HFCC Math Lab

Arithmetic 10 PERCENT PROBLEMS

The meaning of the word percent should be familiar to you. If you score 85% on a test, you know that you scored 85 points out of a possible 100 points. That is, the word “percent” mean parts per hundred. In general: Percent is a part out of 100. CHANGING A PERCENT TO A FRACTION: We can use the idea of percent being part out of 100 to change a percent to a fraction.

Examples:

3% =

3 100

or

47% =

To write a percent as a fraction, use the rule:

47 100

n% =

or

73% =

73 100

n . That is, n parts per hundred. 100

Remember: After changing a percent to a fraction, always reduce the fraction to lowest terms if possible.

Examples:

Change each of the following percents to a fraction reduced to lowest terms.

5 1 = 100 20

1.

5% =

2.

26% =

3.

26 13 = 100 50

1 1 1 1 1 1 % = 2 = ÷ 100 = × = (Remember that a fraction bar means division) 2 100 2 2 100 200 1 1 2 = 37 1 ÷ 100 = 75 × 1 = 75 = 3 37 % = 2 100 2 2 100 200 8 37

4.

5.

16.5% =

16.5 16.5 × 10 165 33 = = = 100 100 ×10 1000 200 -1-

EXERCISE 1: Change each of the following percents to a fraction reduced to lowest terms: 1. 10% 2. 20% 3. 30% 4. 40%

5.

50%

6.

60%

7.

70%

8.

80%

9.

90%

10.

100%

11.

25%

12.

75%

15.

1 12 % 2

16.

1 37 % 2

13.

1 33 % 3

14.

2 66 % 3

17.

1 62 % 2

18.

1 87 % 2

NOTE:

You should be able to mentally convert between the percents and fractions in Exercise 1 above. 2 For example, when you see 40%, you should automatically know that it is equal to the fraction . 5 2 2 When you see the fraction , you should automatically know that it is equal to 66 % . 3 3 Check your answers to Exercise 1 on the answer pages at the end of this handout before memorizing the results.

CHANGING A PERCENT TO A DECIMAL: Examples:

7% =

7 1 = 7× = 7 × .01 = .07 100 100

43.6% = .56% =

43.6 1 = 43.6 × = 43.6 × .01 = .436 100 100

.56 1 = .56 × = .56 × .01 = .0056 100 100

Note: From the examples above, notice that in changing a percent to a decimal, the digits in the number stayed the same. However, the decimal point moved 2 places to the left. To change a percent to a decimal, move the decimal point 2 places to the left and drop the % symbol -2-

Examples:

Change the following percents to decimals:

1.

38% = .38

2.

6.7% = .067

3.

1 % = .5% = .005 2

4.

100% = 1.00 = 1

Note: This means that 100% is all of some quantity. One means the entire quantity.

CHANGING A DECIMAL TO A PERCENT: To change a decimal to a percent, move the decimal point 2 places to the right and write the % symbol.

Examples:

Change the following decimals to percents:

1.

0.35% = 35%

2.

.092 = 9.2%

3.

2.5 = 250%

CHANGING A FRACTION TO A PERCENT: In order to change a fraction to a percent, we must first be able to change a fraction to a decimal. Remember:

To change a fraction to a decimal, divide the numerator by the denominator.

Examples: 1.

Change each of the following fractions to a decimal.

3 = 3 ÷ 4 = .75 4 0.75 4 3.00 28 20 20 0

2.

5 = 5 ÷ 8 = .625 8 0.625 8 5.000 48 20 16 40 40 0

To change a fraction to a percent, change the fraction to a decimal, then change the decimal to a percent. -3-

Examples:

Change each of the following fractions to a percent.

2.

3 = .75 = 75% 4

3.

5 = .625 = 62.5% 8

4.

3 = .6 = 60% 5

5.

1 2=2.5=250% 2

6.

1 1 1 = .33 = 33 % 3 3 3

Note: When we divide 1 by 3, we get a repeating decimal. ( Try it. )

1 = .33333... To 3

change a decimal to a percent, we must move the decimal point two places to the right. Therefore, when the fraction equals a repeating decimal, stop dividing when you have two decimal places and convert the remainder to a fraction.

0.33 3 1.00 9 10 9

This shows that

1

7.

5 2 2 = .41 = 41 % 12 3 3

- 4-

SOLVING BASIC PERCENT PROBLEMS:

1 1 = .33 3 3

There are two methods that can be used to solve basic percent problems similar to the following: What number is 25% of 120? 40 is 65.5% of what number? 15 is what percent of 25?

