Working with Percentages

What if we're asked something like, "What is 15% of 72?" Percentages are out of 100, but, uh, our total here is 72, not 100.

Don't worry. We've got your back.

In this section, we'll show two different methods for solving percentage problems like this one. After you try the two methods, take your pick of which one works best for you. It's your call, boss.

Method 1: Using Ratios and Proportions

All percent problems can be set up as proportions. The tricky part is figuring out what goes where in the proportion. Here are a few things to know when setting it up:

• A proportion is two equal ratios. Both ratios compare a part to a whole. We usually stick the smaller part in the numerator, and we stick the "whole" in the denominator.
• A percent is a proportion where the denominator, or whole, is equal to 100. "Per" means "of" and "cent" is a Latin prefix meaning "hundred." Think of the words century (100 years) and centimeter (one-hundredth of a meter).
• In a percentage problem, our first ratio will have the percentage in the numerator and 100 in the denominator.
• The other "whole" amount in the problem will go in the denominator of the second ratio. An easy way to think about this is that whatever comes after the "of" is the other denominator. Sometimes this is a number, but sometimes it's a variable.
• The word "what" represents the thing we're trying to find, so "what" is our variable.

Sample Problem

What is 15% of 72?

Our first ratio is the percentage. 15% means 15 out of 100, so let's plop that into fraction form.

We want to find 15% of 72. See that word "of"? We said earlier that whatever comes after "of" is our second ratio's denominator, so 72 is a denominator. We're trying to find the numerator of that second ratio, so we'll use the variable x there.

Now we're ready to solve for that missing x. Our denominators are 100 and 72, so multiply both sides by (100)(72). The 100s cancel on the left, and the 72s cancel on the right.

(72)(15) = x(100)
1080 = 100x

Divide both sides by 100 to get x all alone.

1080 ÷ 100 = x
10.8 = x

There we go: 15% of 72 is 10.8.

Method 2: Using Equations

Percentage problems can also be set up as equations. All we need to do is "translate" the sentences. Here's what we should try to remember:

• "What" means the variable, usually x.
• "Is" means "equals."
• "Of" means to multiply.
• Write all percents as decimals or ratios in your equation.
• To write a percent as a decimal, just move the decimal point two places to the left. For example, 45% translates to 0.45, and 99% translates to 0.99.

Sample Problem

A rocket kit is 25% off the original price of $42.50. What is 25% of $42.50?

We're going to shove the words from the above problem into our awesome math translator (we call it the babel fish). The first sentence just tells us information about the problem. The second sentence, the one that asks the question, is what gets translated.

The first word, "what," translates as the variable x. The second word, "is," translates into an equal sign. The 25% translates to a decimal or a ratio. We'll use a decimal this time: 25 outta 100 is the same thing as "25 hundredths," or the decimal 0.25. The word "of" translates into multiplication, and the dollar amount is luckily already a number so it stays.

What is 25% of $42.50? Translation:

x = 0.25 × 42.50

Grab a calculator and multiply those dudes to get x = 10.625. But we're still talkin' about money, so we round that to the nearest cent.

x ≈ $10.63

The rocket kit is on sale for about $10.63 less than it normally costs. Score.