2.04 Percentages of whole numbers

We have learned that "percent" means "per one hundred". In other words, a percent is a ratio where the denominator is 100.

14\% means 14 per 100.

Observe that 14 boxes are shaded blue in this grid of 100 boxes.

We can also write it as a ratio 14:100 and as a fraction \dfrac{14}{100}.

But how do we calculate a percentage of a whole number other than 100?

Benchmark percents are familiar percents that can help solve or estimate problems more efficiently. The most common benchmark percents are 0\%, 5\%, 10\%, 25\%, and 50\%. These benchmark values and their equivalent fractions can be used as a point of reference.

Let's say a school library has 1200 books. If 35 \% of these books are fiction, we can use benchmark percents to find the number of fiction books in the library. We know 35\% = 25\% + 10\%.

We can represent this on a double number line to compare the benchmark percents with their whole number equivalents.

We can see the 25\% of 1200 lines up with 300 on our number line, and 10\% of 1200 lines up with 120. We can add together those values to find 35\%. So 300 + 120 = 420 books in the library are fiction.

Sometimes we can't use benchmark percents to easily find our desired percent. We can also solve percent problems using a proportion by creating two equivalent ratios:\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}

Returning to our library example, if 23\% of the books in the library are textbooks, let's find the number of textbooks in the library. We could use the benchmark percents on our double number line to find 20 \% or 25\% of the 1200 books. But if we want to find exactly 23\%, we can set up a proportion where x represents the number of textbooks.

We found that 276 of the books in the school library are textbooks.

Examples

Example 1

Find 15\% of 20.

Worked Solution

Create a strategy

Find 10\% and 5\% and add them. We can create a double line to help.

Apply the idea

We can first create a number line with benchmark percents.

First, we know half of 100\% is 50\% and half of 20 is 10. So we can add a mark halfway on both number lines and label them.

Next, we know 50\% \div 5 is 10\% and 10 \div 5 is 2.

Finally, we can find 5\% of the total by cutting 10\% in half. Half of 2 is 1, so we can add that to our double number line as well:

We can see 10\% of 20 is 2, and 5\% of 20 is 1. We can add these together to find 15\% of 20.

So 15\% of 20 is 1+2 = 3.

Reflect and check

We can also solve this problem using proportions. Let the unknown be x. In this problem, we know the percent and the whole or starting value.

\displaystyle \dfrac{\text{percent}}{100}	\displaystyle =	\displaystyle \dfrac{\text{part}}{\text{whole}}	Set up proportion
\displaystyle \dfrac{ 15 }{100}	\displaystyle =	\displaystyle \dfrac{x}{20}	Substitute values
\displaystyle 15 \cdot 20	\displaystyle =	\displaystyle 100 \cdot x	Means Extremes Property
\displaystyle 300	\displaystyle =	\displaystyle 100\cdot x	Evaluate the multiplication
\displaystyle 3	\displaystyle =	\displaystyle x	Divide both sides by 100

Watch question walkthrough

Example 2

What percent of 60 is 3?

Worked Solution

Create a strategy

Identify the given numbers and let the unknown be x.

Translate the statement into a proportion.

Apply the idea

We can set up a proportion using two equivalent ratios. In this problem, we know the whole or starting value and the part we are interested in.

\displaystyle \dfrac{\text{part}}{\text{whole}}	\displaystyle =	\displaystyle \dfrac{\text{percent}}{100}	Setup proportion
\displaystyle \dfrac{3}{60}	\displaystyle =	\displaystyle \dfrac{ x }{100}	Substitute values
\displaystyle 3 \cdot 100	\displaystyle =	\displaystyle 60 \cdot x	Means Extremes Property
\displaystyle 300	\displaystyle =	\displaystyle 60 \cdot x	Evaluate the multiplication
\displaystyle 5	\displaystyle =	\displaystyle x	Divide both sides by 60

We have found that 3 is 5\% of 60.

Watch question walkthrough

Example 3

60\% of what number is 120?

Worked Solution

Create a strategy

Identify the given numbers and let the unknown be x.

Translate the statement into a proportion.

Apply the idea

We know the percent and part of the total. We want to find the whole or starting value.

\displaystyle \dfrac{\text{percent}}{100}	\displaystyle =	\displaystyle \dfrac{\text{part}}{\text{whole}}	Setup proportion
\displaystyle \dfrac{ 60 }{100}	\displaystyle =	\displaystyle \dfrac{120}{x}	Substitute values
\displaystyle 60 \cdot x	\displaystyle =	\displaystyle 120 \cdot 100	Means Extremes Property
\displaystyle 60 \cdot x	\displaystyle =	\displaystyle 12000	Evaluate the multiplication
\displaystyle x	\displaystyle =	\displaystyle 200	Divide both sides by 60

We found that 60\% of 200 is 120.

Watch question walkthrough

Example 4

Farrah's bill for dinner is \$45.20. She wants to leave a tip of approximately 20\%. Explain how she could estimate the amount to tip quickly without a calculator.

Worked Solution

Create a strategy

We can use benchmark percent to help us find the tip. We can find 10\% and then double it.

Apply the idea

We know that 10\% is the same as \dfrac{1}{10}. Since we are approximating, we will find 10\% of \$45.00.

\dfrac{1}{10} \cdot \$45.00 = \dfrac{\$45.00}{10} = \$4.50

To find 20\%. we need to double this:

2 \cdot \$4.50 = \$9.00

Farrah should leave a \$9.00 tip on her meal if she wants to approximate 20\% of the bill.

Watch question walkthrough