a 93-pound person weigh on the Moon?

Answer:

The weight of a 93-pound person weigh on the Moon is 15.5 pounds

Explanation:

The ratio of a person weight on earth to moon = 120 : 20

= 120/20 : 1

= 6 : 1

The weight of a 93-pound person weigh on the Moon is 93/6 = 31/2 = 15.5 pounds.

Question 47.

**PROBLEM SOLVING**

Three pounds of lawn seed covers 1800 square feet. How many bags are needed to cover 8400 square feet?

Answer:

The number of bags required to cover 8400 sq ft is 14.

Explanation:

Three pounds of lawn seed covers 1800 square feet.

The ratio of lawn to seed cover = 1800 : 3

= 1800/3 : 1

= 600 : 1

So, 1 seed bag covers 600 sq ft

The number of bags required to cover 8400 sq ft is 8400/600

= 14 bags

Question 48.

**MODELING REAL LIFE**

There are 180 white lockers in a school. There are 3 white lockers for every 5 blue lockers. How many lockers are in the school?

Answer:

There are 480 lockers in the school.

Explanation:

There are 3 white lockers for every 5 blue lockers

= 3 : 5

Let b be the number of blue lockers

3/5 = 180/b

3b = 180 * 5

3b = 900

b = 900/3

b = 300

300 + 180 = 400

So, there are 480 lockers in the school.

**CONVERTING MEASURES** Use a proportion to complete the statement. Round to the nearest hundredth if necessary.

Question 49.

Answer:

3.73 miles

Explanation:

1 km = 0.621371

6 km = 6 * 0.621371

= 3.72823 miles

= 3.73 miles

Question 50.

Answer:

0.66 gal

Explanation:

1 L = 0.264172 gal

2.5 L = 0.264172 * 2.5

= 0.66043

2.5 L = 0.66 gal

Question 51.

Answer:

40.82 kg

Explanation:

1 lb = 0.453592 kgs

90 lb = 0.453592 * 90

= 40.8233

90 lb = 40.82 kg

**SOLVING A PROPORTION** Solve the proportion.

Question 52.

Answer:

x = 3/2

Explanation:

2x/5 = 9/15

Cross-multiply the fractions

15 * 2x = 9 * 5

30x = 45

x = 45/30

x = 3/2

Question 53.

Answer:

d = 12

Explanation:

5/2 = (d – 2)/4

Cross-multiply the fractions

20 = 2(d – 2)

20 = 2d – 4

20 + 4 = 2d

24 = 2d

d = 24/2

d = 12

Question 54.

Answer:

k = 4

Explanation:

4/(k + 3) = 8/14

Cross-multiply the fractions

14 * 4 = 8(k + 3)

56 = 8k + 24

56 – 24 = 8k

32 = 8k

k = 32/8

k = 4

Question 55.

**LOGIC**

It takes 6 hours for 2 people to build a swing set. Can you use the proportion \(\frac{2}{6}=\frac{5}{h}\) to determine the number of hours h it will take 5 people h6to build the swing set? Explain.

Answer:

It will take 15 hours for 5 people to build the swing set.

Explanation:

It takes 6 hours for 2 people to build a swing set

2 people – 6 hours

5 people – ?

We have a proportion,

2/6 = 5/h

2h = 6 * 5

2h = 30

h = 30/2

h = 15

It will take 15 hours for 5 people to build the swing set.

Question 56.

**STRUCTURE**

The ratios a : b and c : d are equivalent. Which of the following equations are proportions? Explain your reasoning.

Answer:

a/d = c/b is not equivalent proportion.

Explanation:

a/b = c/d

ad = bc

b/a = d/c

bc = ad

a/c = b/d

ad = bc

a/d = c/b

ab = cd

c/a = d/b

bc = ad

So, a/d = c/b is not equivalent proportion.

Question 57.

**CRITICAL THINKING**

Consider the proportions \(\frac{m}{n}=\frac{1}{2}\) and \(\frac{n}{k}=\frac{2}{5}\). What is \(\frac{m}{k}\) ? Explain your reasoning.

Answer:

m/k = 1/5

Explanation:

m/n = 1/2

So m = x, n = 2x

n/k = 2/5

n = 2x, k = 5x

So, m/k = x/5x

= 1/5

### Lesson 5.5 Graphs of Proportional Relationships

**EXPLORATION 1**

Representing Relationships Graphically

Work with a partner. The tables represent two different ways that red and blue food coloring are mixed.

a. Represent each table in the same coordinate plane. Which graph represents a proportional relationship? How do you know?

b. Find the unit rate of the proportional relationship. How is the unit rate shown on the graph?

c. What is the multiplicative relationship between x and y for the proportional relationship? How can you use this value to write an equation that relates y and x?

Answer:

Mixture 1

a.

Mixture 2

b. If the line passes through the origin, then x and y are proportional.

Explanation:

a.The points in mixture 1 are A (1, 2), B (2, 4), C (3, 6), D (4, 8)

The points in mixture 2 are E (0, 2), F (2, 4), G (4, 6), H (6, 8)

b. If the line passes through the origin, then x and y are proportional.

**Try It**

**Tell whether x and y are proportional. Explain your reasoning.**

Question 1.

Answer:

The line does not pass through the origin. So, x and y are not proportional.

Explanation:

The line does not pass through the origin. So, x and y are not proportional.

Question 2.

Answer:

x and y are not proportional.

Explanation:

The line pass through the origin. So, x and y are not proportional.

Question 3.

Answer:

x and y are not proportional.

Explanation:

The line pass through the origin. So, x and y are not proportional.

Question 4.

**WHAT IF**

Does your answer change when you use the point (\(\frac{1}{2}\), \(\frac{1}{4}\))to find the speed of the subway car? Explain your reasoning.

Answer:

No.

Explanation:

The point (1/2, 1/4) indicates that the subway car travels 1/4 mile every 1/2 minutes.

1/4 : 1/2

= 2/4 : 1

= 1/2 : 1

So, the unit rate is 1/2 mile per minute

The speed of the subway car is 1/2 mile per minute.

**Self-Assessment for Concepts & Skills**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 5.

**IDENTIFYING A PROPORTIONAL RELATIONSHIP** Use the graph shown to tell whether x and y are proportional. Explain your reasoning.

Answer:

x and y are proportional.

Explanation:

x/y = 0/0

1.5/1 = 15/10 = 3/2

4.5/3 = 45/30 = 3/2

As the line pass through the origin, so, x and y are proportional.

Question 6.

**FINDING A UNIT RATE**

Interpret each plotted point in the graph. Then identify the unit rate, if possible.

Answer:

The unit rate is 1.5

Explanation:

time : distance = 1.5/1

= 15/10 = 3/2

The unit rate is 1.5

**Self-Assessment for Problem Solving**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.

The table shows the temperature (in degrees Fahrenheit), hours after midnight.

a. Describe a proportional relationship between time and temperature shown by the table. Explain your reasoning.

b. Find the temperature 3.5 hours after midnight.

Answer:

b. 161 degrees Fahrenheit

Explanation:

a. The table shows the proportional relationship between hours and temperature.

0.5/44 = 1/88

1/46

1.5/48 = 1/32

The temperature after 3.5 hours, is 3.5 * 46 = 161 degrees Fahrenheit

Question 8.

