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
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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.