Ratio form 2 maths

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Chapter 5Ratio andProportion

Ratio

In mathematics we often compare numbers. Imagine a factory wherethere are 150 men and 100 women working. One way to compare these factsis 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 mento women working in the factory is 150 to 100 or, in reduced form, 3 to 2. Inother words, for every three men working in the factory, there are two women.

Ratio and proportion are useful tools for solving many word problems.

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

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

150 to 100 = 3 to 2 or 150:100 = 3:2 or ~ = 1.100 2The 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.

Example Evelynearns$2400amonth. Shepays$600 amonth in rent. What isthe ratio of her incometo her rent?

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

incomerent

2400 _ 4600 1

The ratio of Evelyn's incometo her rent is4to 1or 4:1or 4 .np, - - - - - - - - -- - - -- - -- - --- - ~ - - ~ -

Always.reduce a ratio to I()westterms. However,when a ratio isanimproper fraction, do r'lotchange it to amixed number.

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138 I MathematicsEXERCISE 1

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

1. 24:30 = 282100:125 =2 . 3. 4_

1.7 4 to 1000 = $560 to $320 =

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 he saves

to the amount he makes?5. There are 24 students in Sam's English class. Four of the studentsspeak 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 ProblemsA 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 examplecarefully.

Example On a test with 20 problems, Maceo got 2 problems wrong. Whatwas the ratio of the number of problems he got right to the totalnumber of problems?

Step 1 Find the number of problems he got right.Subtract the number he got wrong, 2, fromthe total number of problems on the test, 20.

20 - 2=8

Step 2 Make a ratio of thenumber of problems he gotright, 18, to the total number of problems, 20.Then reduce.

right ~ =J . . -total 20 10


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Chapter 5 - Ratio and Proportion 1139

Directions: Solve each problem.1. A GED class of 20 students has'12women.

a.What is the ratio of the number of women to the total numberof students?

b. What is the ratio of the number of men to the total number ofstudents?

c. What is the ratio of the number of men to the number ofwomen?

d. What is the ratio of the number of women to the number ofmen?

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

b. What is the ratio of the number of nonunion workers to thetotal number of workers?

c. What is the ratio of the number of union workers to the numberof nonunion workers?

d. What is the ratio of the total number of workers to the numberof union workers?

3. From a total yearly budget of $18,000,000, the city of McHenryspends $3,000,000 on education. What is the ratio of the amountspent on education to the amount not spent on education?

4. A math test of 50 questions included 15 fraction problems and5 decimal problems. What is the ratio of the total number offraction and decimal problems to the number of questionson the test?

5. There are 1213 registered voters in Paul's village. During the lastelection 887 people actually voted. Which of the following isapproximately the ratio of the number of people who voted to thetotal number of registered voters in the village?(1) 1 to 2(2) 2 to 3(3) 3 to 4(4) 5 to 6

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140 I Mathematics

ProportionA 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. 42proportion,

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. ~X+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 usuallyrepresents 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 in front ofthe missing term.

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

Example 1 Find the missing term in i = 192'Step 1 The cross product of nand 12 is 12n.

The cross product of 8 and 9 is 72.Write a statement that the cross products are equal.

12n =72

Step 2 Divide both sides of the statement by 12. 12n _ 72---12 12The missing term is 6. n=6

Example 2 Solve for c in t= ~.Step 1 The cross product of 3 and c is 3c.

The cross product of 7 and 8 is 56.Write a statement that the cross products are equal.

3c =56

The missing term is 181- .

3c _ 56- --3 3c= 18~3

Step 2 Divide both sides of the statement by 3.


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Example 3 Solve for y in the proportion 5:y =2:8.Step 1 Rewrite the proportion with fractions. The firstterm in each ratio becomes a numerator.

5 _ 2y 8

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.2y= 40

Step 3 Divide both sides of the statement by 2.The missing term is 20. y= 20

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

1 . m 10 3 _ 5 =.Y ~=--6 15 a 6 9 3 7 x2 . 1 _ s 3 _ w 2 _ 4 2 _ 9---3 5 6 5 11 P 8 x3 . 4:e = 6:8 3:7 = 4:y 15:40 = x:60 30:a = 12:16I,

1iIiI IFor problems 4 and 5, choose the answer that is set up correctly.

4 . If~ = then x =7 9'(1) 4 x 79(2) 4 +79(3) 4(9 X 7)(4) 4(9 +7)

5 . If~ = . then c =12 3'(1) 12 X 5 X 3(2)~5x3(3) 5 x 312(4) 3 x 125

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142 I Mathematics

Proportion Word Problems

Example 1

Step 1

Step 2

Step 3

Proportion is a useful tool for solving many word problems. The key tosetting up a proportion is making sure that the amounts being compared are inthe same position on either side of the =sign. Study the examples carefully.

If 12 yards of lumber cost $40, how much do 30 yards of lumbercost?This problem compares yards to cost.Set up two ratios of yards to cost.Here c represents the cost you are looking for.

yards E= 30cost 40 c

Find both cross products. 12c _ 1200----12 12Divide both sides of 12c = 1200 by 12.The cost of 30 yards of lumber is $100.

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 youare looking for represents the unknown. The letter c representscost in the lastexample. The letter m represents menin the nextexample: ' .

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

Step 1

Step 2Step 3

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

men 2 _ m---women 3 480

Find both cross products. 3m _ 960---3 3Divide both sides of 3m =960 by 3.There are 320 men working in the hospital.

m=320

Be sure that the parts in a proportion correspond to the question that isasked. 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 therewere 35 problems on the test?


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Step 1 The ratio in the question is problems wrongto total problems. Carlos got 2 wrong for every5 right. The ratio of wrong to total is 2:7.(2 wrong +5 right =7 total)

wrong 2total 7

Step 2 Set up two ratios of wrong to total.Here w represents problems wrong.2 _ w---7 35

Step 3 Find both cross products. 7w= 70Step 4 .Divide both sides of 7w = 70 by 7.

