Copyright © 2014 Pearson Education, Inc. 5 CHAPTER 5.1 Ratio and Proportions Copyright © 2014 Pearson Education, Inc.

Ratios as Fractions A ratio is the quotient of two quantities. In fact, ratios are no different from fractions, except that ratios are sometimes written using a notation other than fractional notation. For example, the ratio of 1 to 2 can be written as 1 to 2 or or 1: 2 These ratios are all read as “the ratio of 1 to 2.”

Writing a Ratio as a Fraction The order of the quantities is important when writing ratios. To write a ratio as a fraction, write the first number of the ratio as the numerator of the fraction and the second number as the denominator. Copyright © 2014 Pearson Education, Inc.

Writing a Ratio as a Fraction Write the ratio of 12 to 17 using fractional notation. Solution The ratio is Copyright © 2014 Pearson Education, Inc.

Writing a Ratio as a Fraction in Simplest Form Write each ratio as a fraction in simplest form. a. $15 to $10 b. 4 ft to 24 in. c. Solution Write each ratio as a fraction. Convert unlike units to like units. a. b. 2 Copyright © 2014 Pearson Education, Inc.

Writing a Ratio as a Fraction in Simplest Form Write each ratio as a fraction in simplest form. a. $15 to $10 b. 4 ft to 24 in. c. Solution Write each ratio as a fraction. Convert unlike units to like units. c. Copyright © 2014 Pearson Education, Inc.

Using Ratios in Geometry Given the rectangle shown: a. Find the ratio of its width to its length. b. Find the ratio of its length to its perimeter. Solution a. ratio of width to length Copyright © 2014 Pearson Education, Inc.

Using Ratios in Geometry Given the rectangle shown: a. Find the ratio of its width to its length. b. Find the ratio of its length to its perimeter. Solution b. length to perimeter perimeter = 7 + 5 + 7 + 5 = 24 feet Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Solving Proportions An equation stating that two ratios are equal is called a proportion. The first and last numbers in a proportion are the extremes. The middle two numbers are the means. Cross products are the product of the means and also the product of the extremes. Copyright © 2014 Pearson Education, Inc.

Theorem 5.1 Means-Extremes Property Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Solving a Proportion Solve each proportion for the variable. a. b. Solution a. b. The check is left to the student. Copyright © 2014 Pearson Education, Inc.

Copyright © 2014 Pearson Education, Inc. Geometric Means For any two positive numbers a and b, the geometric mean of a and b is the positive number x such that Copyright © 2014 Pearson Education, Inc.

Finding the Geometric Mean What is the geometric mean of 6 and 15? a. 90 b. c. d. 30 Solution Let x be the geometric mean. Use the definition of geometric mean to set up a proportion. Then solve for x. The correct answer is b. Copyright © 2014 Pearson Education, Inc.

Properties of Proportions Copyright © 2014 Pearson Education, Inc.

Using Properties of Proportions Use the Properties of Proportions to write three proportions equivalent to Solution Property 1: Write the reciprocal of each ratio: Property 2: Switch the means: Copyright © 2014 Pearson Education, Inc.

Using Properties of Proportions Use the Properties of Proportions to write three proportions equivalent to Property 3: In each ratio, add the denominator to the numerator: Copyright © 2014 Pearson Education, Inc.

Writing Equivalent Proportions In the diagram, What ratio completes the equivalent proportion Justify your answer. Solution a. Notice the positions of x and y in the first proportion. Now see where x and y appear in the second proportion. Using Property 2: Copyright © 2014 Pearson Education, Inc.

Writing Equivalent Proportions In the diagram, What ratio completes the equivalent proportion Justify your answer. Using cross products: Copyright © 2014 Pearson Education, Inc.

Proportional Segments   Copyright © 2014 Pearson Education, Inc.

Proportional Segments Copyright © 2014 Pearson Education, Inc.