Proportions – Outschool

Proportions Lesson

I. The Basics

Ratios are used to compare quantities and are expressed in a few different ways. For example, the ratio “12 inches to 1 foot” can be expressed as 12:1 or [latex]frac{12}{1}[/latex].

A proportion equates two ratios so you can solve for a missing value. Solve proportions by cross-multiplying.

Example 1: If 3 gallons of milk cost $10.44, how much will 5 gallons cost?

Ratio 1 is 3 gallons: $10.44, or [latex] frac{3}{10.44}[/latex].

Ratio 2 is 5 gallons: $x, or [latex]frac{5}{x}[/latex].

Set up a proportion and cross-multiply:

[latex]frac{3}{10.44} = frac{5}{x}[/latex]
[latex]newline[/latex]

[latex]3x = 52.2[/latex]

[latex]x = [/latex]$[latex]17.40[/latex]

II. More Ratio and Proportion Problems

Example 2: On a certain map, 0.5 inches represents 7 miles. The distance you need to travel is 5.25 inches on the map. How far is this distance in miles?

[latex]frac{0.5}{7}= frac{5.25}{x}[/latex]
[latex]newline[/latex]

[latex]0.5x = 7(5.25)[/latex]
[latex]newline[/latex]

[latex]0.5x = 36.75[/latex]

[latex]x = 73.5[/latex] miles

Example 3: If the ratio a:b is 3:4 and b:c is 6:7, what is the ratio a:c?

In the first ratio, b=4, and in the second ratio, b=6. The least common multiple for 4 and 6 is 12. Scale b up to 12 in both ratios so the ratios can be compared.

Change a:b from 3:4 to 9:12 (scaled up by a factor of 3).

Change b:c from 6:7 to 12:14 (scaled up by a factor of 2).

Since a:b is 9:12 and b:c is 12:14, a:c must be 9:14.