**DIG DEEPER!**

Show how you can use a proportional relationship to plan the heights of the vertical supports of a waterskiing ramp. Then explain how increasing the steepness of the ramp affects the proportional relationship.

Answer:

The vertical supports tell the height of the ramp. As the height of the ramp increases the distance from the ramp is increases.

### Graphs of Proportional Relationships Homework & Practice 5.5

**Review & Refresh**

**Find the value of x so that the ratios are equivalent.**

Question 1.

2 : 7 and 8 : x

Answer:

x = 28

Explanation:

2/7 = 8/x

Cross multiply the fractions

2x = 8 * 7

2x = 56

x = 56/2

x = 28

Question 2.

3 to 2 and x to 18

Answer:

x = 27

Explanation:

3/2 = x/18

Cross multiply the fractions

3 * 18 = 2x

54 = 2x

x = 54/2

x = 27

Question 3.

9 : x and 54 : 8

Answer:

x = 4/3

Explanation:

9/x = 54/8

Cross multiply the fractions

54x = 9 * 8

54x = 72

x = 72/54

x = 4/3

**Find the quotient, if possible.**

Question 4.

36 ÷ 4

Answer:

36 ÷ 4 = 9

Explanation:

The quotient of two integers of the same sign is positive.

36 ÷ 4 = 9

Question 5.

42 ÷ (- 6)

Answer:

42 ÷ (- 6) = -7

Explanation:

The quotient of two integers of the different signs is negative.

42 ÷ (- 6) = -7

Question 6.

– 39 ÷ 3

Answer:

– 39 ÷ 3 = -13

Explanation:

The quotient of two integers of the different signs is negative.

– 39 ÷ 3 = -13

Question 7.

– 44 ÷ (- 4)

Answer:

– 44 ÷ (- 4) = 11

Explanation:

The quotient of two integers of the same sign is positive.

– 44 ÷ (- 4) = 11

**Solve the inequality. Graph the solution.**

Question 8.

– \(\frac{x}{3}\) < 2

Answer:

x > -6

Explanation:

– \(\frac{x}{3}\) < 2

Multiply each side by 3

-x/3 . 3 < 2 . 3

-x < 6

x > -6

Question 9.

\(\frac{1}{3}\)p ≥ 4

Answer:

p ≥ 12

Explanation:

\(\frac{1}{3}\)p ≥ 4

Multiply each side by 3

\(\frac{1}{3}\)3p ≥ 4 . 3

p ≥ 12

Question 10.

– 8 < \(\frac{2}{3}\)n

Answer:

-24 < n

Explanation:

– 8 < \(\frac{2}{3}\)n

Multiply each side by 3/2

-8 x (3/2) < 2n/3 x (3/2)

-24 < n

Question 11.

– 2w ≤ 10

Answer:

w ≥ 5

Explanation:

– 2w ≤ 10

Divide each side by 2

– 2w/2 ≤ 10/2

-w ≤ 5

w ≥ 5

**Concepts, Skills, & Problem Solving**

**REPRESENTING RELATIONSHIPS GRAPHICALLY** Represent the table graphically. Does the graph represent a proportional relationship? How do you know? (See Exploration 1, p. 211.)

Question 12.

Answer:

x and y are not proportional.

Explanation:

The line does not pass through the origin.

So, x and y are not proportional.

Question 13.

Answer:

x and y are proportional.

Explanation:

The line pass through the origin.

So, x and y are proportional.

**IDENTIFYING A PROPORTIONAL RELATIONSHIP** Tell whether x and y are proportional. Explain your reasoning.

Question 14.

Answer:

x and y are proportional.

Explanation:

Compare the values of ratios x to y

x/y = 1/2

2/4 = 1/2

3/6 = 1/2

4/8 = 1/2

The ratios are equivalent.

So, x and y are proportional.

Question 15.

Answer:

x and y are not proportional.

Explanation:

Compare the values of ratios x to y

x/y = -2/0 = 0

-1/2

0/4 = 0

1/6

The ratios are not equivalent

So, x and y are not proportional.

Question 16.

Answer:

x and y are not proportional.

Explanation:

Compare the values of ratios x to y

x/y = -1/-2 = 1/2

0/-1 = 0

1/0 = 0

2/1

All the ratios are not equivalent

So, x and y are not proportional.

Question 17.

Answer:

x and y are proportional.

Explanation:

Compare the values of ratios x to y

x/y = 3/2

6/4 = 3/2

9/6 = 3/2

12/8 = 3/2

All the ratios are equivalent

So, x and y are proportional.

Question 18.

Answer:

x and y are not proportional.

Explanation:

Compare the values of ratios x to y

x/y = 1/3

2/4 = 1/2

3/5

4/6 = 2/3

All the ratios are not equivalent

So, x and y are not proportional.

Question 19.

Answer:

x and y are proportional.

Explanation:

Compare the values of ratios x to y

x/y = 1/0.5 = 2

3/1.5 = 2

5/2.5 = 2

7/3.5 = 2

The ratios are equivalent

So, x and y are proportional.

Question 20.

**YOU BE THE TEACHER**

Your friend uses the graph to determine whether x and y are proportional. Is your friend correct? Explain your reasoning.

Answer:

The line does not pass through the origin.

so, x and y are not proportional.

**FINDING A UNIT RATE** Interpret each plotted point in the graph. Then identify the unit rate.

Question 21.

Answer:

Tnit rate is 1/15

Explanation:

The points are (1, 15), (4, 60)

x/y = 1/15

4/60 = (4 ÷ 4)/(60 ÷ 4)

= 1/15

So, unit rate is 1/15

Question 22.

Answer:

The unit rate is 0.2 seconds per feet.

Explanation:

The points are (1, 5), (6, 30)

x/y = 1/5

6/30 = 1/5

The unit rate is 0.2 seconds per feet.

**IDENTIFYING A PROPORTIONAL RELATIONSHIP** Tell whether x and y are proportional. If so, identify the constant of proportionality. Explain your reasoning.

Question 23.

x – y = 0

Answer:

x and y are proportional.

Explanation:

x – y = 0

x = 0 + y

x = y

So, x and y are proportional.

Question 24.

\(\frac{x}{y}\) = 2

Answer:

x and y are proportional.

Explanation:

\(\frac{x}{y}\) = 2

x = 2y

If y = 0, then x = 2(0) = 0

If y = 1, then x = 2(1) = 2

If y = 2, then x = 2(2) = 4

So, x and y are proportional.

Question 25.

8 = xy

Answer:

x and y are proportional.

Explanation:

8 = xy

y = 8/x

If x = 0, then y = 8/0 = 0

If x = 1, then y = 8/1 = 8

If x = 2, then y = 8/2 = 4

So, x and y are proportional.

Question 26.

x^{2} = y

Answer:

x and y are proportional.

Explanation:

x^{2} = y

If x = 0, then y = 0

If x = 1, then y = 1

If x = 2, then y = 4

So, x and y are proportional.

**WRITING AN EQUATION** The variables and are proportional. Use the values to find the constant of proportionality. Then write an equation that relates x and y.

Question 27.

When y = 72, x = 3.

