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# Applications of Percents

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses applications of percents. By the end of the module students should be able to distinguish between base, percent, and percentage and be able to find the percentage, the percent, and the base.

## Section Overview

• Base, Percent, and Percentage
• Finding the Percentage
• Finding the Percent
• Finding the Base

## Base, Percent, and Percentage

There are three basic types of percent problems. Each type involves a base, a percent, and a percentage, and when they are translated from words to mathemati­cal symbols each becomes a multiplication statement. Examples of these types of problems are the following:

1. What number is 30% of 50? (Missing product statement.)
2. 15 is what percent of 50? (Missing factor statement.)
3. 15 is 30% of what number? (Missing factor statement.)

In problem 1, the product is missing. To solve the problem, we represent the missing product with PP size 12{P} {}.

P = 30% 50 P = 30% 50 size 12{"P "=" 30% " cdot "50"} {}

### Percentage

The missing product PP size 12{P} {} is called the percentage. Percentage means part, or por­tion. In P = 30% 50P = 30% 50 size 12{"P "=" 30% " cdot "50"} {}, PP size 12{P} {} represents a particular part of 50.

In problem 2, one of the factors is missing. Here we represent the missing factor with QQ size 12{Q} {}.

15 = Q 50 15 = Q 50 size 12{"15 "=" Q " cdot "50"} {}

### Percent

The missing factor is the percent. Percent, we know, means per 100, or part of 100. In 15 = Q 5015 = Q 50 size 12{"15 "=" Q " cdot "50"} {}, QQ size 12{Q} {} indicates what part of 50 is being taken or considered. Specifi­cally, 15 = Q 5015 = Q 50 size 12{"15 "=" Q " cdot "50"} {} means that if 50 was to be divided into 100 equal parts, then QQ indicates 15 are being considered.

In problem 3, one of the factors is missing. Represent the missing factor with B B.

15 = 30% B 15 = 30% B size 12{"15 "=" 30% " cdot B} {}

### Base

The missing factor is the base. Some meanings of base are a source of supply, or a starting place. In 15 = 30% B15 = 30% B size 12{"15 "=" 30% " cdot B} {}, BB size 12{B} {} indicates the amount of supply. Specifically, 15 = 30% B15 = 30% B size 12{"15 "=" 30% " cdot B} {} indicates that 15 represents 30% of the total supply.

Each of these three types of problems is of the form

( percentage ) = ( percent ) ( base ) ( percentage ) = ( percent ) ( base ) size 12{ $$"percentage"$$ = $$"percent"$$ cdot $$"base"$$ } {}

We can determine any one of the three values given the other two using the methods discussed in (Reference).

## Finding the Percentage

### Sample Set A

#### Example 1

What number is 30% of 50 ? Missing product statement. (percentage) = (percent) (base) P = 30% 50 Convert 30% to a decimal. P = .30 50 Multiply. P = 15 What number is 30% of 50 ? Missing product statement. (percentage) = (percent) (base) P = 30% 50 Convert 30% to a decimal. P = .30 50 Multiply. P = 15

Thus, 15 is 30% of 50.

#### Example 2

What number is 36% of 95 ? Missing product statement. (percentage) = (percent) (base) P = 36% 95 Convert 36% to a decimal. P = .36 95 Multiply P = 34.2 What number is 36% of 95 ? Missing product statement. (percentage) = (percent) (base) P = 36% 95 Convert 36% to a decimal. P = .36 95 Multiply P = 34.2

Thus, 34.2 is 36% of 95.

#### Example 3

A salesperson, who gets a commission of 12% of each sale she makes, makes a sale of $8,400.00. How much is her commission? We need to determine what part of$8,400.00 is to be taken. What part indicates percentage.

