Ratio Form 2 Maths [PDF]

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Chapter 5

Ratio and Proportion In mathematics we often compare numbers. Imagine a factory where there are 150 men and 100 women working. One way to compare these facts is to subtract. There are 50 more men than women working in the factory. (150 - 100 = 50) Another way to compare these facts is to write a ratio. The ratio of men to women working in the factory is 150 to 100 or, in reduced form, 3 to 2. In other words, for every three men working in the factory, there are two women. Ratio and proportion are useful tools for solving many word problems.

Ratio

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A ratio is a comparison of numbers by division. A ratio can be written with the word to, with a colon (:), or as a fraction. Like a fraction, a ratio always should be reduced. Reducing a ratio is sometimes called simplifying.

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Following are the three ways to write the ratio of the number of men to the number of women working in the factory.

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150 to 100

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The numbers in a ratio must be written in the Iorder the problem asks for. For the example of the factory workers, the ratio 1ofmen to women is 3 to 2, not 2 to 3.

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= 3 to 2 or 150:100 = 3:2 or ~100 = 1. 2

Example

Evelyn earns $2400 a month. She pays $600 a month in rent. What is the ratio of her income to her rent?

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income rent

Make a ratio with her income first (in the numerator) and her rent second. Then reduce.

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2400

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1

The ratio of Evelyn's income to her rent is 4 to 1 or 4:1 or

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4

600

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Always.reduce a ratio to I()west terms. However, when a ratio is an improper fraction, do r'lotchange it to a mixed number.

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137

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EXERCISE

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1 Directions: For problems 1 and 2, simplify each ratio.

1. 24:30 =

200:125 =

28 21

2. 3.4_

4 to 1000 =

$560 to $320 =

1.7

Solve each problem below. 3.

Alvaro makes $600 a week and saves $60 each week. What is the ratio of the amount he makes to the amount he saves?

4.

For Alvaro, in problem 3, what is the ratio of the amount to the amount he makes?

he saves

5. There are 24 students in Sam's English class. Four of the students speak Armenian as a first language. What is the ratio of Armenian speakers to the total number of students in the class? 6.

Anna drove 110 miles on 22 gallons of gas. What is the ratio of the distance she drove to the number of gallons of gas she used?

Two-Step Ratio Problems A problem may not directly state both numbers that you need to set up a ratio. You may have to determine one of the numbers. Read the next example carefully. Example

On a test with 20 problems, was the ratio of the number number of problems?

Maceo got 2 problems wrong. What of problems he got right to the total

Step 1

Find the number of problems he got right. Subtract the number he got wrong, 2, from the total number of problems on the test, 20.

Step 2

Make a ratio of thenumber of problems he got right, 18, to the total number of problems, 20. Then reduce.

20 - 2

right total

=

18

~ = .J..20 10

Chapter

5 - Ratio and Proportion

139

1

Directions: Solve each problem. 1. A G ED class of 20 students has' 12 women. a. What is the of students? b. What is the students? c. What is the women? d. What is the men?

ratio of the number of women to the total number ratio of the number of men to the total number of ratio of the number of men to the number of ratio of the number of women to the number of

2. At Baxter Electronics there are 105 union workers and 45 nonunion workers. a. What is the ratio of the number of union workers to the total number of workers? b. What is the ratio of the number of nonunion workers to the total number of workers? c. What is the ratio of the number of union workers to the number of nonunion workers? d. What is the ratio of the total number of workers to the number of union workers?

3. From a total yearly budget of $18,000,000, the city of McHenry

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spends $3,000,000 on education. What is the ratio of the amount spent on education to the amount not spent on education?

4. A math test of 50 questions included 15 fraction problems and

5 decimal problems. What is the ratio of the total number of fraction and decimal problems to the number of questions on the test?

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5. There are 1213 registered voters in Paul's village. During the last

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election 887 people actually voted. Which of the following is approximately the ratio of the number of people who voted to the total number of registered voters in the village?

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(1) 1 to 2

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(2) 2 to 3

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(3) 3 to 4

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(4) 5 to 6

140 I

Mathematics

Proportion A proportion is a statement that says two ratios (or two fractions) are equal. The statement 2:4 = 1:2 is a proportion. The statement 1. = 1..is also a .42 proportion, You learned that the cross products

of equal fractions are equal.

For example, the cross products are 2 X 2

=

4 and 4 X 1

=

4. ~

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Each of the four numbers in a proportion is called an element or a term. In many proportion problems one term is missing. A letter usually represents the missing term. -

R U L E -.----------.-

To solve a proportion, follow these steps: 1. Write a statement with two equal cross products. 2. Divide both sides of the statement by the number the missing term.

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in front of

Note: This is an example of writing in the language of algebra. If a missing term in a proportion is represented by the letter n and the other term that makes the cross product with n is 12, then the cross product of 12 X n is 12n. Example 1

Step 1

Step 2

Find the missing term in The cross product The cross product Write a statement

i=

9 1 2'

of nand 12 is 12n. of 8 and 9 is 72. that the cross products

Divide both sides of the statement

12n

Step 1

Step 2

Solve for c in

72

are equal. 72 12

12n _

---

by 12.

