# Solving Proportions by Cross Multiplication Objective To introduce and use cross multiplication to solve proportions.

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## by Cross Multiplication

### 2/22/11 10:43 AM

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Date Time

Math Message

For Part a of each problem, write = or ≠ in the answer box. For Part b, calculate the cross products.

1. a. 3 _ 5

### =

6 _ 10 2. a. 7 _ 8 _ 23 b. b. 3. a. 4. a. 6_9 _12 8 b. b. 5. a. 2_8 _10 4 6. a. 10_12 _58 b. b. 7. a. 1_4 _20 5 8. a. 5_7 _1521 b. b. 9. a. 10_16 _48 10. a. 3 _ 5 10 _ 15 b. b.

11. What pattern can you find in Parts a and b in the problems above?

3 5 106

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10 ∗ 3 = = 5 ∗ 6 114 115 7 8 23

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3 ∗ 7 = = 8 ∗ 2 2 3 69

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9 ∗ 2 = = 3 ∗ 6 2 8 104

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= 8 ∗ 4 10 ∗ 2 = 1 4 205

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= 4 ∗ 5 20 ∗ 1= 4 8 10 16

### 64

8 ∗ 10 = =16 ∗ 4 6 9 128

### 72

12 ∗ 6 = = 9 ∗ 8 5 8 10 12

### 60

= 12 ∗ 5 8 ∗ 10 = 5 7 1521

### 105

21 ∗ 5 = = 7 ∗ 15 3 5 1015

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= 5 ∗ 10 15 ∗ 3 =

2 _ 3 6_9

LESSON

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Date Time

114 115

LESSON

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Use cross multiplication to solve these proportions.

Example: _ 4 6 = p _ 15 15 ∗ 4 = 6 ∗p 60 = 6p _ 60 6 =p 10 =p 1. _3 6 = _10 y

2. _7 21 = 3_ c

3. _m 20 = _2 8

4. _2 10 = 5_z

5. _15 9 = _12k

6. 10_12 = d_9

7. _2 9 = _54 t

8. _4 10 = 26_z

9. _3 4 = _r 28

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10. 16_ p = 128_ 40

### 5

11. _51 102 = _6 h

### 12

12. _8 j = _72 192

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4 6 p 15 15 ∗ 4 = = 6 ∗p 278_323_EMCS_S_G6_U08_576442.indd 288 2/21/11 4:41 PM

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continued

LESSON

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Date Time

114 115

For Problems 13–16, set up a proportion and solve it using cross multiplication. Show how the units cancel. Then write the answer.

Example: Jessie swam 6 lengths of the pool in 4 minutes. At this rate,

how many lengths will she swim in 10 minutes?

Proportion:

10 minutes ∗ 6 lengths = 4 minutes ∗n lengths 60 minutes ∗ lengths = 4 minutes ∗n lengths

=n lengths 15 lengths=n lengths Answer: Jessie will swim

### 15

lengths in 10 minutes.

13. Belle bought 8 yards of ribbon for \$6. Solution: How many yards could she buy for \$9?

\$72 ∗ yards

\$6

=

### 12

6 lengths 4 minutes n lengths 10 minutes = Solution: 6_4 = _ 10 n 6 4 n 10 10 ∗ 6 = = 4 ∗n 60 minutes ∗ lengths 4 minutes 278_323_EMCS_S_G6_MJ2_U08_576442.indd 289 3/9/11 11:11 AM

Date Time LESSON

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14. Before going to France, Maurice Solution: exchanged \$25 for 20 euros. At that

exchange rate, how many euros could he get for \$80?

Answer: Maurice could get euros for \$80.

15. One gloomy day, 4 inches of rain Solution: fell in 6 hours. At this rate, how

many inches of rain had fallen after 4 hours?

16. Adelio’s apartment building has Solution: 9 flights of stairs. To climb to the

top floor, he must go up 144 steps. How many steps must he go up to climb 5 flights?

= =

=

continued

LESSON

### _

\$1,600 \$25∗ euros

16 hours

6 hours ∗inches

_

720 steps

### __

9 flights ∗flights

### 80 steps

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Date Time

continued

LESSON

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Set up a proportion for each problem and solve it using cross multiplication.

