module - ratio, proportion, and percent - northern...

Pre-Algebra

Ratio, Proportion, and Percent

Copyright This publication © The Northern Alberta Institute of Technology 2002. All Rights Reserved.

LAST REVISED June, 2007


1

Ratio, Proportion, and Percent Statement of Prerequisite Skills Complete all previous TLM modules before beginning this module.

Required Supporting Materials Access to the World Wide Web. Internet Explorer 5.5 or greater.Macromedia Flash Player.

Rationale Why is it important for you to learn this material? Ratio, proportion, and percent are basic math skills that the student will encounter in many applied situations. These skills are also essential to a beginning algebra student.

Learning Outcome When you complete this module you will be able to… Solve problems using ratio, proportion, and percent.

Learning Objectives 1. Determine equivalent ratios and solve. 2. Change percent to fractions. 3. Change fractions to percent. 4. Change percent to decimals. 5. Change decimals to percent. 6. Solve percent questions. 7. Solve percent error in measurement problems.

Connection Activity Consider the many times you have encountered fractions or percentages in daily life:

• 1/3 off regular cost • 7% gst • Top 5% of the class • What percentage of your paycheck do you spend in rent?

Can you think of other applications of ratio, proportion, and percent?

http://www.microsoft.com/windows/ie/default.asp

http://www.macromedia.com/shockwave/download/download.cgi?P1_Prod_Version=ShockwaveFlash


OBJECTIVE ONE When you complete this objective you will be able to… Determine equivalent ratios and solve.

Exploration Activity A ratio is a comparison of two quantities. The ratio of one number to another is the first number divided by the second number. That is, the ratio of a to b is:

ba

Therefore, a ratio is a comparison of numbers by division.

EXAMPLE 1

92a) The ratio of 2 to 9 is

37b) The ratio of 7 to 3 is

NOTE: A PROPORTION is a statement of equality between two ratios;

32

64i.e. = is a proportion.

2


EXAMPLE 2 If a car travels 80 km in 2 hours, the ratio of distance to time is:

hkm

280

reducing this gives us;

hkm

140

and

hkm

hkm

140

280

=

The ratios are equal. CHECK: To see if the ratios are equal, perform the cross products.

140

280

=

If this is true, then:

8080240180

=×=×

The cross products are equal, therefore the ratios are equal.

The general statement for the equality of 2 ratios is:

dc

ba

=if

then cbda ⋅=⋅ Notice the proportion has 4 components which are a, b, c, and d. We use ratios to solve problems when we are given 3 of these 4 components.

3


EXAMPLE 3

dx

ba

=

If we are given the values for a, b, d, then we could solve for x.

bdax

dabx

⋅=

⋅=⋅

EXAMPLE 4 The ratio of a given number to 3 is the same as the ratio of 16 to 6. Find the given number.

1. Maintain proper order; i.e. use given number to 3 and 16 to 6

616

3=

numbergiven

2. Let x = given number

616

3=

x

3. If these ratios are equal then

( ) ( )

( )

8

6316

3166

=

=

=

x

x

x

4. Check by using cross products in original proportion

( ) ( )

4848

16386

616

38

=

=

=

The cross products are equal, therefore: x = 8 is correct.

4


EXAMPLE 5 On a blueprint the scale is 1 km to 25 cm. What is the actual distance between 2 points, if they are 5 cm apart?

1. Maintain order i.e. km to cm

5251 x

=

2. let x = actual distance and write ratios

( ) ( )

km2.0

km51

255

2551

=

=

=

=

x

x

x

x

CHECK:

correct is km 2.0

55

)2.0(25)1(5

52.0

251

=

=

=

=

x

5


EXAMPLE 6 A cedar board 8 m long is cut into two pieces that are in the ratio 1:4. Find the length of each piece. SOLUTION: Total number of units is 1 + 4 = 5

Total = 5

4 1

Therefore: 5 - total number of parts 8 - total length of board Therefore the ratio is either:

totalpiecelarger

ortotal

piecesmaller

54

51 or=

let x = the length of the shorter piece. Therefore:

6.1

58

851

=

=

=

x

x

Shorter piece = 1.6 m Longer piece = 6.4 m

6


Experiential Activity One I. Solve the given proportions for x.

85

2=

x1.

127

3=

x 2.

9158 x

= 3.

II. Solve the given problems by setting up the proper proportion.

4. The ratio of a number to 15 is the same as the ratio of 17 to 60. Find the number.

5. The ratio of a number to 40 is the same as the ratio of 7 to 16. Find the number

6. 908 g = 2 lb; what weight in grams is 10 lbs?

7. Medication contains 2 substances, A and B, in the ratio of 3 to 5 respectively. If there is 200 mg of substance B, how many mg of substance A is there?

8. A 6 m length of pipe is cut into 2 parts that are in the ratio 8 to 1. Find the length

of each part. Show Me.

9. A 5 m length of 2 by 10 planking is to be cut into 2 parts that are in the ratio of 4 to 3. Find the length of each part.