What number is 12% of 8,400.00 ? Missing product statement. (percentage) = (percent) (base) P = 12% 8,400.00 Convert to decimals. P = .12 8,400.00 Multiply. P = 1008.00 What number is 12% of 8,400.00 ? Missing product statement. (percentage) = (percent) (base) P = 12% 8,400.00 Convert to decimals. P = .12 8,400.00 Multiply. P = 1008.00

Thus, the salesperson's commission is $1,008.00. #### Example 4 A girl, by practicing typing on her home computer, has been able to increase her typing speed by 110%. If she originally typed 16 words per minute, by how many words per minute was she able to increase her speed? We need to determine what part of 16 has been taken. What part indicates percentage. What number is 110% of 16 ? Missing product statement. (percentage) = (percent) (base) P = 110% 16 Convert to decimals. P = 1.10 16 Multiply. P = 17.6 What number is 110% of 16 ? Missing product statement. (percentage) = (percent) (base) P = 110% 16 Convert to decimals. P = 1.10 16 Multiply. P = 17.6 Thus, the girl has increased her typing speed by 17.6 words per minute. Her new speed is 16 + 17.6 = 33.616 + 17.6 = 33.6 size 12{"16 "+" 17" "." "6 "=" 33" "." 6} {} words per minute. #### Example 5 A student who makes$125 a month working part-time receives a 4% salary raise. What is the student's new monthly salary?

With a 4% raise, this student will make 100% of the original salary + 4% of the original salary. This means the new salary will be 104% of the original salary. We need to determine what part of $125 is to be taken. What part indicates percentage. What number is 104% of 125 Missing product statement. (percentage) = (percent) (base) P = 104% 125 Convert to decimals. P = 1.04 125 Multiply. P = 130 What number is 104% of 125 Missing product statement. (percentage) = (percent) (base) P = 104% 125 Convert to decimals. P = 1.04 125 Multiply. P = 130 Thus, this student's new monthly salary is$130.

#### Example 6

An article of clothing is on sale at 15% off the marked price. If the marked price is $24.95, what is the sale price? Since the item is discounted 15%, the new price will be 100% - 15% = 85%100% - 15% = 85% size 12{"100% - 15% "=" 85%"} {} of the marked price. We need to determine what part of 24.95 is to be taken. What part indicates percentage. What number is 85% of$ 24.95 . Missing product statement. (percentage) = (percent) (base) P = 85% 24.95 Convert to decimals. P = .85 24.95 Multiply. P = 21.2075 Since this number represents money, we'll round to 2 decimal places P = 21.21 What number is 85% of $24.95 . Missing product statement. (percentage) = (percent) (base) P = 85% 24.95 Convert to decimals. P = .85 24.95 Multiply. P = 21.2075 Since this number represents money, we'll round to 2 decimal places P = 21.21 Thus, the sale price of the item is$21.21.

### Practice Set A

#### Exercise 1

What number is 42% of 85?

35.7

#### Exercise 3

An assembly line worker can assemble 14 parts of a product in one hour. If he can increase his assembly speed by 35%, by how many parts per hour would he increase his assembly of products?

4.9

## Finding the Percent

### Sample Set B

#### Example 7

15 is what percent of 50 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 15 = Q 50 15 is what percent of 50 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 15 = Q 50

Recall that (missing factor) = (product) ÷ (known factor).

Q = 15÷50 Divide. Q = 0.3 Convert to a percent. Q = 30% Q = 15÷50 Divide. Q = 0.3 Convert to a percent. Q = 30%

Thus, 15 is 30% of 50.

#### Example 8

4.32 is what percent of 72 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 4.32 = Q 72 4.32 is what percent of 72 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 4.32 = Q 72

Q = 4.32÷72 Divide. Q = 0.06 Convert to a percent. Q = 6% Q = 4.32÷72 Divide. Q = 0.06 Convert to a percent. Q = 6%

Thus, 4.32 is 6% of 72.

#### Example 9

On a 160 question exam, a student got 125 correct answers. What percent is this? Round the result to two decimal places.

We need to determine the percent.

125 is what percent of 160 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 125 = Q 160 125 is what percent of 160 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 125 = Q 160

Q = 125÷160 Divide. Q = 0.78125 Round to two decimal places. Q = .78 Q = 125÷160 Divide. Q = 0.78125 Round to two decimal places. Q = .78

Thus, this student received a 78% on the exam.

