7.6 Applications of percents

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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 $P$ .

$P=\text{30%}\cdot \text{50}$

Percentage

The missing product $P$ is called the percentage . Percentage means part , or por­tion . In $P=\text{30%}\cdot \text{50}$ , $P$ represents a particular part of 50.

In problem 2 , one of the factors is missing. Here we represent the missing factor with $Q$ .

$\text{15}=Q\cdot \text{50}$

Percent

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

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

$\text{15}=\text{30%}\cdot B$

Base

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

Each of these three types of problems is of the form

$\left(\text{percentage}\right)=\left(\text{percent}\right)\cdot \left(\text{base}\right)$

We can determine any one of the three values given the other two using the methods discussed in [link] .

Sample set a

$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{↓}}{\text{is}}& \underset{\text{↓}}{30%}& \underset{\text{↓}}{\text{of}}& \underset{\text{↓}}{50}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{↓}}{\text{(percentage)}}& \underset{\text{↓}}{\text{=}}& \underset{\text{↓}}{\text{(percent)}}& \underset{\text{↓}}{\cdot }& \underset{\text{↓}}{\text{(base)}}& \\ P& =& 30%& \cdot & 50& \text{Convert 30% to a decimal.}\hfill \\ P& =& .30& \cdot & 50& \text{Multiply.}\hfill \\ P& =& 15& & & \end{array}$

Thus, 15 is 30% of 50.

$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{↓}}{\text{is}}& \underset{\text{↓}}{36%}& \underset{\text{↓}}{\text{of}}& \underset{\text{↓}}{95}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{↓}}{\text{(percentage)}}& \underset{\text{↓}}{\text{=}}& \underset{\text{↓}}{\text{(percent)}}& \underset{\text{↓}}{\cdot }& \underset{\text{↓}}{\text{(base)}}& \\ P& =& 36%& \cdot & 95& \text{Convert 36% to a decimal.}\hfill \\ P& =& .36& \cdot & 95& \text{Multiply}\hfill \\ P& =& 34.2& & & \end{array}$

Thus, 34.2 is 36% of 95.

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 .

$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{↓}}{\text{is}}& \underset{\text{↓}}{12%}& \underset{\text{↓}}{\text{of}}& \underset{\text{↓}}{8,400.00}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{↓}}{\text{(percentage)}}& \underset{\text{↓}}{\text{=}}& \underset{\text{↓}}{\text{(percent)}}& \underset{\text{↓}}{\cdot }& \underset{\text{↓}}{\text{(base)}}& \\ P& =& 12%& \cdot & 8,400.00& \text{Convert to decimals.}\hfill \\ P& =& .12& \cdot & 8,400.00& \text{Multiply.}\hfill \\ P& =& 1008.00& & & \end{array}$

Thus, the salesperson's commission is \$1,008.00.

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 .

$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{↓}}{\text{is}}& \underset{\text{↓}}{110%}& \underset{\text{↓}}{\text{of}}& \underset{\text{↓}}{16}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{↓}}{\text{(percentage)}}& \underset{\text{↓}}{\text{=}}& \underset{\text{↓}}{\text{(percent)}}& \underset{\text{↓}}{\cdot }& \underset{\text{↓}}{\text{(base)}}& \\ P& =& 110%& \cdot & 16& \text{Convert to decimals.}\hfill \\ P& =& 1.10& \cdot & 16& \text{Multiply.}\hfill \\ P& =& 17.6& & & \end{array}$

Thus, the girl has increased her typing speed by 17.6 words per minute. Her new speed is $\text{16}+\text{17}\text{.}\text{6}=\text{33}\text{.}6$ words per minute.

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