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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:
In problem 1 , the product is missing. To solve the problem, we represent the missing product with $P$ .
$P=\text{30\%}\cdot \text{50}$
In problem 2 , one of the factors is missing. Here we represent the missing factor with $Q$ .
$\text{15}=Q\cdot \text{50}$
In problem 3 , one of the factors is missing. Represent the missing factor with $B$ .
$\text{15}=\text{30\%}\cdot B$
Each of these three types of problems is of the form
$(\text{percentage})=(\text{percent})\cdot (\text{base})$
We can determine any one of the three values given the other two using the methods discussed in [link] .
$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{\u2193}}{\text{is}}& \underset{\text{\u2193}}{30\%}& \underset{\text{\u2193}}{\text{of}}& \underset{\text{\u2193}}{50}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{\u2193}}{\text{(percentage)}}& \underset{\text{\u2193}}{\text{=}}& \underset{\text{\u2193}}{\text{(percent)}}& \underset{\text{\u2193}}{\cdot}& \underset{\text{\u2193}}{\text{(base)}}& \\ P& =& 30\%& \cdot & 50& \text{Convert 30\% to a decimal.}\hfill \\ P& =& .30& \cdot & 50& \text{Multiply.}\hfill \\ P& =& 15& & & \end{array}$
Thus, 15 is 30% of 50.
$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{\u2193}}{\text{is}}& \underset{\text{\u2193}}{36\%}& \underset{\text{\u2193}}{\text{of}}& \underset{\text{\u2193}}{95}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{\u2193}}{\text{(percentage)}}& \underset{\text{\u2193}}{\text{=}}& \underset{\text{\u2193}}{\text{(percent)}}& \underset{\text{\u2193}}{\cdot}& \underset{\text{\u2193}}{\text{(base)}}& \\ P& =& 36\%& \cdot & 95& \text{Convert 36\% to a decimal.}\hfill \\ P& =& .36& \cdot & 95& \text{Multiply}\hfill \\ P& =& 34.2& & & \end{array}$
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 .
$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{\u2193}}{\text{is}}& \underset{\text{\u2193}}{12\%}& \underset{\text{\u2193}}{\text{of}}& \underset{\text{\u2193}}{\mathrm{8,400.00}}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{\u2193}}{\text{(percentage)}}& \underset{\text{\u2193}}{\text{=}}& \underset{\text{\u2193}}{\text{(percent)}}& \underset{\text{\u2193}}{\cdot}& \underset{\text{\u2193}}{\text{(base)}}& \\ P& =& 12\%& \cdot & \mathrm{8,400.00}& \text{Convert to decimals.}\hfill \\ P& =& .12& \cdot & \mathrm{8,400.00}& \text{Multiply.}\hfill \\ P& =& 1008.00& & & \end{array}$
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 .
$\begin{array}{cccccc}\underbrace{\text{What number}}& \underset{\text{\u2193}}{\text{is}}& \underset{\text{\u2193}}{110\%}& \underset{\text{\u2193}}{\text{of}}& \underset{\text{\u2193}}{16}\text{?}& \text{Missing product statement.}\hfill \\ \underset{\text{\u2193}}{\text{(percentage)}}& \underset{\text{\u2193}}{\text{=}}& \underset{\text{\u2193}}{\text{(percent)}}& \underset{\text{\u2193}}{\cdot}& \underset{\text{\u2193}}{\text{(base)}}& \\ P& =& 110\%& \cdot & 16& \text{Convert to decimals.}\hfill \\ P& =& 1.10& \cdot & 16& \text{Multiply.}\hfill \\ P& =& 17.6& & & \end{array}$
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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