Experiential Activity One Answers 1. 1.25

2. 1.75

3. 4.8

4. 4.25

5. 17.5

6. 4540

7. 120

8. 0.67 m, 5.33 m

9. 2.86 m, 2.14 m

7


OBJECTIVE TWO When you complete this objective you will be able to… Change percent to fractions.

Exploration Activity Percent To this point we have used fractions and decimals for representing parts of a unit or quantity. Now we will consider the concept of percent and shall find that percentages are useful in numerous applications. The word percent means by the hundred. Therefore, percent represents a decimal fraction with a denominator of 100. The symbol % is used to denote percent.

EXAMPLE 1 For 5% the denominator is 100

1005Write the fraction with a numerator 5 and get

201 Reduce the fraction and get =

EXAMPLE 2

:%43 the denominator is 100.

1004

3

43 = . Numerator is

1001

43

×= . Write fraction

4003 . Reduce and apply rules for dividing fractions

8


EXAMPLE 3

215 %: the denominator is 100.

100215

215 = Numerator is

100211 Write fraction =

20011

1001

211

=× Reduce =

9


Experiential Activity Two Change the following percent to fractions.

31 % 2. 1. 50%

3. 6% 4. 12%

54

5125% 6. % 5.

214

212

10

7. % 8. % Show Me.

9. 30% 10. 81 %

Experiential Activity Two Answers

3001

21 2. 1.

503

253 4. 3.

1251

25063 6. 5.

2009

401 8. 7.

1039. 10.

8001


OBJECTIVE THREE When you complete this objective you will be able to… Change fractions to percent.

Exploration Activity

Fractions

EXAMPLE 1

53Change to a percent. Use ratio and proportion.

3 is to 5 as a number is to 100 (% means per hundred)

Let x = a number

51003

10035

10053

⋅=

⋅=⋅

=

x

x

x

Solve for x

10060so, = 60%

53 = 60% and

11


EXAMPLE 2

65 . So 5 is to 6 as a number is to 100. Change

Let x = a number so we get,

10065 x

=

Write ratios 10056 ⋅=⋅ x

61005 ⋅

=x

Solve for x

x = 83.3

1003.83so, = 83.3%

65 = 83.3% and

12


13

Experiential Activity Three Change the following to percent. 1. 3/4 2. 1/100 3. 1/8 4. 4/5 Show Me. 5. 1/50 6. 1/4

Experiential Activity Three Answers 1. 75%

2. 1%

3. 12.5%

4. 80%

5. 2%

6. 25%


OBJECTIVE FOUR When you complete this objective you will be able to… Change percent to decimals.


EXAMPLE 1 Change 25% to a decimal. Write it as a fraction with denominator = 100 Divide by 100

25.010025

=

EXAMPLE 2

43Write % as a decimal. Write it as a fraction with denominator = 100.

10043

Simplify the fraction

1001

43

×=

4003

=

= 0.0075

14


Experiential Activity Four Change the following percents to decimals.

1. 75%

53 % 2.

4143. %

212 % 4.

3155. %

516 % Show Me.

15

6.

Experiential Activity Four Answers 1. 0.75 2. 0.006 3. 0.0425 4. 0.025 5. 0.0533 6. 0.062


OBJECTIVE FIVE When you complete this objective you will be able to… Change decimals to percent.


EXAMPLE 1 0.5 = _______________ % (multiply by 100)

0.5 = 50% CHECK: Change percent to decimal by dividing by 100

1005050% = = 0.5


1.1 = 110% CHECK: Change percent to decimal by dividing by 100

1.1100110%110 ==


0.062 = 6.2% CHECK: Change percent to decimal by dividing by 100

062.0100

2.6%2.6 ==

16


17

Experiential Activity Five Change the following decimals to a percent. 1. 0.1 2. 0.8 3. 0.25 4. 4.5 5. 0.125 Show Me. 6. 0.05

Experiential Activity Five Answers 1. 10% 2. 80% 3. 25% 4. 450% 5. 12.5% 6. 5%


OBJECTIVE SIX When you complete this objective you will be able to… Solve percent questions.

Exploration Activity All problems using percent will be done using ratios. Therefore all percent problems can be grouped into 3 types.

TYPE I: Calculate the percent of a quantity. Example: Find 30% of 75 TYPE II: Determine what percent one quantity is of another. Example: 25 is what percent of 60? TYPE III: Determine the quantity from percentage and percent. Example: 25 is 50% of what number?

The following model will be used to solve all percentage problems:

( )( )

( )( )43

21

=

These are 4 positions; one of them is always taken up by the number 100 because percent is always based out of 100.

( ) ( )( )43

1001

=

Percent always goes over 100.

( )( )43

100%

=

18


Finding the percent of a number the number always goes in position 4.

( )numberaof3

100%

=

In position (3) we find the answer.

numberaofanswer

=100%

This is the model we will use to solve percent problems.