Answer:

y = 24x

Explanation:

y/x = 72/3

y/x = 24

y = 24x

Question 28.

When y = 20, x = 12.

Answer:

5x = 3y

Explanation:

y/x = 20/12

y/x = 5/3

Cross multiply the fractions

3y = 5x

Question 29.

When y = 45, x = 40.

Answer:

9x = 8y

Explanation:

y/x = 45/40

y/x = 9/8

Cross multiply the fractions

8y = 9x

Question 30.

**MODELING REAL LIFE**

The table shows the profit y for recycling x pounds of aluminum. Find the profit for recycling 75 pounds of aluminum.

Answer:

The profit for recycling 75 pounds of aluminum is $33.75

Explanation:

The profit is $4.50 for recycling 10 pounds of aluminum

= 4.50 : 10

= 4.50/10 : 1

= 0.45 : 1

The profit is 40.45 for recycling 1 pound of aluminum.

The profit for recycling 75 pounds of aluminum is $135/4

Question 31.

**MODELING REAL LIFE**

The graph shows the cost of buying concert tickets. Tell whether x and y are proportional. If so, find and interpret the constant of proportionality. Then find the cost of 14 tickets.

Answer:

x and y are proportional.

The cost for buying 14 tickets is $182

13x = y

Explanation:

The line in the graph passes through the origin. so, x and y are proportional.

The cost for buying 2 tickets is 26

= 26 : 2

=13 : 1

The cost for buying 1 ticket is $13.

So, the cost for buying 14 tickets is 14 * 13 = $182

x/y = 2/26

x/y = 1/13

Cross multiply the fractions

13x = y

Question 32.

**REASONING**

The graph of a proportional relationship passes through (12, 16) and (1, y) Find y.

Answer:

y = 4/3

Explanation:

12/16 = 1/y

3/4 = 1/y

Cross multiply the fractions

3y = 4

y = 4/3

Question 33.

**PROBLEM SOLVING**

The amount of chlorine in a swimming pool is proportional to the volume of water. The pool has 2.5 milligrams of chlorine per liter of water. How much chlorine is in the pool?

Answer:

The amount of chlorine in the pool is 75708.225 milligrams.

Explanation:

The amount of chlorine in a swimming pool varies directly with the volume of water. The pool has 2.5 milligrams of chlorine per liter of water.

1 gallon = 3.78541 litre

8000 gallons = 8000 * 3.78541 = 30283.29 litres

The amount of chlorine in the pool = 30283.29 * 2.5

= 75708.225 milligrams.

Question 34.

**DIG DEEPER!**

A vehicle travels 250 feet every 3 seconds. Find the value of the ratio, the unit rate, and the constant of proportionality. How are they related?

Answer:

The ratio is 250 : 3

The unit rate is 250/3 : 1

3x = 250y

Explanation:

A vehicle travels 250 feet every 3 seconds

= 250 : 3

= 250/3 : 1

The unit rate is 250/3 : 1

x/y = 250/3

Cross multiply the proportions

3x = 250y

### Lesson 5.6 Scale Drawings

**EXPLORATION 1**

Creating a Scale Drawing

Work with a partner. Several sections in a zoo are drawn on 1-centimeter grid paper as shown. Each centimeter in the drawing represents 4 meters.

a. Describe the relationship between the lengths of the fences in the drawing and the actual side lengths of the fences.

b. Describe the relationship between the areas of the sections in the drawing and the actual areas of the sections.

c. Are the relationships in parts (a) and (b) the same? Explain your reasoning.

d. Choose a different distance to represent each centimeter on a piece of 1-centimeter grid paper. Then create a new drawing of the sections in the zoo using the distance you chose. Describe any similarities or differences in the drawings.

**Try It**

Question 1.

What is the actual distance between Traverse City and Marquette?

Answer:

The distance between Traverse City and Marquette is 150 miles

Explanation:

Use the centimeter ruler to find the distance on the map between traverse city and Marquette.

the map distance is about 3 centimeters

Use the scale 1 cm : 50 mi and the ratio 3 cm : d mi to write and solve a proportion

1/50 = 3/d

d = 50 * 3

d = 150

So, the distance between Traverse City and Marquette is 150 miles

Question 2.

A drawing has a scale of 1 mm : 20 cm. What is the scale factor of the drawing?

Answer:

The scale factor is 1/200

Explanation:

Write the scale with the same units. Use the factor 1 cm = 10 mm

1 mm : 20 cm = 1 mm : (20 • 10) mm

= 1 : 200

So, the scale factor is 1/200

**Self-Assessment for Concepts & Skills**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 3.

**VOCABULARY**

In your own words, explain the meaning of the scale and scale factor of a drawing or model.

Answer:

A scale factor is a number that multiplies times a given quantity to produce a smaller or longer version of the original number. It is also defined as the ratio of a drawing or blueprint to the actual object or distance.

The measurements in scale drawings are proportional to the measurements of the actual object. The scale is the ratio that compares the measurements of the drawings with the actual measurements.

Question 4.

**FINDING AN ACTUAL DISTANCE**

Consider the scale drawing of Balanced Rock in Arches National Park. What is the actual height of the structure?

Answer:

h = 1/975.36

Explanation:

scale drawing is 1 cm : 32 ft

1 foot = 30.48 cm

So, 32 ft = 975.36 cm

1 cm : 32 ft = 1 : 975.36

h = 1/975.36

Question 5.

**FINDING A SCALE FACTOR**

A drawing has a scale of 3 in. : 2 ft. What is the scale factor of the drawing?

Answer:

The scale factor is 1/8

Explanation:

3 in. : 2 ft

1 foot = 12 inches

2 ft = 24 inches

The scale is 3 : 24 inches = 1/8

Question 6.

**REASONING**

Describe the scale factor of a model that is (a) larger than the actual object and (b) smaller than the actual object.

Answer:

If the scale of the factor is larger than the actual object, then every point is considered larger in the image. It means we represent a lion in a smaller box.

If the scale of the factor is less than the actual object, it means we represent the pen as a larger box.

**Self-Assessment for Problem Solving**

Solve each exercise. Then rate your understanding of the success criteria in your journal.

Question 7.

A scale drawing of the Parthenon is shown. Find the actual perimeter and area of the rectangular face of the Parthenon. Then recreate the scale drawing with a scale factor of 0.2. Find the perimeter and area of the rectangular face in your drawing.

Answer:

The actual perimeter of the rectangular face of the Parthenon is 26 ft

The actual area of the rectangular face of the Parthenon is 36 sq ft

the perimeter of the rectangular face in your drawing is 5.2 ft

area of the rectangular face in your drawing is 1.44 sq ft.

Explanation:

Perimeter of the rectangular face of the Parthenon = 2(l + b)

= 2(4 + 9) = 13 * 2

= 26 ft

Area of the rectangular face of the Parthenon = l * b

= 4 * 9

= 36 sq ft

Scale factor = 0.2

The length of the rectangular face in your drawing = 0.2 * 9

= 1.8

The breadth of the rectangular face in your drawing = 4 * 0.2

= 0.8

Perimeter is 2(1.8 + 0.8) = 5.2

Area is 0.8 * 1.8 = 1.44

Question 8.