#### Example 10

A bottle contains 80 milliliters of hydrochloric acid (HCl) and 30 milliliters of water. What percent of HCl does the bottle contain? Round the result to two decimal places.

We need to determine the percent. The total amount of liquid in the bottle is

80 milliliters + 30 milliliters = 110 milliliters 80 milliliters+30 milliliters=110 milliliters.

80 is what percent of 110 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 80 = Q 110 80 is what percent of 110 ? Missing factor statement. (percentage) = (percent) (base) [(product) = (factor) ⋅ (factor)] 80 = Q 110

Q = 80÷110 Divide. Q = 0.727272. . . Round to two decimal places. Q 73% The symbol "≈" is read as "approximately." Q = 80÷110 Divide. Q = 0.727272. . . Round to two decimal places. Q 73% The symbol "≈" is read as "approximately."

Thus, this bottle contains approximately 73% HCl.

#### Example 11

Five years ago a woman had an annual income of $19,200. She presently earns$42,000 annually. By what percent has her salary increased? Round the result to two decimal places.

We need to determine the percent.

42,000 is what percent of 19,200 ? Missing factor statement. (percentage) = (percent) (base) 42,000 = Q 19,200 42,000 is what percent of 19,200 ? Missing factor statement. (percentage) = (percent) (base) 42,000 = Q 19,200

Q = 42,000÷19,200 Divide. Q = 2.1875 Round to two decimal places. Q = 2.19 Convert to a percent. Q = 219% Convert to a percent. Q = 42,000÷19,200 Divide. Q = 2.1875 Round to two decimal places. Q = 2.19 Convert to a percent. Q = 219% Convert to a percent.

Thus, this woman's annual salary has increased 219%.

### Practice Set B

#### Exercise 5

99.13 is what percent of 431?

23%

#### Exercise 6

On an 80 question exam, a student got 72 correct answers. What percent did the student get on the exam?

90%

#### Exercise 7

A bottle contains 45 milliliters of sugar and 67 milliliters of water. What fraction of sugar does the bottle contain? Round the result to two decimal places (then express as a percent).

40%

## Finding the Base

### Sample Set C

#### Example 12

15 is 30% of what number? Missing factor statement. (percentage) = (percent) (base) [(percentage) = (factor) ⋅ (factor)] 15 = 30% B Convert to decimals. 15 = .30 B [(missing factor) = (product) ÷ (known factor)] 15 is 30% of what number? Missing factor statement. (percentage) = (percent) (base) [(percentage) = (factor) ⋅ (factor)] 15 = 30% B Convert to decimals. 15 = .30 B [(missing factor) = (product) ÷ (known factor)]

B = 15÷.30 B = 50 B = 15÷.30 B = 50

Thus, 15 is 30% of 50.

Try Exercise 8 in Section 11.

#### Example 13

56.43 is 33% of what number? Missing factor statement. (percentage) = (percent) (base) 56.43 = 33% B Convert to decimals. 56.43 = .33 B Divide. 56.43 is 33% of what number? Missing factor statement. (percentage) = (percent) (base) 56.43 = 33% B Convert to decimals. 56.43 = .33 B Divide.

B = 56.43÷.33 B = 171 B = 56.43÷.33 B = 171

Thus, 56.43 is 33% of 171.

Try Exercise 8 in Section 11.

#### Example 14

Fifteen milliliters of water represents 2% of a hydrochloric acid (HCl) solution. How many milliliters of solution are there?

We need to determine the total supply. The word supply indicates base.

15 is 2% of what number? Missing factor statement. (percentage) = (percent) (base) 15 = 2% B Convert to decimals. 15 = .02 B Divide. 15 is 2% of what number? Missing factor statement. (percentage) = (percent) (base) 15 = 2% B Convert to decimals. 15 = .02 B Divide.

B = 15÷.02 B = 750 B = 15÷.02 B = 750

Thus, there are 750 milliliters of solution in the bottle.

Try Exercise 9 in Section 11.