12

The missing term is 6. Example 2

=

n=6

t = ~.

The cross product The cross product Write a statement

of 3 and c is 3c. of 7 and 8 is 56. that the cross products

Divide both sides of the statement The missing term is 181- .

by 3.

3c

=

56

3c 3

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56 3

are equal. -

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c= 18~

3

Chapter

5 - Ratio and Proportion

Solve for y in the proportion 5:y = 2:8.

Example 3 Step 1

Rewrite the proportion with fractions. The first term in each ratio becomes a numerator.

Step 2

The cross product of y and 2 is 2y. The cross product of 5 and 8 is 40. Write a statement that the cross products are equal.

Step 3

Divide both sides of the statement

5 _ 2 y 8

2y= 40

by 2. y= 20

The missing term is 20.

EXERCISE

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3 Directions: For problems 1-3, find the missing term in each proportion. m 1. --6

2.

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3 _ 5 a 6

10 15

1 _ s 3 5

3. 4:e = 6:8

±=.Y 9

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2 _ 4 11 P

2 _ 9 8 x

3:7 = 4:y

15:40 = x:60

30:a = 12:16

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For problems 4 and 5, choose the answer that is set up correctly.

4. If ~7 = ±9' then x = (1) 4 x

f,

7

9

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(2) 4 + 7 9

(3) 4(9

X

7)

(4) 4(9 + 7)

5. If ~12 = (1) 12

.£ 3' X

(2)~

5x3

(3) 5 x

3

12

(4) 3 x 5

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12

then c = 5

X

3

x

3 _ w 6 5

1

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~=± 7

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Mathematics

Proportion Word Problems Proportion is a useful tool for solving many word problems. The key to setting up a proportion is making sure that the amounts being compared are in the same position on either side of the = sign. Study the examples carefully. Example 1

If 12 yards of lumber cost $40, how much do 30 yards of lumber cost?

Step 1

This problem compares yards to cost. Set up two ratios of yards to cost. Here c represents the cost you are looking for.

Step 2

Find both cross products.

Step 3

Divide both sides of 12c = 1200 by 12. The cost of 30 yards of lumber is $100.

E=

yards cost

40

30 c

12c _ 12

1200 12

----

c = $100

T'lP

. Ariy letter can represent

the unknown you.are looking for in a -proportlon. In these examples, the.first letter of the quantity you are looking for represents the unknown. The letter c represents cost in the lastexample. The letter m represents menin the next example: ' .

Example 2

The ratio of the number of men to the number of women working in the county hospital is 2:3. If 480 women work in the hospital, how many men work there?

Step 1

This problem compares men to women. Set up two ratios of men to women. Here m represents the number of men.

Step 2

Find both cross products.

Step 3

Divide both sides of 3m = 960 by 3. There are 320 men working in the hospital.

men

women

2 _ m 3 480

---

3m _

--3

m

=

960

3

320

Be sure that the parts in a proportion correspond to the question that is asked. Read the next example carefully. Example 3

Carlos got 2 problems wrong for every 5 problems right on a test. How many problems did Carlos get wrong if there were 35 problems on the test?

Chapter

5 - Ratio

Step 1

The ratio in the question is problems wrong to total problems. Carlos got 2 wrong for every 5 right. The ratio of wrong to total is 2:7. (2 wrong + 5 right = 7 total)

Step 2

Set up two ratios of wrong to total. Here w represents problems wrong.

Step 3

Find both cross products.

Step 4

. Divide both sides of 7w Carlos got 10 problems

143

and Proportion

wrong

2

total

7

2 _ 7

35

---

1

w

7w= 70 7w _

= 70 by 7.

---

70

7

7

w= 10

wrong.

Some problems may ask you to choose the correct set-up for a proportion problem. Manny drove 110 miles in 2 hours. Which expression shows the distance he can travel in 5 hours if he drives at the same speed?

Example 4

(1)

5

x

2

(2) 110 x 2

110

,

(3) 110X5

5

Step 1

Set up two ratios of miles to hours. Here m represents miles.

Step 2

Write both cross products, but write the cross product of 110 and 5 as a set-up.

Step 3

Divide both sides by 2. Choice (3) is correct.

,s

(4) 110+5

2

2 110 _ m

miles hours

2m

2

=

5

110

X

5

m=110X5 2

In the last example, notice that choice (4) is wrong because it shows the sum of 110 and 5 rather than the product.

EXERCISE

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4 Directions: Solve each problem.

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1.

For every $13 that Helen earns, she takes home $10. Helen's gross pay each month is $1950. How much does she take home each month?

2.

Pat's softball team won 5 games for every 3 games they lost. Altogether, the team played 32 games. How many games did they win?

3.

The ratio of the number of men to the number of women working at Apex, Inc., is 7:2. Altogether, there are 360 workers at the company. How many of the workers are women?

