Working with Ratios and Proportions

Let a and b be two real numbers, where b 0. We write the ratio of a to b, as a:b or . Often, problems involving ratios require us to set up a proportion. A proportion is an equation in which two ratios are set equal to each other.

Example 1

In a certain classroom, the ratio of boys to girls is 4 to 7. If there are 35 girls in the class, then how many total students are there?

Solution1

We need to find the number of boys in the class to determine the total number of students. To solve the problem, set up a proportion using the ratio of boys to girls: . Let’s abbreviate to to make the computation easier. Cross-multiply to solve the equation:

There are 20 boys in the classroom and therefore, a total of 35 + 20, or 55 students.

Example 2

The legend on a map says .5 inch represents 14 miles. How many inches are needed to represent 105 miles?

Solution 2
Set up a proportion using the ratio of inches to miles, and solve in the same method as the previous example.

This means 3.75 inches on the map represent 105 miles.

Example 3

The exchange rate this week is 1 dollar for .85 euros. If I exchange 75 dollars, how many euros will I receive?

Solution3
Set up a proportion using the ratio of dollars to euros, and solve as above.

I will receive 63.75 euros under that exchange rate.