6.5 Solve proportions and their applications  (Page 4/12)

For example, $\text{60\%}=\frac{60}{100}$ and we can simplify $\frac{60}{100}=\frac{3}{5}.$ Since the equation $\frac{60}{100}=\frac{3}{5}$ shows a percent equal to an equivalent ratio, we call it a percent proportion    . Using the vocabulary we used earlier:

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

We could also say:

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

Translate to a proportion. What number is $\text{75\%}$ of $90?$

Solution

If you look for the word "of", it may help you identify the base.

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=\text{number}$ .	$\frac{n}{90}=\frac{75}{100}$

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Translate to a proportion. $19$ is $\text{25\%}$ of what number?

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=\text{number}$ .	$\frac{19}{n}=\frac{25}{100}$

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Translate to a proportion. What percent of $27$ is $9?$

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $p=\text{percent}$ .	$\frac{9}{27}=\frac{p}{100}$

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Translate and solve percent proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.

Translate and solve using proportions: What number is $\text{45\%}$ of $80?$

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 100.
Simplify.
Write a complete sentence that answers the question.	36 is 45% of 80.
Check if the answer is reasonable.	Yes. 45 is a little less than half of 100 and 36 is a little less than half 80.

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In the next example, the percent is more than $100,$ which is more than one whole. So the unknown number will be more than the base.

Translate and solve using proportions: $\text{125\%}$ of $25$ is what number?

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n=$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 100.
Simplify.
Write a complete sentence that answers the question.	125% of 25 is 31.25.
Check if the answer is reasonable.	Yes. 125 is more than 100 and 31.25 is more than 25.

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Percents with decimals and money are also used in proportions.