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1441

Mathematics

4. The ratio of good parts to defective parts coming off the assembly line at Apex, Inc., is 20:1. Every day the factory produces 10,500 parts. How many of these parts are defective? 5. The ratio of the number of workers who voted to strike to the number of workers who voted not to strike at Apex was 3:2. If 360 workers voted, how many voted to strike? 6. Recently 300 people in Central County took a civil service examination. For every 6 people who took the exam,S people passed. How many people passed the exam? 7. The picture shown at the right is to be enlarged. The short side will measure 20 inches. Find the measurement of the long side. 4 in. 6 in.

8. The illustration below shows the ratio of blue paint to gray paint in a special color mix. How many gallons of gray paint are needed to make a total of 30 gallons of mix?

9. A recipe calls for 2 cups of sugar for every 3 cups of flour. Which expression below shows the number of cups of sugar a cook needs with 12 cups of flour? (1)

2

x

3

12

(2) 3 + 12 2

(3) 3x12 2

(4) 2 x 12 3

(5) 2+12 3

10. Apples cost 90 cents a dozen. Which expression below represents the cost, in cents, of 8 apples? (1) (12 x 8) x 90 (2) 90 x 8 12

(3) 129~ 8 (4) 90 x 12 8

(5) 90+12 8

Chapter 5 - Ratio and Proportion

Ratio and Proportion PART I

8.

Directions: Use a calculator to solve problems 1-10. 1. Simplify the ratio 48:60. 2. Write the ratio 1.6 to 6.4 in simplest terms.

145

1

Review

One seat was empty for every 4 that were occupied at an open meeting in the town hall about a proposal to build a new firehouse. The meeting room in the town hall can seat 140 people. How many seats were empty at the meeting? (1) 21 (2) 28

3. Express the ratio 75:35 in reduced form.

(3) 35

(4) 42 4. Solve for x in the proportion

x:9

5. Solve for s in the proportion

f=

6.

= 12:36. 25°.

Find the value of n in 8:n = 5:18.

Choose the correct answer for problems 7 and 8. 7. Among the seniors at Cripple Creek High School, 2 students said that they planned to leave Cripple Creek within 2 years for every 3 students who said that they planned to stay. There are 110 seniors at the school. How many of them said that they plan to leave within 2 years? (1) 22 (2) 33 (3) 44 (4) 55 (5) 66

(5) 49

For problems 9 and 10, mark each answer on the corresponding number grid. 9. The Towsons planted a 35-acre field that yielded 3150 bushels of wheat. At the same rate, how many acres would they need to produce 1890 bushels?

146

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Mathematics

10. At the Central County Municipal Airport the ratio of delayed flights to flights that leave on time is 2:7. During a normal week there are 108 scheduled flights leaving the Central County Airport. How many of those flights are likely to be delayed?

PART

II

Directions: calculator.

Solve problems 11-20 without

a

For problems 11-13, refer to the following information. Write each answer in fraction form and mark your answer on the corresponding number grid. A survey shows that 42 families on Maple Avenue own their homes and 28 families rent their homes.

11. What is the ratio of the number of families who own their homes to the number who rent?

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12. What is the ratio of the number of families who rent their homes to the total number of families on Maple Avenue?

Chapter

13. Find the ratio of the number of families who own their homes to the total number who live on Maple Avenue.

5 - Ratio and Proportion

1147

16. The scale on a map says that 2 inches = 150 miles. If two cities are actually 325 miles apart, how many inches apart will they be on the map? (1) 3.! 4

(2) 4.! 3

(3)

5

Z 8

(4) 7 (5) 15 17. How many hours will a plane take to go 1200 miles if it travels 450 miles in 2 hours? (1) 2 ~

Choose the correct answer for each of the following problems. , 1

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14. Which of the following ratios is not equal to the ratio 12:36? (1) 3:9

f

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4

(2) 4.! 2

(3) 5.! 3

(4) 9 (5) 12

(2) 5:15 (3) 9:36 (4) 10:30 (5) 16:48 15. If 6 feet of wire cost $3.40, how much do 9 feet of wire cost?

(2) 21 (3) 28

(2) $2.55

(4) 35

(3) $3.40

(5) 42

(5)$5.10

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(1) 14

(1)$1.70

(4) $4.60

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18. To make a certain color of paint, the ratio of green paint to white paint is 5:2. How many gallons of green paint are required to mix with 14 gallons of white paint?

148 I

Mathematics

19. A snapshot that wfs 3 inches wide and 5 inches long was enlarged to be 12 inches long. Which expression represents the width of the enlargement?

20. A worker can make 15 motor parts in 2 hours. Which expression represents the time the worker needs to make 100 parts?

(1)3+12 5

(1)2X100 15

(2) 5 x 12 3

(2) 15x100 2

(3) 3 x 12 5

(3) 15x2 100

(4) 3 x 5 12

(4) 15 + 2 100 2 (5) 15 x 100

(5) 5+12 3

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You should have gotten at least 16 problems right on this exercise. If you did not get 16 right, review your ratio and proportion skills before you go on. If you got 16 or more right, correct any problem you got wrong. Then go on to the next chapter.

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