Method I:

Note:

Translating to an Equation to Solve a Percent Problem

To solve a percent problem: 1. Translate the sentence to an equation 2. Solve the equation

1.

When translating words ot an equation, remember that:

“is” means “=” “of” usually means to multiply 2.

We will let the letter “n” stand for “what number” or “what percent”.

3.

Before multiplying by a percent, we need to change the percent to a decimal or a fraction.

Examples:

1.

Translate to an equation and find the number.

What number is 25% of 120? ↓

↓

n

= .25 × 120

n

=

Answer:

2.

↓

↓

↓

30

30 is 25% of 120

32.5% of 24 is what number? ↓ .325

↓ ↓ ↓ × 24 = 7.8

Answer:

=

↓ n n

32.5% of 24 is 7.8 -5-

3.

What number is 125% of 340? ↓ n

↓ ↓ = 1.25

n

=

Answer:

Examples:

1.

425 is 125% of 340

16 is 50% of what number? ↓

↓

↓

↓

16 = .50 × 16 .50 × n = .50 .50 32 = n Answer:

n

16 is 50% of 32

4.2% of what number is 2.73? ↓

↓

↓

.042 ×

↓

↓

n = 2.73 .042 × n 2.73 = .042 .042 n = 65

Answer:

3.

425

Translate to an equation and find the number.

↓

2.

↓ ↓ × 340

4.2% of 65 is 2.73

240 is 325% of what number? ↓

↓

↓

↓

↓

240 = 3.25 × 240 3.25 × n = 3.25 3.25 780 = n

n

Note: 73.85 is rounded off.

Answer:

240 is 325% of 73.85 -6-

Examples:

Translate to an equation and find the percent.

1.

15 is what percent of 25? ↓ ↓

↓

15 = n 15 n × 25 = 25 25 .6 = n and

Answer:

2.

↓

↓

×

25

.6 = 60% 15 is 60% of 25

What percent of 90 is 60? ↓ n

↓ ×

↓ ↓ ↓ 90 = 60

n × 90 60 = 90 90 n

and

Answer:

3.

.66

= .66

2 2 = 66 % 3 3

2 66 % of 90 is 60 3

12 is what percent of 8? ↓

↓

12 =

↓ n

↓ ↓ × 8

12 n × 8 = 8 8 1.5 = n and 1.5 = 150%

Answer: 12 is 150% of 8. -7-

2 3

Method II:

Use a Proportion to Solve a Percent Problem

To solve a percent problem, set up the following proportion, then solve for the unknown quantity.

The Percent The Part = 100 The Whole

Note: 1.

Examples: 1.

Percent is a part out of 100. Therefore, in the proportion above, the percent corresponds to the part, and 100 corresponds to the whole.

2.

The words or quantity that follows “of” is the whole.

3.

If you don’t remember how to set up or solve a proportion, see the Learning Lab Handouts: Arithmetic 7 – Ratio and Proportion Arithmetic 8 – Proportion Word Problems

Translate to a proportion and find the number. What number is 18% of 35?

Given:

The Percent = 18 The Whole = 35

Find:

The Part

Set up the proportion:

Let n = The Part

The Percent The Part = 100 The Whole

18 n = 100 35 Solve the proportion:

Answer:

100 × n = 18 × 35 100n = 630 100n 630 = 100 100 n = 6.3

6.3 is 28% of 35.

-8-

2.

125% of 484 is what number?

Given:

The Percent = 125 The Whole = 484

Find:

The Part

Set up the proportion:

Let n = The Part

The Percent = 100 125 = 100

The Part The Whole n 484

100 × n = 125 × 484

Solve the proportion:

100 × n = 60500 100 × n 60500 = 100 100 n = 605 Answer: 3.

125% of 484 is 605.

25 is 40% of what number?

Given:

The Part = 25 The Percent = 40

Find:

The Whole

Set up the proportion:

Let n = The Whole

The Percent The Part = 100 The Whole 40 25 = 100 n

Solve the proportion:

40 × n = 100 × 25 40 × n = 2500

40 × n 2500 = 40 40 n = 62.5 Answer:

25 is 40% of 62.5.

- 9-

4.

24.6% of what number is 103.32?

Given:

The Percent = 24.6 The Part = 103.32

Find:

The Whole

Set up the proportion:

Solve the proportion:

Let n = The Whole

The Percent The Part = 100 The Whole 24.6 103.32 = 100 n 24.6 × n = 100 ×103.32 24.6 × n = 10332 24.6 × n 10332 = 24.6 24.6 n = 420

Answer:

Examples: 1.