**DIG DEEPER!**

You are in charge of creating a billboard advertisement that is 16 feet long and 8 feet tall. Choose a product. Create a scale drawing of the billboard using words and a picture. What is the scale factor of your design?

Answer:

The scale factor is 2

Explanation:

The details of the billboard advertisement is 16 : 8

= 2 : 1

1 unit is 4 units on the scale

So, the billboard advertisement is 16 * 4 : 8 * 4

= 64 : 32

= 2 : 1

So, the scale factor is 2

### Scale Drawings Homework & Practice 5.6

**Review & Refresh**

**Tell whether x and y are proportional. Explain your reasoning.**

Question 1.

Answer:

x and y are not proportional

Explanation:

Compare the values of ratios x to y

10/5 = 2

9/4 = 2.25

8/3 = 2.6666

7/2 = 3.5

The ratios are not equivalent

So, x and y are not proportional

Question 2.

Answer:

x and y are proportional

Explanation:

Compare the values of ratios x to y

6/7

12/14 = 6/7

18/21 = 6/7

24/28 = 6/7

The ratios are equivalent

So, x and y are proportional

**Simplify the expression.**

Question 3.

7p + 6p

Answer:

7p + 6p = 13p

Explanation:

The given expression is 7p + 6p

= 13p

Question 4.

8 + 3d – 17

Answer:

8 + 3d – 17 = 3d – 9

Explanation:

The given expression is 8 + 3d – 17

= 3d – 9

Question 5.

– 2 + \(\frac{2}{5}\)b – \(\frac{1}{4}\)b + 6

Answer:

– 2 + \(\frac{2}{5}\)b – \(\frac{1}{4}\)b + 6 = 4 + 3b/20

Explanation:

The given expression is – 2 + \(\frac{2}{5}\)b – \(\frac{1}{4}\)b + 6

= 4 + (8b – 5b)/20

= 4 + 3b/20

**Write the word sentence as an inequality.**

Question 6.

A number is less than – 3.

Answer:

n < -3

Explanation:

Let us the number as n

n < -3

Question 7.

7 plus a number z is more than 5.

Answer:

7 + z > 5, z > -2

Explanation:

more than means > symbol

7 + z > 5

z > 5 – 7

z > -2

Question 8.

The product of a number m and 6 is no less than 30.

Answer:

6m > 30

Explanation:

No less than means >

m . 6 > 30

6m > 30

Divide both sides by 6

6m/6 > 30/6

m > 30/6

**Concepts, Skills, & Problem Solving**

**CREATING A SCALE DRAWING** Each centimeter on the 1-centimeter grid paper represents 8 inches. Create a proportional drawing of the figure that is larger or smaller than the figure shown. (See Exploration 1, p. 217.)

Question 9.

Answer:

The base of the traingle is 16 inches

Explanation:

1 cm = 8 inches

The base of triangle = 2 cm

= 2 * 8 = 16 inches

Actually, 1 cm = 0.393701 inches

The size of drawing/real size = 1/8

2/real size = 1/8

16 = real size

Question 10.

Answer:

The bottom surface is 24 inches, the height is 24 inches, the top surface is 8 inches.

Explanation:

1 cm = 8 inches

Bottom surface is 3 cm

= 3 * 8 = 24 inches

Height is 3 cm

= 3 * 8 = 24 inches

Top surface = 1 cm

= 1 * 8 = 8 inches

**FINDING AN ACTUAL DISTANCE** Use the map in Example 1 to find the actual distance between the cities.

Question 11.

Kalamazoo and Ann Arbor

Answer:

The distance between Kalamazoo and Ann Arbor is 100 miles.

Explanation:

Use the centimeter ruler to find the distance on the map between Kalamazoo and Ann Arbor

The map distance is about 2 cm

Use the scale 1 cm : 50 mi and the ratio 2 cm : d mi to write and solve a proportion

1/50 = 2/d

d = 50 * 2

d = 100

So, the distance between Kalamazoo and Ann Arbor is 100 miles.

Question 12.

Lansing and Flint

Answer:

The distance between Lansing and Flint is 50 miles.

Explanation:

Use the centimeter ruler to find the distance on the map between Lansing and Flint

The map distance is about 1 cm

Use the scale 1 cm : 50 mi and the ratio 1 cm : d mi to write and solve a proportion

1/50 = 1/d

d = 50 * 1

d = 50

So, the distance between Lansing and Flint is 50 miles.

Question 13.

Grand Rapids and Escanaba

Answer:

The distance between Grand Rapids and Escanaba is 200 miles.

Explanation:

Use the centimeter ruler to find the distance on the map between Grand Rapids and Escanaba

The map distance is about 4 cm

Use the scale 1 cm : 50 mi and the ratio 4 cm : d mi to write and solve a proportion

1/50 = 1/d

d = 50 * 4

d = 200

So, the distance between Grand Rapids and Escanaba is 200 miles.

Question 14.

Saginaw and Alpena

Answer:

The distance between Saginaw and Alpena is 150 miles.

Explanation:

Use the centimeter ruler to find the distance on the map between Saginaw and Alpena

The map distance is about 3 cm

Use the scale 1 cm : 50 mi and the ratio 4 cm : d mi to write and solve a proportion

1/50 = 1/d

d = 50 * 3

d = 150

So, the distance between Saginaw and Alpena is 150 miles.

**USING A SCALE** Find the missing dimension. Use the scale 1 : 12.

Answer:

Explanation:

15. 1 inch = 12 inches

So, 6.25 x 12 = 75 inches

16. 1 ft = 12 inches

x in = 15 * 12

x in = 180 inches

17. 1 m = 100 cm

32 cm = 0.32 m

32 * 1 = .32 * 12

= 3.84 m

18. 1 ft = 0.333333 yard

5.4 ft = 5.4 * 0.333333

= 1.8 yard * 12 = 21.6 yard

19. 1 cm = 10 mm

21 cm = 210 mm

210 * 12 = 2520 mm

**FINDING A SCALE FACTOR** Use a centimeter ruler to find the scale and the scale factor of the drawing.

Question 20.

Answer:

1 cm = 30 m

Explanation:

Using the centimeter ruler, the length is 4 cm

So, 4 cm = 120 m

1 cm = 120/4

1 cm = 30 m

Question 21.

Answer:

1 cm = 8 mm

Explanation:

Using the centimeter ruler, the length is 3 cm

So, 3 cm = 24 mm

1 cm = 24/3

= 8 mm

Question 22.

**CRITICAL THINKING**

You know the length and the width of a scale model. What additional information do you need to know to find the scale of the model? Explain.

Answer:

You need to know the scale factor to know the scale of the model.

Question 23.

**MODELING REAL LIFE**

Central Park is a rectangular park in New YorkCity.

a. Find the perimeter and the area of the scale drawing of Central Park.

b. Find the actual perimeter and area of Central Park.

