What if you knew that 25% of a number was equal to 24? How could you find that number? After completing this Concept, you'll be able to use the percent equation to solve problems like this one.
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CK-12 Foundation: 0314S The Percent Equation (H264)
Guidance
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.
Example A
Find 25% of $80.
Solution
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 .
Example B
Express $90 as a percentage of $160.
Solution
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%.
Example C
$50 is 15% of what total sum?
Solution
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 $333.33.
Watch this video for help with the Examples above.
CK-12 Foundation: The Percent Equation
Vocabulary
- A percent is simply a ratio with a base unit of 100—for example, @$\begin{align*}13\% = \frac{13}{100}\end{align*}@$ .
- The percent equation is @$\begin{align*}\text{Rate} \times \text{Total} = \text{Part}\end{align*}@$ , or R% of Total is Part.
- The percent change equation is @$\begin{align*}\text{Percent change} = \frac{\text{final amount - original amount}}{\text{original amount}} \times 100\%.\end{align*}@$ A positive percent change means the value increased , while a negative percent change means the value decreased .
Guided Practice
$96 is 12% of what total sum?
Solution:
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 $800.
Explore More
Find the following.
- 30% of 90
- 27% of 19
- 16.7% of 199
- 11.5% of 10.01
- 0.003% of 1,217.46
- 250% of 67
- 34.5% of y
- 17.02% of y
- x% of 280
- a% of 0.332
- @$\begin{align*}y\%\end{align*}@$ of @$\begin{align*}3x\end{align*}@$
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 .