Carlos got 10 problems wrong.7w _ 70---7 7w= 10

Some problems may ask you to choose the correct set-up for a proportionproblem.

Example 4 Manny drove 110 miles in 2 hours. Which expression shows thedistance he can travel in 5 hours if he drives at the same speed?(1) 5 x 2110 (2) 110 x 25 (3) 110X52 (4) 110+52

Step 1 Set up two ratios of miles to hours.Here m represents miles.

miles 110 _ mhours 2 5

Step 2 Write both cross products, but write thecross product of 110 and 5 as a set-up.

2m =110 X 5

s,,Step 3 Divide both sides by 2.

Choice (3) is correct.m=110X5 2

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

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EXERCISE 4Directions: Solve each problem.

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

i2. Pat's softball team won 5 games for every 3 games they lost.

Altogether, the team played 32 games. How many games didthey win?

iI{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 thecompany. How many of the workers are women?


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1441 Mathematics

4. The ratio of good parts to defective parts coming off theassembly line at Apex, Inc., is 20:1. Every day the factoryproduces 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. If360 workers voted, how many voted to strike?6. Recently 300 people in Central County took a civil serviceexamination. For every 6 people who took the exam,S peoplepassed. How many people passed the exam?7. The picture shown at the right is to be enlarged.The short side will measure 20 inches. Findthe measurement of the long side. 4 in.

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

9. A recipe calls for 2 cups of sugar for every 3 cups of flour. Whichexpression below shows the number of cups of sugar a cookneeds with 12 cups of flour?(1) 2 x 312(2) 3 +122(3) 3x122(4) 2 x 123(5) 2+123

10. Apples cost 90 cents a dozen. Which expression below representsthe cost, in cents, of 8 apples?(1) (12 x 8) x 90(2) 90 x 812(3) 129~ 8(4) 90 x 128(5) 90+128


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Ratio and Proportion ReviewPART I 8. One seat was empty for every 4 that were

occupied at an open meeting in the townhall about a proposal to build a newfirehouse. The meeting room in the townhall can seat 140 people. How many seatswere empty at the meeting?

Directions: Use a calculator to solveproblems 1-10.

1. Simplify the ratio 48:60.2. Write the ratio 1.6 to 6.4 in simplest terms. (1) 21(2) 28

(3) 35(4) 42(5) 49

3. Express the ratio 75:35 in reduced form.4. Solve for x in the proportion x: 9 = 12:36.5. Solve for s in the proportion f = 25. For problems 9 and 10, mark each answeron the corresponding number grid.6. Find the value of n in 8:n = 5:18.Choose the correct answer for problems 7and 8.

9. The Towsons planted a 35-acre field thatyielded 3150 bushels of wheat. At the samerate, how many acres would they need toproduce 1890 bushels?7. Among the seniors at Cripple Creek High

School, 2 students said that they planned toleave Cripple Creek within 2 years for every3 students who said that they planned tostay. There are 110 seniors at the school.How many of them said that they plan toleave within 2 years?(1) 22(2) 33(3) 44(4) 55(5) 66


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146 I Mathematics

10. At the Central County Municipal Airport theratio of delayed flights to flights that leaveon time is 2:7. During a normal week thereare 108 scheduled flights leaving the CentralCounty Airport. How many of those flightsare likely to be delayed?

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PART IIDirections: Solve problems 11-20 without acalculator.For problems 11-13, refer to the followinginformation. Write each answer in fraction formand mark your answer on the correspondingnumber grid.A survey shows that 42 families on MapleAvenue own their homes and 28 families renttheir homes.

11.What is the ratio of the number of familieswho own their homes to the number whorent?

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


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13. Find the ratio of the number of families whoown their homes to the total number wholive on Maple Avenue.

16. The scale on a map says that 2 inches =150 miles. If two cities are actually 325 milesapart, how many inches apart will they be onthe map?(1)3.!4(2) 4.!3(3) 5 Z8( 4) 7(5) 15

17. How many hours will a plane take to go1200 miles if it travels 450 miles in 2 hours?

Choose the correct answer for each of thefollowing problems.

(1) 2~4(2) 4.!2(3) 5.!3( 4) 9(5) 12

14.Which of the following ratios is not equalto the ratio 12:36?(1) 3:9(2) 5:15(3) 9:36(4) 10:30(5) 16:48

18. To make a certain color of paint, the ratio ofgreen paint to white paint is 5:2. How manygallons of green paint are required to mixwith 14 gallons of white paint?(1) 14(2) 21(3) 28(4) 35(5) 42

15. If 6 feet of wire cost $3.40, how much do9 feet of wire cost?(1)$1.70(2) $2.55(3) $3.40(4) $4.60(5)$5.10


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148 I Mathematics

19. A snapshot that wfs 3 inches wide and5 inches long was enlarged to be 12 incheslong. Which expression represents the widthof the enlargement?(1)3+125(2) 5 x 123(3) 3 x 125(4) 3 x 512(5) 5+123

20. A worker can make 15 motor parts in2 hours. Which expression representsthe time the worker needs to make100 parts?(1)2X10015(2) 15x1002(3) 15x2100(4) 15+2100

2(5) 15 x 100

';i > ~4'ftiVt(}~rsiu~e.Q,! Jea 9,e r~p. -" . "" ." "". .- ." -s -r= . ,,, , r< ,~ ... ,You should have gotten at least 16 problems right on this exercise. If you did notget 16 right, review your ratio and proportion skills before you go on. If you got 16or more right, correct any problem you got wrong. Then go on to the next chapter.

Ratio form 2 maths

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