Answer:

a. The perimeter of the scale drawing of Central Park = 26 cm

The area of the scale drawing of Central Park = 30 sq cm

b. The perimeter of the scale drawing of Central Park = 8320 cm

The area of the scale drawing of Central Park = 3072000 sq cm

Explanation:

a. The length of the central park is 10 cm

The breadth of the central park is 3 cm

The perimeter of the scale drawing of Central Park = 2(10 + 3)

= 2(13)

= 26 cm

The area of the scale drawing of Central Park = 10 * 3

= 30 sq cm

b. 1 cm = 320 m

The length of the central park is 10 * 320 = 3200

The breadth of the central park is 3 * 320 = 960

The perimeter of the scale drawing of Central Park = 2(3200 + 960)

= 8320 cm

The area of the scale drawing of Central Park = 3200 * 960

= 3072000 sq cm

Question 24.

**PROBLEM SOLVING**

In a blueprint,each square has a side length of \(\frac{1}{4}\) inch.

a. Ceramic tile costs $5 per square foot. How much does it cost to tile the bathroom?

b. Carpet costs $18 per square yard. How much does it cost to carpet the bedroom and living room?

Answer:

a. The cost to the bathroom = $0.06075

b. The cost to carpet the bedroom and living room = $0.02430

Explanation:

Square side length = 1/4 inch

1 feet = 12 inch

1 inch : 16 ft = 1 : 16 * 12 = 1 : 192

a. one square side length is 1/4

1 foot = 12 inch

So, the length of the bathroom = 4(1/4) = 1 = 1 * 0.0833333

= 0.0833333

The breadth of the bathroom = 7(1/4) = 7/4

= (7 * 0.0833333)/4

= 0.583333/4

= 0.14583325

The area of bathroom = 0.0833333 * 0.14583325

= 0.01215 sq in

Ceramic tile costs $5 per square foot

The cost to the bathroom = 0.01215 * 5 = $0.06075

b. 1 inch = 0.0277778 yard

length of bathroom + living room = 1 * 0.0277778 = 0.0277778

breadth of bathroom + living room = 7/4 (0.0277778)

= 0.04861115

The area of bathroom + living room = 0.0277778 * 0.04861115

= 0.00135031080247

Carpet costs $18 per square yard.

The cost to carpet the bedroom and living room = 0.00135031080247 * 18 = $0.02430

**REPRODUCING A SCALE DRAWING** Recreate the scale drawing so that it has a scale of 1 cm : 4 m.

Question 25.

Answer:

Explanation:

1 m = 100 cm

Actual length = 1 cm * (1 cm / 400 cm)

= 1/400 cm

Actual breadth = 8 m * (1 cm / 4 m)

= 2 cm

Question 26.

Answer:

Explanation:

1 cm = 100 m

Actual length = 1 cm * (1 cm / 400 cm)

= 1/400 cm

Actual breadth = 2 m * ( 1 cm / 4 m)

= 1/2 cm

Question 27.

**DIG DEEPER!**

Make a conjecture about the relationship between the scale factor of a drawing and the quotients . Explain your reasoning.

Answer:

drawing perimeter/ actual perimeter = scale factor

drawing area/actual area = scale factor

### Ratios and Proportions Connecting Concepts

**Using the Problem-Solving Plan**

Question 1.

The table shows the toll y(in dollars) for traveling x miles on a turnpike. You have $8.25 to pay your toll. How far can you travel on the turnpike?

Understand the problem.

The table shows the tolls for traveling several different distances on a turnpike. You have $8.25 to pay the toll. You are asked to find how far you can travel on the turnpike with $8.25 for tolls.

Make a plan.

First, determine the relationship between x and y and write an equation to represent the relationship. Then use the equation to determine the distance you can travel.

Solve and check.

Use the plan to solve the problem. Then check your solution.

Answer:

You can travel 55 miles with $8.25 for tolls.

Explanation:

x : y = 40 : 6

= 40/6 : 1

= 20/3 : 1

You need to pay $1 for 20/3 miles

So, 20/3 : 1 = x : 8.25

8.25 (20/3) = x

55 = x

So, you can travel 55 miles with $8.25 for tolls.

Question 2.

A company uses a silo in the shape of a rectangular prism to store bird seed. The base of the silo is a square with side lengths of 20 feet. Are the height and the volume of the silo proportional? Justify your answer.

Answer:

The height and the volume of the silo proportional

Explanation:

The base of the silo is a square with side lengths of 20 feet.

The area of silo = side² = 20²

= 400 sq ft

Height = h

Volume = lwh

= 400h

h : 400h = 1 : 400

The height and the volume of the silo proportional

Question 3.

A rectangle is drawn in a coordinate plane as shown. In the same coordinate plane, create a scale drawing of the rectangle that has a vertex at (0, 0) and a scale factor of 3.

Answer:

The length of rectangle = 2/3

Breadth = 1/3

Explanation:

The length of rectangle = 2 units

The breadth of rectangle = 1 unit

length : breadth = 2 : 1

length = 2breadth

The scale factor = 3

The length of rectangle = 2/3

Breadth = 1/3

**Performance Task**

Mixing Paint

At the beginning of this chapter, you watched a STEAM Video called “Painting a Large Room.” You are now ready to complete the performance task related to this video, available at BigIdeasMath.com. Be sure to use the problem-solving plan as you work through the performance task.

### Ratios and Proportions Chapter Review

**Review Vocabulary**

Write the definition and give an example of each vocabulary term.

**Graphic Organizers**

You can use an Example and Non-Example Chart to list examples and non-examples of a concept. Here is an Example and Non-Example Chart for scale factor.

Choose and complete a graphic organizer to help you study the concept.

1. ratio

2. equivalent ratios

3. rate

4. unit rate

5. equivalent rates

6. proportion

7. cross products

8. proportional

9. scale

**Chapter Self-Assessment**

As you complete the exercises, use the scale below to rate your understanding of the success criteria in your journal.

**5.1 Ratios and Ratio Tables (pp. 183–188)**

Learning Target: Understand ratios of rational numbers and use ratio tables to represent equivalent ratios.

Write the ratio. Then find and interpret the value of the ratio.

Question 1.

salt: flour

Answer:

1/2 : 2

Explanation:

salt : flour = 1/2 cup : 2 cups

Question 2.

water to flour

Answer:

3/4 cup : 2 cups

Explanation:

water to flour = 3/4 : 2

Question 3.

salt to water

Answer:

1/2 cup : 3/4 cup

Explanation:

salt to water = 1/2 cup : 3/4 cup

**Find the missing values in the ratio table. Then write the equivalent ratios.**

Question 4.

Answer:

The equivalent ratios are 3/2 : 1/2, 3 : 1, 9/2 : 3/2, 6 : 2

Explanation:

The missing values in the ratio table are

3/2 • x = 3

x = 3 • 2/3

x = 2

Milk1 = 2 • 1/2 = 1

1/2 • y = 3/2

y = 3

Flour1 = 3/2 • 3

= 9/2

1/2 • z = 2

z = 4

Flour2 = 3/2 • 4

= 6

The equivalent ratios are 3/2 : 1/2, 3 : 1, 9/2 : 3/2, 6 : 2

Question 5.

Answer:

The equivalent ratios are 45 : 0.75, 135 : 2.25, 180 : 3, 90 : 1.5

Explanation:

The missing values in the ratio table are

45 • x = 135

x = 135/45

x = 3

Hours1 = 0.75 • 3

= 2.25

0.75 • y = 3

y = 3/0.75

y = 4

Miles1 = 45 • 4 = 180

45 • z = 90

z = 90/45 = 2

Hours2 = 0.75 • 2 = 1.5

The equivalent ratios are 45 : 0.75, 135 : 2.25, 180 : 3, 90 : 1.5

Question 6.

