5 1 Ratios Rates Proportions Ratio The ratio

5. 1 Ratios, Rates, & Proportions • Ratio: The ratio of 2 numbers a and b is the quotient of the numbers. The numbers a & b are referred to as the terms of the ratio. • The ratio of 2 numbers a and b may be written in a variety of ways: a a to b a: b b • In writing the ratio of 2 numbers, express the ratio (fraction) in simplest form: 50 or 1 : 2 100 • Units must be commensurable (convertible to the same unit of measure). 12/11/2021 Section 5. 1 Jones/Nack 1

Ratio Example • Loren is 30 years old & Lucas is 10 years old. How do their ages compare? Obviously, Loren is 20 years older than Lucas. It is sometimes desirable, however, to compare 2 numbers by determining how many times larger (or smaller) one number is compared to a second number. • We divide the first number by the second number: Loren’s age = 30 = 3 Lucas’ age 10 1 • Loren is 3 times as old as Lucas. The result of dividing 2 numbers is called a ratio. 12/11/2021 Section 5. 1 Jones/Nack 2

Ratios (cont’d) • If, however, a ratio is formed in order to determine how many times larger or smaller one value is than another, both quantities must be expressed in the same unit of measurement. • Ex. If the length of AB is 2 feet and the length of XY is 16 inches, then to determine how many times larger AB is compared to XY, we must convert one of the units of measurement into the other. • 2 feet = 24 inches: AB = 24 inches = 3 XY 16 inches 2 12/11/2021 Section 5. 1 Jones/Nack 3

Rate • A quotient that compares two quantities that are incommensurable. • For example, if a person travels 120 miles in a car in 3 hours, then the ratio of the distance traveled to the time traveled is: 120 miles = 40 miles 3 hours 1 hour The value 40 miles/hr is the average rate of speed. Ex. 1 a, c, e, Ex 2. 12/11/2021 Section 5. 1 Jones/Nack 4

Proportions • An equation that states that 2 ratios are equal is called a proportion. • Each term of a proportion is given a special name according to its position in the proportion. We say that “a is to be as c is to d. ” – The first and last terms (a and d) of the proportion are the extremes – the second and third terms (b and c) are the means. 12/11/2021 Section 5. 1 Jones/Nack 5

Property 1: (Means-Extremes Property) • In a proportion , the product of the means equals the product of the extremes (the cross products of a proportion are equal) a d=b c • Forming this cross products is sometimes referred to as cross-multiplying. • Examples 3, 4 p. 221 12/11/2021 Section 5. 1 Jones/Nack 6

Mean Proportional (Geometric Mean) • If the second & third terms of a proportion are the same, then either term is referred to as the mean proportional or geometric mean between the first & fourth terms of the proportions: Ex 5 p. 222 Extended Ratios compares more than 2 quantities: a: b: c should be represented by ax: bx: cx. Ex: 6, 7 p. 223 12/11/2021 Section 5. 1 Jones/Nack 7

Algebraic Properties of Proportions • Property 2: The means or the extremes or both may be interchanged to form an equivalent proportion : If then (provided a, b, c, & d are nonzero numbers. ) d=c d=b a=b b a c d • Property 3: If the denominator is added or subtracted from the numerator on each side of the proportion, then an equivalent proportion results: a+b=c+d a-b =c–d b d 12/11/2021 Section 5. 1 Jones/Nack 8

Using Proportions in Triangles • Example 8 p. 225 Given: AB = AC =BC DE DF EF Find the lengths of DF and EF: Set up the proportion: 4=5=6 10 x y Solve for x in the first ratio pair. Solve for y, using the first and third ratio pair. 12/11/2021 Section 5. 1 Jones/Nack 4 5 6 x 10 y 9