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Percent Equations

Equations to solve for rates, totals, and parts.

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Percent Equations

Percent Equations 

The percent equation is often used to solve problems. It goes like this:

\begin{align*}& \text{Rate} \times \text{Total} = \text{Part}\\ & \qquad \qquad \text{or}\\ & R\% \ \text{of Total is Part}\end{align*}

Rate is the ratio that the percent represents (\begin{align*}R\%\end{align*} in the second version).

Total is often called the base unit.

Part is the amount we are comparing with the base unit.

 

Finding a Percent of a Total 

Find 25% of $80.

We are looking for the part. The total is $80. ‘of’ means multiply. \begin{align*}R\%\end{align*} is 25%, so we can use the second form of the equation: 25% of $80 is Part, or \begin{align*}0.25 \times 80 = \text{Part}\end{align*}.

\begin{align*}0.25 \times 80 = 20\end{align*}, so the Part we are looking for is $20.

Expression Values and Percentages 

Express $90 as a percentage of $160.

This time we are looking for the rate. We are given the part ($90) and the total ($160). Using the rate equation, we get \begin{align*}\text{Rate} \times 160 = 90\end{align*}. Dividing both sides by 160 tells us that the rate is 0.5625, or 56.25%.

Finding the Total Sum 

$50 is 15% of what total sum?

This time we are looking for the total. We are given the part ($50) and the rate (15%, or 0.15). Using the rate equation, we get \begin{align*}0.15 \times \text{Total} = \$50\end{align*}. Dividing both sides by 0.15, we get \begin{align*}\text{Total} = \frac{50}{0.15} \approx 333.33\end{align*}. So $50 is 15% of $333.33.

 

Example

Example 1

$96 is 12% of what total sum?

This time we are looking for the total. We are given the part ($96) and the rate (12%, or 0.12). Using the rate equation, we get \begin{align*}0.12 \times \text{Total} = \$96\end{align*}. Dividing both sides by 0.15, we get \begin{align*}\text{Total} = \frac{96}{0.12}=800\end{align*}. So $96 is 12% of $800.

Review 

Find the following.

  1. 30% of 90
  2. 27% of 19
  3. 16.7% of 199
  4. 11.5% of 10.01
  5. 0.003% of 1,217.46
  6. 250% of 67
  7. 34.5% of y
  8. 17.02% of y
  9. x% of 280
  10. a% of 0.332
  11. \begin{align*}y\%\end{align*} of \begin{align*}3x\end{align*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 3.14. 

Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook® resource, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9613.

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Vocabulary

Percent Equation

The percent equation can be stated as: "Rate times Total equals Part," or "R% of Total is Part."

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