The cost for 16 ounces of cheese is $3.20. What is the cost for 20 ounces of cheese?

Answer:

The cost for 20 ounces of cheese is $3.95

Explanation:

The cost for 16 ounces of cheese is $3.20

The cost for 1 ounce of cheese is 3.20/16

The cost for 20 ounces of cheese is 20(3.20/16)

= 63.2/16

= 3.95

**5.2 Rates and Unit Rates (pp. 189–194)**

Learning Target: Understand rates involving fractions and use unit rates to solve problems.

**Find the unit rate.**

Question 7.

289 miles on 10 gallons

Answer:

28.9 miles per 1 gallon

Explanation:

289 miles on 10 gallons

= 289 : 10

= 289/10 : 1

= 28.9 : 1

Question 8.

6 \(\frac{2}{5}\) revolutions in 2\(\frac{2}{3}\) seconds

Answer:

2.4 revolutions in 1 second

Explanation:

6 \(\frac{2}{5}\) revolutions in 2\(\frac{2}{3}\) seconds = 32/5 : 8/3

= 32/5 • 3/8 : 1

= 12/5 : 1

Question 9.

You can mow 23,760 square feet in \(\frac{1}{2}\) hour. How many square feet can you mow in 2 hours? Justify your answer.

Answer:

95040 sq ft you can mow in 2 hours.

Explanation:

You can mow 23,760 square feet in \(\frac{1}{2}\) hour

= 23760 : 1/2

= 23760 • 2 : 1

= 47520 : 1

So you can mow 47520 sq ft in 1 hour

The number of sq ft you can mow in 2 hours = 47520 • 2

= 95040

**Tell whether the rates are equivalent. Justify your answer.**

Question 10.

60 centimeters every 2.5 years

30 centimeters every 15 months

Answer:

The rates are equivalent.

Explanation:

60 centimeters every 2.5 years = 60 : 2.5

= 600 : 25

= 24 : 1

24 cm every year

= 24 : 12

= 2 : 1

2 cm every month

30 centimeters every 15 months = 30 : 15

= 2 : 1

So, The rates are equivalent.

Question 11.

2.56 per \(\frac{1}{2}\) pound

$0.48 per 6 ounces

Answer:

The ratios are not equivalent.

Explanation:

2.56 per \(\frac{1}{2}\) pound = 2.56 : 1/2

= 2.56 * 2 : 1

= 5.12 : 1

1 pound = 16 ounces

$0.48 per 6 ounces = 0.48 : 6

= 0.48 : 6 * 16

= 0.48 : 96

= 1 : 200

So, the ratios are not equivalent.

**5.3 Identifying Proportional Relationships (pp. 195–202)**

Learning Target: Determine whether two quantities are in a proportional relationship.

**Tell whether the ratios form a proportion.**

Question 12.

4 to 9 and 2 to 3

Answer:

The rates do not form a proportion.

Explanation:

4 to 9 = 4 : 9

2 to 3 = 2 : 3

So, the rates do not form a proportion.

Question 13.

12 : 22 and 18 : 33

Answer:

The rates form a proportion.

Explanation:

12 : 22 = 12 : 22

= 6 : 11

18 : 33 = 18 : 33

= 6 : 11

So, the rates form a proportion.

Question 14.

\(\frac{1}{2}\) : 2 and \(\frac{1}{4}\) : \(\frac{1}{10}\)

Answer:

The rates do not form a proportion

Explanation:

\(\frac{1}{2}\) : 2 = 1 : 4

\(\frac{1}{4}\) : \(\frac{1}{10}\) = 1/4 : 1/10

= 10 : 4

= 5 : 2

So, the rates do not form a proportion.

Question 15.

3.2 to 8 and 1.2 to 3

Answer:

The ratios are equivalent.

Explanation:

3.2 to 8 = 3.2 : 8

= 2/5

1.2 to 3 = 1.2 : 3

= 2/5

So, the ratios are equivalent.

Question 16.

Tell whether x and y are proportional.

Answer:

The x and y are proportional

Explanation:

x/y = 1/4

3/12 = (3 • 1)/(4 • 3) = 1/4

6/24 = (6 • 1)/(6 • 4) = 1/4

8/32 = (8 • 1)/(8 • 4) = 1/4

So, the x and y are proportional.

Question 17.

You can type 250 characters in 60 seconds. Your friend can type 375 characters in 90 seconds. Do these rates form a proportion? Explain.

Answer:

The rates form a proportion.

Explanation:

You can type 250 characters in 60 seconds = 250 : 60

= (25 • 10) : (6 • 10) = 25 : 6

Your friend can type 375 characters in 90 seconds = 375 : 90

= (25 • 15) : (6 • 15)

= 25 : 6

So, the rates form a proportion.

**5.4 Writing and Solving Proportions (pp. 203–210)**

Learning Target: Use proportions to solve ratio problems.

**Solve the proportion. Explain your choice of method.**

Question 18.

\(\frac{3}{8}=\frac{9}{x}\)

Answer:

x = 24

Explanation:

\(\frac{3}{8}=\frac{9}{x}\)

Cross multiply the fractions

3x = 9 * 8

x = 3 * 8

x = 24

Question 19.

\(\frac{x}{4}=\frac{2}{5}\)

Answer:

x = 8/5

Explanation:

\(\frac{x}{4}=\frac{2}{5}\)

Cross multiply the fractions

5x = 2 * 4

5x = 8

x = 8/5

Question 20.

\(\frac{5}{12}=\frac{y}{15}\)

Answer:

y = 75/12

Explanation:

\(\frac{5}{12}=\frac{y}{15}\)

Cross multiply the fractions

15 * 5 = 12y

75 = 12y

y = 75/12

Question 21.

\(\frac{s+1}{4}=\frac{4}{8}\)

Answer:

s = 1

Explanation:

\(\frac{s+1}{4}=\frac{4}{8}\)

Cross multiply the fractions

8(s + 1) = 4 * 4

8s + 8 = 16

8s = 16 – 8

8s = 8

s = 1

**Use the table to write a proportion.**

Question 22.

Answer:

6/12 = 8/m, m = 12

Explanation:

The ratios are proportional.

6/12 = 8/m

Cross multiply the fractions

6m = 8 * 12

6m = 96

m = 96/8

m = 12

Question 23.

Answer:

15/h = 18/2.5

Explanation:

The ratios are proportional.

So, 15/h = 18/2.5

Question 24.

Find the value of x so that the ratios 8 : 20 and 6 : x are equivalent.

Answer:

x = 15

Explanation:

8/20 = 6/x

Cross multiply the fractions

8x = 6 * 20

8x = 120

x = 120/8

x = 15

Question 25.

Swamp gas consists primarily of methane, a chemical compound consisting of a 1 : 4 ratio of carbon to hydrogen atoms. If a sample of methane contains 1564 hydrogen atoms, how many carbon atoms are present in the sample?

Answer:

The number of carbon atoms present in the sample is 391.