24.6% of 420 is 103.32.

Translate to a proportion and find the percent. 36 is what percent of 72?

Given:

The Part = 36 The Whole = 72

Find:

The Percent

Set up the proportion:

Let n = The Percent

The Percent The Part = 100 The Whole n 36 = 100 72

Solve the proportion:

72 × n = 100 × 36 72 × n = 100 × 36

72 × n = 3600 72 × n 3600 = 72 72 n = 50 Answer:

36 is 50% of 72.

-10-

Note: When you use the Proportion Method to find a percent, you do NOT need to move the decimal point in your answer. This is because the variable n represents the percent already. 2.

What percent of 35.8 is 4.475?

Given:

The Whole = 35.8 The Part = 4.475

Find:

The Percent

Set up the proportion:

Let n = The Percent

The Percent The Part = 100 The Whole n 4.475 = 100 35.8

Solve the proportion:

n × 35.8 = 100 × 4.475 n × 35.8 = 447.5 n × 35.8 447.5 = 35.8 35.8 n = 12.5

Answer: 3.

12.5% of 35.8 is 4.475.

What percent of 250 is 500?

Given:

The Whole = 250 The Part = 500

Find:

The Percent

Set up the proportion:

Let n = The Percent

The Percent The Part = 100 The Whole n 500 = 100 250

Solve the proportion:

n × 250 = 100 × 500 n × 250 = 5000 n × 250 5000 = 250 250 n = 200

Answer:

200% of 250 is 500. -11-

Note: With practice, you should be able to use both methods to solve percent problems. EXERCISE 2: Translate to an equation and find the answer. 1.

What is 26% of 250?

2.

82 is 20.5% of what number?

3.

87 is what percent of 29?

4.

33 is 220% of what number?

5.

What is 96% of 75?

EXERCISE 3: Translate to a proportion and find the answer. 6.

What percent of 344 is 43?

7.

What is 235% of 4.4?

8.

14 is 0.5% of what number?

9.

What is 6.5% of 300?

10.

15 is what percent of 5000?

EXERCISE 4: Find an answer in whichever way you choose. 11.

38 is what percent of 95?

12.

21 is 24% of what number?

13.

What percent of 4 is 12?

14.

What number is 20 % of 120?

15.

25% of what number is 11?

Note: The answers and selected solutions to the Exercises are found on the following pages.

-12-

Answers and selected solutions: EXERCISE 1: 1.

10% =

1 10

4.

40% =

40 2 = 100 5

7.

70% =

7 10

10.

100% =

13.

15.

17.

2.

5.

8.

100 =1 100

1 1 33 % = 3 3

20% =

3.

30% =

3 10

1 2

6.

60% =

60 3 = 100 5

80 4 = 100 5

9.

90% =

9 10

12.

75% =

50% =

80% =

11.

20 2 = 100 5

25% =

1 4

2 2 2 200 1 2 66 % = 3 = 66 ÷ 100 = × = 3 100 3 3 100 3 66

14.

1 1 12 % = 2 8

1 1 2 = 37 1 ÷ 100 = 75 × 1 = 3 16. 37 % = 2 100 2 2 100 8

1 5 62 % = 2 8

1 1 2 = 87 1 ÷ 100 = 175 × 1 = 7 18. 87 % = 2 100 2 2 100 8

37

87

EXERCISE 2: 1.

75 3 = 100 4

n = .26 × 250 n = 65

2.

82 is 20.5% of 400

4.

33 is 220% of 15.

65 is 26% of 250.

3. 87 = n × 29 87 =n 29 3=n 87 is 300% of 29 -13-

5. n = .96 × 75 n = 72 72 is 96% of 75

EXERCISE 3: 6.

12.5% of 344 is 43.

235 n = 100 4.4 100 × n = 235 × 4.4

7.

100 × n = 1034

8.

14 is 0.5% of 2800.

100 × n 1034 = 100 100 n = 10.34 10.34 is 235% of 4.4

9.

6.5 n = 100 300 100 × n = 6.5 × 300 100 × n = 1950

10.

15 is 0.3% of 5000

100 × n 1950 = 100 100 n = 19.5 19.5 is 6.5% of 300.

EXERCISE 4: 11.

38 is 40% of 95.

12.

21 is 24% of 87.5.

14.

24 is 20% of 120.

15.

25% of 44 is 11.

-14-

13.

300% of 4 is 12.