﻿ RATIOS - Ratios and Proportions - TOPICS IN ARITHMETIC - SAT SUBJECT TEST MATH LEVEL 1 ﻿

## CHAPTER 4 Ratios and Proportions

• Ratios

• Proportions

• Exercises

ratio is a fraction that compares two quantities that are measured in the same units. The first quantity is the numerator, and the second quantity is the denominator.

### RATIOS

For example, if in right ABC, the length of leg  is 6 inches and the length of leg  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 . Like any fraction, a ratio can be reduced and can be converted to a decimal or a percent.

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  and  are both equivalent to the ratio . In fact, there are infinitely many possibilities for the lengths.

The important thing to observe is that the length of  can be any multiple of 3 as long as the length of  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.

 TACTICC1 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.

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  12x = 360  x = 30

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

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