Copyright © Cengage Learning. All rights reserved. Rational Expressions and Equations; Ratio and Proportion 6

Copyright © Cengage Learning. All rights reserved. Section 6.7 Ratios

3 Objectives Express a ratio in simplest form. Translate an English sentence to a ratio. Write a ratio as a unit cost. Write a ratio as a rate

4 Express a ratio in simplest form 1.

5 Express a ratio in simplest form Ratios appear often in real-life situations. For example, To prepare fuel for a Lawnboy lawnmower, gasoline must be mixed with oil in the ratio of 50 to 1. To make 14-karat jewelry, gold is mixed with other metals in the ratio of 14 to 10. At Rock Valley College, the ratio of students to faculty is 16 to 1. Ratios give us a way to compare numerical quantities.

6 Express a ratio in simplest form Ratios A ratio is a comparison of two numbers by their indicated quotient. In symbols, If a and b are two numbers, the ratio of a to b is. Comment The denominator b cannot be 0 in the fraction, but b can be 0 in the ratio a to b.

7 Express a ratio in simplest form Some examples of ratios are,, and The fraction can be read as “the ratio of 7 to 9.” The fraction can be read as “the ratio of 21 to 27.” The fraction can be read as “the ratio of 2,290 to 1,317.” Because and represent equal numbers, they are equal ratios.

8 Example Express each phrase as a fraction in simplest form. a. the ratio of 15 to 12 b. the ratio of 0.3 to 1.2 Solution: a. The ratio of 15 to 12 can be written as the fraction. After simplifying, the ratio is.

9 b. The ratio of 0.3 to 1.2 can be written as the fraction. We can simplify this fraction as follows: Example – Solution Multiply. Simplify. To clear the decimals, multiply both numerator and denominator by 10. cont’d

10 Translate an English sentence to a ratio 2.

11 Example – Student / Faculty Ratios At a college, there are 2,772 students and 154 faculty members. Write a fraction in simplified form that expresses the ratio of students per faculty member. Solution: The ratio of students to faculty is 2,772 to 154. We can write this ratio as the fraction and simplify it.

12 Example – Solution The ratio of students to faculty is 18 to 1. cont’d

13 Write a ratio as a unit cost 3.

14 Write a ratio as a unit cost The unit cost of an item is the ratio of its cost to its quantity. For example, the unit cost (the cost per pound) of 5 pounds of gourmet coffee priced at $41.75 is given by The unit cost is $8.35 per pound. $41.75  5 = $8.35

15 Example – Shopping Pizza sauce comes packaged in a 14-ounce jar, which sells for $1.49, or in a 30-ounce jar, which sells for $3.32. Which is the better buy? Solution: To find the better buy, we must find each unit cost. The unit cost of the 14-ounce jar is Change $1.49 to 149 cents.

16 Example– Solution The unit cost of the 30-ounce jar is Since the unit cost is less when the pizza sauce is packaged in 14-ounce jars, that is the better buy. cont’d Change $3.32 to 332 cents.

17 Write a ratio as a rate 4.

18 Write a ratio as a rate When ratios are used to compare quantities with different units, they often are called rates. For example, if we drive 413 miles in 7 hours, the average rate of speed is the quotient of the miles driven to the length of time of the trip.

19 Write a ratio as a rate The rate can be expressed in any of the following forms:, 59 miles per hour, 59 miles/hour, or 59 mph

20 Example– Hourly Pay Find the hourly rate of pay for a student who earns $370 for working 40 hours. Solution: We can write the rate of pay as and simplify by dividing 370 by 40. The rate is $9.25 per hour.