17. Sarah uses 5 scoops of coffee beans to brew Solution: 8 cups of coffee. How many scoops of beans

does Sarah use per cup?

Answer: Sarah uses scoop(s) of beans per cup of coffee.

18. Jeremiah ran 1 1_

4 miles in 12 minutes. At this Solution:

pace, how long would it take him to run 5 miles?

Answer: It would take Jeremiah minutes to run 5 miles.

19. It took Zach 12 days to read a book that was Solution: 186 pages long. If he read the same amount

each day, how many pages did he read in one week?

20. At sea level, sound travels 0.62 mile in 3 seconds. Solution: What is the speed of sound in miles per hour?

(Hint: First find the number of seconds in 1 hour.)

Answer: Sound travels at the rate of miles per hour.

2,232

3

3

3

=

5

8

=

5

8

14

14

60 1 _ 14

=

14

1,30212

=

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### Math Boxes

2. A boat traveled 128 kilometers in 4 hours.

Fill in the rate table.

At this rate, how far did the boat travel in 2 hours 15 minutes?

LESSON

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Date Time

5. Insert parentheses to make each number sentence true. a. 0.01 ∗ 7 + 9 / 4 = 0.04 b. _ 4 5 ∗ 25 - 10 / 2 = 15 c. _64 / 5 + 3 ∗ 3 = 3 d. 5 ∗ 102+ 102 2 = 700 1. Which rate is equivalent to 70 km in

2 hr 30 min? Fill in the circle next to the best answer.

A. 35 km in 75 min

B.70,000 m in 230 min

C. 140 km in 4 hr 30 min

D. 1,400 m in 300 min

3. A bag contains 1 red counter, 2 blue counters, and 1 white counter. You pick 1 counter at random. Then you pick a second counter without replacing the first counter.

a. Draw a tree diagram to show all possible counter combinations.

b. What is the probability of picking 1 red counter and 1 white counter (in either order)?

4. Add or subtract. a.-303 + (-28) = b. = 245 - 518 c. =-73 + 89 d. 280 - (-31) = 110 111 156 109–111 95 96 247

2

12

1

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6 distance (km) 24 72 144 hours 3 _ 4 1 1 _ 2

3 1

4

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Name Date Time

LESSON

### Double Number Lines

Howie is making tamales. He used 8 cups of filling to make 4 dozen tamales. How much filling does he need to make 10 dozen tamales?

The double number line below can be used to help solve this problem. Notice that the scale at the top of the number line is labeled in dozens of tamales. The scale at the bottom of the number line is labeled in cups of filling. Find the mark for 8 cups of filling. Notice how it lines up with 4 dozen tamales. This represents the information given in the problem. The per-unit rate is also shown on the number line: Howie uses 2 cups of filling per 1 dozen tamales, so the mark for 1 dozen tamales lines up with the mark for 2 cups of filling. This information was used to complete the double number line.

Dozens of tamales Cups of filling 0 0 1 2 2 4 3 6 4 8 5 10 6 12 7 14 8 16 9 18 10 20

The problem asks how much filling Howie needs to make 10 dozen tamales. Find the mark for 10 dozen tamales. Then find the number on the cups-of-filling scale that lines up with this mark. It’s 20, so Howie needs 20 cups of filling to make 10 dozen tamales.

1. A marine animal trainer noted that the aquarium’s newest beluga whale ate 150 pounds of food in 3 days. The whale was fed the same amount of food each day.

a. How many pounds of food does the whale eat per day?

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pounds

b. Use your answer to Part a to fill in the blanks on the top scale of the double number line below.

### 50

Pounds of food Days 0 0 1 2 3 4 5 6 7 300 150

c. If he continues to eat at this rate, how many pounds of food will the whale eat in 5 days?

### 250

pounds

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py g g p STUDY LINK

### Calculating Rates

Name Date Time

If necessary, draw a picture, find a per-unit rate, make a rate table, or use a proportion to help you solve these problems.

1. A can of worms for fishing costs \$2.60. There are 20 worms in a can. a. What is the cost per worm?

b. At this rate, how much would 26 worms cost?