Explanation:

Swamp gas consists primarily of methane, a chemical compound consisting of a 1 : 4 ratio of carbon to hydrogen atoms

The number of carbon atoms present in the sample is 1564/4

= 391

**5.5 Graphs of Proportional Relationships (pp. 211–216)**

Learning Target: Represent proportional relationships using graphs and equations.

Question 26.

Tell whether x and y are proportional. Explain your reasoning.

Answer:

x and y are proportional

Explanation:

The line passes through origin.

So, x and y are proportional

Question 27.

The graph shows the number of visits your website received over the past 6 months. Interpret each plotted point in the graph. Then identify the unit rate.

Answer:

50 : 1

Explanation:

Number of visitors : months = 150 : 3

= 150/3 : 1

= 50 : 1

Tell whether x and y are proportional. If so, identify the constant of proportionality. Explain your reasoning.

Question 28.

x + y = 6

Answer:

x and y are not proportional.

Explanation:

x + y = 6

y = 6 – x

If x = 1, then y = 6 – 1 = 5

If x = 2, then y = 6 – 2 = 4

If x = 3, y = 6 – 3 = 3

1/5, 2/4, 3/3 are not equivalent ratios

So, x and y are not proportional.

Question 29.

y – x = 0

Answer:

x and y are proportional.

Explanation:

y – x = 0

y = x

If x = 1, then y = 1

If x = 2, then y = 2

If x = 3, then y = 3

1/2, 2/2, 3/3 are equivalent ratios

So, x and y are proportional.

Question 30.

\(\frac{x}{y}\) = 20

Answer:

x and y are proportional.

Explanation:

\(\frac{x}{y}\) = 20

x = 20y

If y = 1, then x = 20(1) = 20

If y = 2, then x = 20(2) = 40

If y = 3, then x = 20(3) = 60

20/1, 40/2, 60/3 are equivalent ratios

So, x and y are proportional.

Question 31.

x = y + 2

Answer:

x and are not proportional.

Explanation:

x = y + 2

If y = 1, then x = 1 + 2 = 3

If y = 2, then x = 2 + 2 = 4

If y = 3, then x = 3 + 2 = 6

3/1, 4/2, 6/3 are not equivalent

So, x and are not proportional.

Question 32.

The variables x and y are proportional. When y = 4, x = \(\frac{1}{2}\). Find the constant of proportionality. Then write an equation that relates x and y.

Answer:

x = 8y

Explanation:

x/y = k

(1/2)/4 = k

k = 1/8

x/y = 8

x = 8y

**5.6 Scale Drawings (pp. 217–222)**

Learning Target: Solve problems involving scale drawings.

**Find the missing dimension. Use the scale factor 1 : 20.**

Answer:

Explanation:

33. Scale factor = 1/20

Use the scale 1 : 20

90(1/20) = 9/2 = 4.5

34. x/20 = 3.75

x = 3.75 * 20

x = 75

Actual legth = 75 ft

**Use a centimeter ruler to find the scale and the scale factor of the drawing.**

Question 35.

Answer:

The scale is 7.62, scale factor is 39.3701 in

Explanation:

The length of drawing = 10 cm

10 cm = 30 in

1 cm = 0.393701

10 cm = 39.3701 in

10 cm : 30 in = 1 cm : 3 in

3 inch = 7.62

The actual length is 39.3701 inch

So, the scale is 7.62, scale factor is 39.3701 in

Question 36

Answer:

The scale is 19.05, scale factor is 19.05

Explanation:

Uisng the centimeter ruler, length is 4 cm

4 cm = 7.5 in

1 cm = 0.393701

4 cm = 1.5748 in

4 cm : 7.5 in

7.5 inch = 19.05

4 cm : 7.5 inch = 4 cm : 19.05 cm

the scale is 19.05, scale factor is 19.05

Question 37.

A scale model of a lighthouse has a scale of 1 in.: 8ft. The scale model is 20 inches tall. How tall is the lighthouse?

Answer:

Explanation:

A scale model of a lighthouse has a scale of 1 in.: 8ft

The scale model is 20 inches tall

### Ratios and Proportions Practice Test

**Find the unit rate.**

Question 1.

84 miles in 12 days

Answer:

7 miles in 1 day

Explanation:

84 miles in 12 days

= 84 : 12

= 84/12 : 1

= 7 : 1

Question 2.

2 \(\frac{2}{5}\) kilometers in 3\(\frac{3}{4}\) minutes

Answer:

48 kilometers in 75 minutes.

Explanation:

2 \(\frac{2}{5}\) kilometers in 3\(\frac{3}{4}\) minutes = 12/5 : 15/4

= 12/5 • 4/15 : 1

= 48/75 : 1

**Tell whether the ratios form a proportion.**

Question 3.

1 to 0.4 and 9 to 3.6

Answer:

The ratios form a proportion.

Explanation:

1 to 0.4 = 1 : 0.4

= 5/2

9 to 3.6 = 9 : 3.6

= 5/2

So, the ratios form a proportion.

Question 4.

2 : \(\frac{8}{3}\) and \(\frac{2}{3}\) : 6

Answer:

The ratios do not form a proportion.

Explanation:

2 : \(\frac{8}{3}\) = 6 : 8

= 3/4

\(\frac{2}{3}\) : 6 = 2 : 18

= 1/9

So, the ratios do not form a proportion.

**Tell whether and are proportional. Explain your reasoning.**

Question 5.

Answer:

x and y are proportional

Explanation:

x/y = 2/10 = 1/5

4/20 = 1/5

6/30 = 1/5

8/40 = 1/5

So, x and y are proportional

Question 6.

Answer:

x and y are not proportional.

Explanation:

x/y = 1/3, 3/7, 5/11, 7/15

So, x and y are not proportional.

Question 7.

Use the table to write a proportion.

Answer:

1/30 = 8/m

Explanation:

the ratios are proportional

so, 6/180 = 8/m

1/30 = 8/m

**Solve the proportion.**

Question 8.

\(\frac{x}{8}=\frac{9}{4}\)

Answer:

x = 18

Explanation:

\(\frac{x}{8}=\frac{9}{4}\)

Cross multiply the fractions

4x = 9 * 8

4x = 72

x = 72/4

x = 18

Question 9.

\(\frac{17}{4}=\frac{y}{6}\)

Answer:

y = 51/2

Explanation:

\(\frac{17}{4}=\frac{y}{6}\)

Cross multiply the fractions

17 * 6 = 4y

102 = 4y

y = 102/4

y = 51/2

**Tell whether and are proportional. If so, identify the constant of proportionality. Explain your reasoning.**

Question 10.

xy – 11 = 5

Answer:

x, y are not proportional.

Explanation:

xy – 11 = 5

xy = 5 + 11

xy = 16

x = 16/y

Question 11.

\(\frac{y}{x}\) = 8

Answer:

x and y are proportional

Explanation:

\(\frac{y}{x}\) = 8

y = 8x

So, x and y are proportional

Question 12.

A recipe calls for \(\frac{2}{3}\) cup flour for every cup sugar. Write the ratio of sugar to flour. Then find and interpret the value of the ratio.

Answer:

The ratio of sugar to flour is 3 : 2.

