If you need to calculate someone’s *salary increase* or *raise* based on the percentage of the raise, you can do this using the distributive property. Here’s an example:

Alison’s salary was $40,000 last year, and at the end of the year she received a 5% raise. What will she earn this year?

To solve this problem, first realize that Alison got a raise. So whatever she makes this year, it will be more than she made last year. The key to setting up this type of problem is to think of percent increase as “100% of last year’s salary plus 5% of last year’s salary.” Here’s the word equation:

This year’s salary = 100% of last year’s salary + 5% of last year’s salary

Before you continue, you should be familiar with the distributive property. It says that multiplying a number by the sum of two other numbers in parentheses is the same thing as multiplying by the numbers in parentheses individually and then adding their products. In other words,

3(1 + 5) = 3(1) + 3(5)

If you evaluate both sides of the equation, here’s what you get. You can see that they’re equal:

3(6) = 3 + 15

18 = 18

The property also works for subtraction:

5(6 – 4) = 5(6) – 5(4)

5(2) = 30 – 20

10 = 10

In percent terms, you can say that the following statements are true:

Salary (100% + 5%) = Salary (100%) + Salary (5%)

Salary (100% – 5%) = Salary (100%) – Salary (5%)

What’s the point? Well, you can solve percent problems either way, but adjusting the percents first is often easier. You may be able to do the addition or subtraction in your head, and as soon as you do the multiplication, you have your answer.

Now, returning to the above problem, you can just add the percentages:

This year’s salary = (100% + 5%) of last year’s salary = 105% of last year’s salary

Change the percent to a decimal and the word *of* to a multiplication sign; then fill in the amount of last year’s salary:

This year’s salary = 1.05 $40,000

Now you’re ready to multiply:

This year’s salary = $42,000

So Alison’s new salary is $42,000.