2. An 11-ounce bag of chips costs \$1.99.

a. What is the cost per ounce, rounded to the nearest cent?

6. A 1-pound bag of candy containing 502 pieces costs 16.8 cents per ounce. What is the cost of 1 piece of candy? Circle the best answer. 1.86 cents 2.99 cents 0.33 cent cent

7. Mr. Rainier’s car uses about 1.6 fluid ounces of gas per minute when the engine is idling. One night, he parked his car but forgot to turn off the motor. He had just filled his tank. His tank holds 12 gallons.

About how many hours will it take before his car runs out of gas? Explain what you did to find the answer.

Sources: 2201 Fascinating Facts; Everything Has Its Price

b. What is the cost per pound, rounded to the nearest cent?

3. Just 1 gram of venom from a king cobra snake can kill 150 people. At this rate, about how many people would 1 kilogram kill?

4. A milking cow can produce nearly 6,000 quarts of milk each year. At this rate, about how many gallons of milk could a cow produce in 5 months?

5. A dog-walking service costs \$2,520 for 6 months. What is the cost for 2 months? For 3 years?

111–116 Try This

1 _ 2

1,536 oz

1.6 oz per min

960 min

60 min per hour

### 16 hours

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LESSON

Name Date Time

### Ingredients for Peanut Butter Fudge

1. The list at the right shows the ingredients used

to make peanut butter fudge but not how much of each ingredient is needed. Use the following clues to calculate the amount of each ingredient needed to make 1 pound of peanut butter fudge. Record each amount in the ingredient list.

Clues

Use 20 cups of sugar to make 10 pounds of fudge.

You need cups of milk to make 5 pounds of fudge.

You need 15 cups of peanut butter to make 48 pounds of fudge. (Hint: 1 cup = 16 tablespoons)

An 8-pound batch of fudge uses 1 cup of corn syrup.

Use 6 teaspoons of vanilla for each 4 pounds of fudge.

Use teaspoon of salt for each 4 pounds of fudge.

2. Suppose you wanted to make an 80-pound batch of fudge. Record how much of each ingredient you would need.

Use the following equivalencies and your ingredient lists to complete each problem.

3 teaspoons = 1 tablespoon

16 tablespoons = 1 cup

3. cups of peanut butter are needed for 80 pounds of fudge.

4.

### 10

cups of corn syrup are needed for 80 pounds of fudge.

5.

### 40

tablespoons of vanilla are needed for 80 pounds of fudge.

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Ingredient List for 80 Pounds of Peanut Butter Fudge

cups of sugar tablespoons of corn syrup

cups of milk teaspoons of vanilla

tablespoons of peanut butter teaspoons of salt

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3 _ 34 1 _ 2

Peanut Butter Fudge (makes 1 pound) cups of sugar cup of milk tablespoons of peanut butter tablespoons of corn syrup teaspoons of vanilla teaspoon of salt

3

4

1 2 1

8

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Name Date Time

continued LESSON

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For Problems 2–4, fill in the blanks on the double number lines and use them to help you solve the problem.

2. Jamie is ordering supplies for his dog-washing business. Last week, he washed 24 dogs and used 4 bottles of shampoo. Jamie uses the same amount of shampoo for each dog he washes.

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Dogs Bottles of shampoo 0 0 4

a. How many dogs can he wash with one bottle of shampoo?

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dogs

b. How many bottles of shampoo should he order if he expects to wash 30 dogs this week?

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bottles

3. A craft store has skeins of yarn on special. They are selling 2 skeins for \$5.

### \$15

Skeins Cost 0 \$0 \$5

a. What is the cost per skein of yarn?

### \$2.50

b. Holly needs 6 skeins of yarn to make an afghan. How much will the yarn cost?

### \$15

4. Katie rode her bicycle to work today. The 8-mile ride took her 40 minutes.

### 5

Miles Minutes 0 8 0

a. On average, how long does it take Katie to ride one mile?

### 5

minutes

b. At that rate, how long will it take her to ride 11 miles to get from work to her sister’s house?

### 55

minutes

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