#### Example 15

In a particular city, a sales tax of 612612 size 12{6 { {1} over {2} } } {}% is charged on items purchased in local stores. If the tax on an item is $2.99, what is the price of the item? We need to determine the price of the item. We can think of price as the starting place. Starting place indicates base. We need to determine the base. 2.99 is 612% of what number? Missing factor statement. (percentage) = (percent) (base) 2.99 = 612% B Convert to decimals. 2.99 = 6.5% B 2.99 = .065 B [(missing factor) = (product) ÷ (known factor)] 2.99 is 612% of what number? Missing factor statement. (percentage) = (percent) (base) 2.99 = 612% B Convert to decimals. 2.99 = 6.5% B 2.99 = .065 B [(missing factor) = (product) ÷ (known factor)] B = 2.99÷.065 Divide. B = 46 B = 2.99÷.065 Divide. B = 46 Thus, the price of the item is$46.00.

Try Exercise 10 in Section 11.

#### Example 16

A clothing item is priced at $20.40. This marked price includes a 15% discount. What is the original price? We need to determine the original price. We can think of the original price as the starting place. Starting place indicates base. We need to determine the base. The new price,$20.40, represents 100%- 15% = 85%100%- 15% = 85% size 12{"100% - 15% "=" 85%"} {} of the original price.

20.40 is 85% of what number? Missing factor statement. (percentage) = (percent) (base) 20.40 = 85% B Convert to decimals. 20.40 = .85 B [(missing factor) = (product) ÷ (known factor)] 20.40 is 85% of what number? Missing factor statement. (percentage) = (percent) (base) 20.40 = 85% B Convert to decimals. 20.40 = .85 B [(missing factor) = (product) ÷ (known factor)]

B = 20.40÷.85 Divide. B = 24 B = 20.40÷.85 Divide. B = 24

36.28%

### Exercise 49

The distance from the sun to the earth is approx­imately 93,000,000 miles. The distance from the sun to Pluto is approximately 860.2% of the dis­tance from the sun to the Earth. Approximately, how many miles is Pluto from the sun?

### Exercise 50

The number of people on food stamps in Maine in 1975 was 151,000. By 1980, the number had decreased to 59,200. By what percent did the number of people on food stamps decrease? (Round the result to the nearest percent.)

61

### Exercise 51

In Nebraska, in 1960, there were 734,000 motor-vehicle registrations. By 1979, the total had in­creased by about 165.6%. About how many motor-vehicle registrations were there in Ne­braska in 1979?

### Exercise 52

From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?

19,500

### Exercise 53

In 1950, in the United States, there were 1,894 daily newspapers. That number decreased to 1,747 by 1981. What percent did the number of daily newspapers decrease?

### Exercise 54

A particular alloy is 27% copper. How many pounds of copper are there in 55 pounds of the alloy?

14.85

### Exercise 55

A bottle containing a solution of hydrochloric acid (HCl) is marked 15% (meaning that 15% of the HCl solution is acid). If a bottle contains 65 milliliters of solution, how many milliliters of water does it contain?

### Exercise 56

A bottle containing a solution of HCl is marked 45%. A test shows that 36 of the 80 milliliters contained in the bottle are hydrochloric acid. Is the bottle marked correctly? If not, how should it be remarked?

Marked correctly

### Exercises For Review

#### Exercise 57

((Reference)) Use the numbers 4 and 7 to illustrate the commutative property of multiplication.

#### Exercise 58

((Reference)) Convert 145145 size 12{ { {"14"} over {5} } } {} to a mixed number.

##### Solution

2 4 5 2 4 5 size 12{2 { {4} over {5} } } {}

#### Exercise 59

((Reference)) Arrange the numbers 712712 size 12{ { {7} over {"12"} } } {}, 5959 size 12{ { {5} over {9} } } {} and 4747 size 12{ { {4} over {7} } } {} in increasing order.

#### Exercise 60

((Reference)) Convert 4.006 to a mixed number.

##### Solution

4 3 500 4 3 500 size 12{4 { {3} over {"500"} } } {}

#### Exercise 61

((Reference)) Convert 7878 size 12{ { {7} over {8} } } {} % to a fraction.

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