Explanation:

A recipe calls for \(\frac{2}{3}\) cup flour for every cup sugar

Flour : sugar = 2/3 : 1

Sugar to flour = 1 : 2/3

= 3 : 2

Question 13.

The graph shows the number of cycles of a crosswalk signal during the day and during the night.

a. Write equations that relate x and y for both the day and night periods.

b. Find how many more cycles occur during the day than during the night for a six-hour period.

Answer:

a. The equation for day is 40 x = y

the equation for night is 30x = y

b. 60 more cycles occur during the day than during the night for a six-hour period.

Explanation:

a. For day, 2 hours is 80 cycles

x/y = 2/80 = 1/40

40x = y

For night, 3 hours is 90

x/y = 3/90 = 1/30

30x = y

a. For day, the number of cycles for 3 hours is 90

90 : 3 = 30 : 1

So, 30 cycles per 1 hour

The number of cycles per 6 hours is 30 * 6 = 180

For night,

the number of cycles is 80 for 2 hours

80 : 2 = 40 : 1

The number of cycles for 6 hours is 40 * 6 = 240

240 – 180 = 60

Question 14.

An engineer is using computer-aided design (CAD) software to design a component for a space shuttle. The scale of the drawing is 1 cm : 60 in. The actual length of the component is 12.75 feet. What is the length of the component in the drawing?

Answer:

The length of the component in the drawing is 25 cm

Explanation:

Write the proportion relating the model length to the actual length. the scale has inches for the actual length units so you must use 12.5 feet = 15 inches in the proportion

model/actual = 1 cm/60 in = x cm/150 in

60x = 150

x = 150/6

= 25

Question 15.

A specific shade of green glaze is made of 5 parts blue glaze to 3 parts yellow glaze. A glaze mixture contains 25 quarts of blue glaze and 9 quarts of yellow glaze. How can you fix the mixture to make the specific shade of green glaze?

Answer:

25 parts blue to 15 parts yellow

Explanation:

25 blue to 9 yellow

The rate is not equal to the specific rate, 5 parts blue to 3 parts yellow

25 parts blue to 15 parts yellow

### Ratios and Proportions Cumulative Practice

Question 1.

The school store sells 4 pencils for $0.80. What is the unit cost of a pencil?

A. $0.20

B. $0.80

C. $3.20

D. $5.00

Answer:

A. $0.20

Explanation:

The school store sells 4 pencils for $0.80

= 0.80 : 4

= 0.80/4 : 1

= 0.2 : 1

Question 2.

What is the simplified form of the expression?

3x – (2x – 5)

F. x – 5

G. x + 5

H. 5x – 5

I. – x – 5

Answer:

G. x + 5

Explanation:

The given expression is 3x – (2x – 5)

= 3x – 2x + 5

= x + 5

Question 3.

Which fraction is equivalent to – 1.25?”

Answer:

B.

Explanation:

-1.25 = -125/100

= -5/4

-1(1/4) = -5/4

Question 4.

What is the value of for the proportion \(\frac{8}{12}=\frac{x}{18}\)?

Answer:

x = 12

Explanation:

\(\frac{8}{12}=\frac{x}{18}\)

Cross multiply the fractions

8 * 18 = 12x

144 = 12x

x = 144/12

x = 12

Question 5.

What inequality is represented by the graph?

F. x – 3 < 7

G. x + 6 ≤ 10

H. – 5 + x < – 1 I. x – 8 > – 4

Answer:

G. x + 6 ≤ 10

Explanation:

the graph represents x ≤ 4.

Solve all the given options

x – 3 < 7

x < 7 + 3

x < 10

G. x + 6 ≤ 10

x ≤ 10 – 6

x ≤ 4

H. – 5 + x < – 1

x < -1 + 5

x < 4

I. x – 8 > – 4

x > -4 + 8

x > 4

Question 6.

What is the missing value in the ratio table?

Answer:

B. 30

Explanation:

2/3 • x = 10/3

x = 10/3 • 3/2

x = 5

6 • 5 = 30

Question 7.

Which expression shows factoring 12x + 54 using the GCF?

F. 2(6x + 27)

G. 3(4x + 18)

H. 6(2x + 9)

I. 12(x + \(\frac{9}{2}\))

Answer:

H. 6(2x + 9)

Explanation:

12x = 3 • 2 • 2 • x

54 = 2 • 3 • 9

The common prime factors are 3 • 2 = 6

So, the GCF of 12x + 54 is 6

12x + 54 = 6(2x) + 6(9)

= 6(2x + 9)

Question 8.

The distance traveled by a high-speed train is proportional to the number of hours traveled. Which of the following is not a valid interpretation of the graph?

A. The train travels 0 kilometers in 0 hours.

B. The unit rate is 200 kilometers per hour.

C. After 4 hours, the train is traveling 800 kilometers per hour.

D. The train travels 800 kilometers in 4 hours.

Answer:

B. The unit rate is 200 kilometers per hour.

Explanation:

The distance traveled by a high-speed train is proportional to the number of hours traveled

200 proportional to 1 = 200 : 1

Question 9.

Which graph represents a number that is at most – 2?

Answer:

H.

Explanation:

At most means less than or equal to

x ≤ -2

Question 10.

A map of the state where your friend lives has the scale \(\frac{1}{2}\) in. : 10 mi.

Part A Your friend measured the distance between her town and the state capital on the map. Her measurement was 4\(\frac{1}{2}\) inches. Based on your friend’s measurement, what is the actual distance (in miles) between her town and the state capital? Show your work and explain your reasoning.

Part B Your friend wants to mark her favorite campsite on the map. She knows that the campsite is 65 miles north of her town. What distance on the map (in inches) represents an actual distance of 65 miles? Show your work and explain your reasoning.

Answer:

A. 90 miles

B. 3(1/4) inch

Explanation:

A. Write the proportion relating the model distance on the map to the actual distance where x is the actual distance. Then solve the proportion by cross multiplying

model/actual = 1/2/10 = 4(1/2)/x

0.5/10 = 4.5/x

0.5x = 45

x = 45/0.5

x = 90 miles

B. Write the proportion relating the model distance to the actual distance where y is the model distance.

0.5/10 = y/65

cross multiply

10y = 0.5 * 65

10y = 32.5

y = 32.5/10

y = 3.25

= 3(1/4) inch

Question 11.

What is the value of the expression – 56 ÷ (- 8)?

Answer:

– 56 ÷ (- 8) = 7

Explanation:

– 56 ÷ (- 8)

The quotient of two negative integers is positive

56 ÷ 8 = 7

Question 12.

The quantities and are proportional. What is the missing value in the table?

A. 38

B. 42

C. 46

D. 56

Answer:

D. 56

Explanation:

Two quantities are proportional

5/7x = 4

x = 4 (7/5)

x = 28/5

Missing value = 10(28/5) = 2(28) = 56

Question 13.

To begin a board game, you place a playing piece at START. On your first three turns, you move ahead 8 spaces, move back 3 spaces, and then move ahead 2 spaces. How many spaces are you from START?

F. 2

G. 3

H. 7

I. 13

Answer:

H. 7

Explanation:

For the first three turns, you move ahead 8 spaces and move back 3 spaces

= 8 – 3 = 5

move ahead 2 spaces

5 + 2 = 7