7.1.1 Ratio and Proportion
Key Objectives • Write and simplify ratios. • Use proportions to solve problems. Key Terms • A ratio compares two numbers by division. • A proportion is an equation stating that two ratios are equal. • In the proportion =
a b c d , the values a and d are the extremes . The values b and c are the means . • The product of the extremes and the product of the means are called the cross products . Theorems, Postulates, Corollaries, and Properties • Cross Products Property In a proportion, if =
a b
c d and b and d ≠ 0, then ad = bc . c d is equivalent to the following: =
a b
• Properties of Proportions The proportion
=
= ad bc b a d c a c = b d
A proportion can be expressed in different ways that all produce the same cross products because they all express the same relationship between the factors. Substituting numbers for the variables will show that the different forms all express the same relationship. Example 1 Writing Ratios
The slope of a line is expressed here as a ratio. The coordinates of the endpoints of the line are given. The slope of a line is defined as the ratio of the rise of the line (change in y ) to the run of the line (change in x ). The slope is the difference in y values ( y 2 − y 1 ) over the difference in x values ( x 2 − x 1 ). To express the slope of the given line, substitute the appropriate coordinate values into the formula for slope and simplify. The solution yields a slope of 1/4.
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