TYPE I Problems

EXAMPLE 1 Find 30% of 75. One position is taken up by 100. Percent always goes over 100.

??

10030

=

We are finding 30% of the number 75. Answer in last position.

7510030 answer

=

Replace the word answer with the variable x,

7510030 x

=

Solve:

( ) ( )( )7530100 =x 2250100 =x

1002250

=x

5.22=x Therefore: 30% of 75 = 22.5

19


EXAMPLE 2 Find 60% of 35. Place the 100. Then 60% goes over 100.

??

10060

=

60% of the number 35 goes where? Answer = x, goes where?

3510060 x

=

Solve

( )( )3560100 =x ( )( )

1003560

=x Fill

20

Therefore: 60% of 35 = ____________

in the blank.

All problems finding the percent of a certain number are called TYPE 1.

TYPE II Problems

numberaofanswer

=100%

EXAMPLE 1 36 is what percent of 48? Fill in the 100. Percent over 100. We do not know this value. Let it = x.

4836

100=

x

(48)(x) = (36)(100)

)48()100)(36(

=x

x = 75

36 is 75% of 48.


EXAMPLE 2 18 is what percent of 72? Place 100. Percent over 100.

7218

100=

x

( )( )1001872 =x

x = 25 Fill in the blank. Therefore: 18 is ___________ % of 72.

TYPE III Problems

numberaofanswer

=100%

EXAMPLE 1 30 is 50% of what number? Place the 100. Percent over 100. 50% of a number. We do not know the number, therefore let x = the number.

x30

10050

=

( )( ) ( )( )1003050 =x Fill in the blanks. x = ______________

Therefore: 30 is 50% of ______________ ?

21


EXAMPLE 2 18 is 25% of what number? Place the 100.

Fill in the blanks. % over 100.

=

Let x = the number

( )( ) ( )( )1001825 =x

x = ___________ Therefore: 18 is 25% of ______________ ?

22


Experiential Activity Six Solve for the following:

1. 52% of 30 2. 37% of 125 3. 65% of 75 4. 3.5% of 17.2 5. 5 is what percent of 250? 6. 3.6 is what percent of 48? Show Me. 7. 0.14 is what percent of 3.5? 8. 6.5 is what percent of 30? 9. 7 is 30% of what number? 10. 18 is 75% of what number? Show Me. 11. 240 is 10% of what number? 12. 65 is 90% of what number?

The following exercise is a review of the 3 types of percent problems just presented in this module. Complete the table as shown in number 1 and solve for the indicated unknown.

Type Problem Model Solution I, II, III

5214

100=

x x = ? 1. 14 is what % of 52 II

2. 30% of 10

3. 65 is 30% of a number

4. 1.75% of a number is 7

5. 20% of 65

6. 12 is what % of 175?

7. 75 is 95% of what number?

8. 95% of 6000

9. 55 is what % of 90?

10. 3% of a number is 17

23


24

Experiential Activity Six Answers 1. 15.6 2. 46.25 3. 48.75 4. 0.602 5. 2% 6. 7.5% 7. 4% 8. 21.6667% 9. 23.3333 10. 24 11. 2400 12. 72.2222

ANSWERS for review exercise.

1. 26.9231% 2. 3 3. 216.6667 4. 400 5. 13 6. 6.8571% 7. 78.9474 8. 5700 9. 61.1111% 10. 566.6667


OBJECTIVE SEVEN When you complete this objective you will be able to… Solve percent error in measurement problems.

Exploration Activity The percent error in a measurement is calculated from:

100% ×−

=valuetrue

valuetruevaluemeasurederror

EXAMPLE 1 In a laboratory experiment a student determined the velocity of sound to be 352 m/s. The true value under the same conditions is 343 m/s. Determine the percent error in the measurement. Solution:

True value = 343 m/s Measured value = 352 m/s

Substituting into the above equation, we get

100343

343352% ×−

=error

= 2.62%

Notice the answer is positive. If the measurement were less than the true value the answer would have been negative.

25


26

Experiential Activity Seven 1. An airport runway is measured to be 5362 m in length. Its true value was

supposed to be 5400 m. Find the percent error in the length of the runway. 2. A surveyor's tape reads 100 m. However on a particular day it is actually

100.12 m in length. Find the percent error in its length. 3. A grocer's scale reads 3 kg on an item that is actually 2.85 kg. Find the

percent error in the measurement. Is the customer getting a deal? 4. If the present length of a steel rail is 13.0 m, what will be its length after a 0.5

percent expansion caused by heating? 5. The volume of a gas is measured to be 58.5 ml. If this is 6% lower than the

true volume, what is the true volume? Show Me.

Experiential Activity Seven Answers 1. −0.7037%

2. 0.1199%

3. 5.2632%; no

4. 13.065 m

5. 62.2340 ml

Practical Application Activity Complete the ratio, proportion, and percent module assignment in TLM.

Summary This module introduced the student to the basic concepts of ratio, proportion, and percent.

module - ratio, proportion, and percent - northern...

Documents