Ratios

For example, if in right ΔABC the length of leg $\overline{AC}$ is 6 inches and the length of leg $\overline{BC}$ is 8 inches, we say that the ratio of AC to BC is 6 to 8, which is often written as 6 : 8 but is just the fraction $\frac{6}{8}$. Like any fraction, a ratio can be reduced and can be converted to a decimal or a percent.

$\begin{array}{c}AC \mathrm{to} BC=6 \mathrm{to} 8=6 : 8=\frac{6}{8}\\ AC \mathrm{to} BC=3 \mathrm{to} 4=3 : 4=\frac{3}{4}=0.75=75\%\end{array}$

TIP

Ratios can always be written as a fraction.

If you know that AC = 6 inches and BC = 8 inches, you know that the ratio of AC to BC is 6 to 8. However, if you know that the ratio of AC to BC is 6 to 8, you cannot determine how long either side is. They may be 6 and 8 inches long but not necessarily. Their lengths, in inches, may be 60 and 80 or 300 and 400 since $\frac{60}{80}$ and $\frac{300}{400}$ are both equivalent to the ratio $\frac{6}{8}$. In fact, there are infinitely many possibilities for the lengths.

The important thing to observe is that the length of $\overline{AC}$ can be any multiple of 3 as long as the length of $\overline{BC}$ is the same multiple of 4.

Key Fact C1

If two numbers are in the ratio of a : b, then for some number x, the first number is ax and the second number is bx.

TACTIC C1 In any ratio problem, write x after each number and use some given information to solve for x.

EXAMPLE 1: In a right triangle, the ratio of the length of the shorter leg to the length of the longer leg is 5 to 12. If the length of the hypotenuse is 65, what is the perimeter of the triangle?

Draw a right triangle and label it with the given information; then use the Pythagorean theorem.

$\begin{array}{c}{\left(5x\right)}^2+{\left(12x\right)}^2={\left(65\right)}^2\Rightarrow 25x^2+144x^2=\mathrm{4,225}\\ \Rightarrow 169x^2=\mathrm{4,225}\Rightarrow x^2=25\Rightarrow x=5\end{array}$

So, AC = 5(5) = 25, BC = 12(5) = 60, and the perimeter equals 25 + 60 + 65 = 150.

Ratios can be extended to 3 or 4 or more terms. For example, we can say that the ratio of freshmen to sophomores to juniors to seniors in a school band is 3 : 4 : 5 : 4. This means that for every 3 freshmen in the band there are 4 sophomores, 5 juniors, and 4 seniors.

NOTE

TACTIC C1 applies to extended ratios as well.

EXAMPLE 2: What is the degree measure of the largest angle of a quadrilateral if the measures of the four angles are in the ratio of 2 : 3 : 3 : 4?

Let the measures of the four angles be 2x, 3x, 3x, and 4x. Use the fact that the sum of the measures of the angles in any quadrilateral is 360°.

$2x+3x+3x+4x=360\Rightarrow 12x=360\Rightarrow x=30$

The measure of the largest angle is 4(30) = 120°.