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6.7: Proportions to Find Base b

Difficulty Level: At Grade Created by: CK-12
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Kelsey wants to buy a new necklace for the school dance. She learns that her favorite store is having a one-day sale and has knocked down the price of all its necklaces to $9.95. The necklace Kelsey wants has been reduced by 47%. What is its non-sale price?

In this concept, you will learn to use cross products of proportions to find a base, \begin{align*}b\end{align*}b.

Using Proportions to Find Base, b

There are many different ways to figure out a part of a whole, but one way is by using proportions.

Cross products can be used to solve for the base as well as the percent and the amount.

When you see the phrase “Of what number?” you know that you are going to be solving for \begin{align*}b\end{align*}b, the base. You can then use the same proportion to solve for \begin{align*}b\end{align*}b.

Let’s take a look at an example.

33 is 15% of what number?

Remember that the number following the word “of” is the base. Since there is no number after the word “of”, you need to find the base \begin{align*}(b)\end{align*}(b). 33 is the amount \begin{align*}(a)\end{align*}(a) and 15 is the percent \begin{align*}(p)\end{align*}(p).

\begin{align*}\begin{array}{rcl} \frac{a}{b} & = & \frac{p}{100}\\ \frac{33}{b} & = & \frac{15}{100}\\ 15b & = & 33(100)\\ 15b & = & 3,300\\ \frac{15b}{15} & = & \frac{3,300}{15}\\ b & = & 220 \end{array}\end{align*}ab33b15b15b15b15b======p1001510033(100)3,3003,30015220 

The answer is 33 is 15% of 220.

Let’s look at another example.

6 is 17% of what number. Round your answer to the nearest whole number?

To figure this out, let’s write a proportion.

\begin{align*}\frac{6}{b} = \frac{17}{100}\end{align*}6b=17100

Next, use cross products to solve for \begin{align*}b\end{align*}b.

\begin{align*}\begin{array}{rcl} 17b & = & 6(100)\\ 17b & = & 600\\ b & = & 35.29 \end{array}\end{align*}17b17bb===6(100)60035.29

The answer is rounded to the nearest whole number, 6 is 17% of 35.

Examples

Example 1

Earlier, you were given a problem about Kelsey and her new necklace.

For one day, the necklace she wants has been reduced to $9.95.

The discount applied is 47%. What is the regular price of the necklace?

First, figure out what you’re trying to find. If the discount is 47%, the sale price is \begin{align*}100\% - 47\%\end{align*}100%47% or 53% of the regular price. So you’re trying to find “$9.95 is 53% of what number?”

Now, let’s write a proportion.

\begin{align*}\frac{9.95}{b} = \frac{53}{100}\end{align*}9.95b=53100

Next, use cross products to solve for \begin{align*}b\end{align*}b.

\begin{align*}\begin{array}{rcl} 53b & = & 9.95(100)\\ 53b & = & 995\\ b & = & \$ 18.77 \end{array}\end{align*}53b53bb===9.95(100)995$18.77

The answer is the regular price of the necklace is $18.77. 

Example 2

Six students in Miss Lang’s third period math class got A’s on their math test. This was 24% of the class.

How many students are in Miss Lang’s third period math class?

You can think of this problem as “6 is 24% of what number?” First, let’s set up the proportion.

\begin{align*}\frac{6}{b} = \frac{24}{100}\end{align*}6b=24100

Next, use cross products to solve for \begin{align*}b\end{align*}b.

\begin{align*}\begin{array}{rcl} 24b & = & 6(100)\\ 24b & = & 600\\ b & = & 600 \div 24\\ b & = & 25 \end{array}\end{align*}24b24bbb====6(100)600600÷2425

The answer is there are 25 students in Miss Lang’s third period math class.

Example 3

6 is 25% of what number?

First, let’s write a proportion.

\begin{align*}\frac{6}{b} = \frac{25}{100}\end{align*}

Next, use cross products to solve for \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} 25b & = & 6(100)\\ 25b & = & 600\\ b & = & 24 \end{array}\end{align*}

The answer is 6 is 25% of 24.

Example 4

12 is 8% of what number?

First, let’s write a proportion.

\begin{align*}\frac{12}{b} = \frac{8}{100}\end{align*}

Next, use cross products to solve for \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} 8b & = & 12(100)\\ 8b & = & 1,200\\ b & = & 150 \end{array}\end{align*}

The answer is that 12 is 8% of 150.

Example 5

22 is 11% of what number?

First, let’s write a proportion.

\begin{align*}\frac{22}{b} = \frac{11}{100}\end{align*}

Next, use cross products to solve for \begin{align*}b\end{align*}.

\begin{align*}\begin{array}{rcl} 11b & = & 22(100)\\ 11b & = & 2,200\\ b & = & 200 \end{array}\end{align*}

The answer is that 22 is 11% of 200.

Review

Find each missing base.

  1. 5 is 10% of what number?
  2. 7 is 10% of what number?
  3. 10 is 20% of what number?
  4. 16 is 40% of what number?
  5. 8 is 25% of what number?
  6. 14 is 50% of what number?
  7. 25 is 5% of what number?
  8. 4 is 80% of what number?
  9. 18 is 25% of what number?
  10. 9 is 3% of what number?
  11. 15 is 20% of what number?
  12. 18 is 13% of what number?
  13. 15 is 12.5% of what number?
  14. 18 is 55% of what number?
  15. 22 is 5.5% of what number?

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.7.

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Vocabulary

Percent

Percent means out of 100. It is a quantity written with a % sign.

Proportion

A proportion is an equation that shows two equivalent ratios.

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