# 6.13: Percent Equation to Find the Base, b

**At Grade**Created by: CK-12

**Practice**Percent Equation to Find the Base b

A digital camera and SD card are bundle priced at $65.98. This price represents a 30% discount off the price of buying the digital camera and SD card separately. If you bought the digital camera and SD card separately instead of bundled, how much would you pay?

In this concept, you will learn how to use the percent equation to find the base, \begin{align*}b\end{align*}

### Using the Percent Equation to Find the Base

When you know the percent and the amount, you can use the percent equation to find the base, \begin{align*}b\end{align*}

Let’s look at how to apply the percent equation to this problem.

3 is 50% of what number?

First, the words “of what number” let you know that you are looking for the base.

Next, insert the other given information into our equation. First, let’s change the percent to a decimal.

\begin{align*}p = 50 \% = 0.50\end{align*}

Then, work on the percent equation, filling in the given values.

\begin{align*}\begin{array}{rcl}
a &=& p \times b \\
3 &=& 0.50b
\end{array}\end{align*}

Next, use the inverse operation and solve for \begin{align*}b\end{align*}

\begin{align*}
\begin{array}{rcl}
\frac{3}{0.5} &=& b \\
6 &=& b
\end{array}
\end{align*}

The answer is 3 is 50% of 6.

Let’s look at another example.

9 is 75% of what number?

First, let’s change 75% to a decimal.

\begin{align*}75 \% = 0.75\end{align*}

Next, let’s fill the given values into the equation.

\begin{align*}9 = 0.75b\end{align*}

Now you can solve for \begin{align*}b\end{align*}

\begin{align*}\begin{array}{rcl}
\frac{9}{0.75} &=& b \\
12 &=& b
\end{array}\end{align*}

The answer is 9 is 75% of 12.

### Examples

#### Example 1

Earlier, you were given a problem about the bundled digital camera and SD card.

It was priced at $65.98, which was 30% off the total cost of buying both items separately. If you bought the digital camera and SD card separately instead of bundled, how much would you pay?

First, a 30% discount means that $65.98 is 70% \begin{align*}(100 \% - 30 \%)\end{align*}

Now, let’s change 70% to a decimal.

\begin{align*}70 \% = 0.70\end{align*}

Next, let’s fill the given values into the equation.

\begin{align*}65.98 = 0.70b\end{align*}

Then you can solve for \begin{align*}b\end{align*}

\begin{align*}\begin{array}{rcl}
\frac{65.98}{0.70} &=& b \\
94.26 &=& b
\end{array}
\end{align*}

The answer is the cost of buying the two items separately would be $94.26.

#### Example 2

The Cougars basketball team won 21 games. If that number is 70% of the games that the team played, how many games did the Cougars play?

Think: 70% is the percent. 21 is the amount of games that they won. You want to find the base.

First, take 70% and write it as a decimal.

\begin{align*}70 \% = 0.70\end{align*}

Next, write the given information into an equation and solve it using the inverse operation.

\begin{align*}\begin{array}{rcl}
21 &=& 0.70 b \\
\frac{21}{0.7} &=& b \\
30 &=& b
\end{array}\end{align*}

The answer is the team played 30 games in all.

#### Example 3

8 is 20% of what number?

First, let’s change 20% to a decimal.

\begin{align*}20 \% = 0.20\end{align*}

Next, let’s fill the given values into the equation.

\begin{align*}8 = 0.20b\end{align*}

Now you can solve for \begin{align*}b\end{align*}

\begin{align*}\begin{array}{rcl}
\frac{8}{0.20} &=& b \\
40 &=& b
\end{array}\end{align*}

The answer is 8 is 20% of 40.

#### Example 4

15 is 30% of what number?

First, let’s change 30% to a decimal.

\begin{align*}30 \% = 0.30\end{align*}

Next, let’s fill the given values into the equation.

\begin{align*}15 = 0.30b\end{align*}

Now you can solve for \begin{align*}b\end{align*}

\begin{align*}
\begin{array}{rcl}
\frac{15}{0.30} &=& b \\
50 &=& b
\end{array}\end{align*}

The answer is 15 is 30% of 50.

#### Example 5

22 is 40% of what number?

First, let’s change 40% to a decimal.

\begin{align*}40 \% = 0.40\end{align*}

Next, let’s fill the given values into the equation.

\begin{align*}22 = 0.40b\end{align*}

Now you can solve for \begin{align*}b\end{align*} using the inverse operation.

\begin{align*} \begin{array}{rcl} \frac{22}{0.40} &=& b \\ 55 &=& b \end{array}\end{align*}

The answer is 22 is 40% of 55.

### Review

Use the percent equation to find each base.

- 5 is 10% of what number?
- 15 is 30% of what number?
- 18 is 20% of what number?
- 12 is 50% of what number?
- 15 is 40% of what number?
- 14 is 20% of what number?
- 80 is 25% of what number?
- 60 is 30% of what number?
- 45 is 40% of what number?
- 16 is 25% of what number?
- 60 is 18% of what number?
- 25 is 30% of what number?
- 21 is 15% of what number?
- 55 is 10% of what number?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 6.13.

### Resources

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Amount

In a proportion, the amount is the part of the base that is being calculated.Base

In the context of the percent equation, the base is the part of the whole from which the amount is calculated.Decimal

In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).Equation

An equation is a mathematical sentence that describes two equal quantities. Equations contain equals signs.Inverse Operation

Inverse operations are operations that "undo" each other. Multiplication is the inverse operation of division. Addition is the inverse operation of subtraction.Percent

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

In this concept, you will use the percent equation to find the base, b.

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