Ratios & Proportions Lesson 8.1

Ratio: a ratio is a quotient of two numbers. a:b a to b a÷b Always given in lowest terms. Slope of a line is a ratio between two points. (rise over run)

Proportions: two or more ratios set equal to each other. a:b = c:d = a is the first term b is the second term c is the third term d is the fourth term

Product and Ratio Theorems In a product containing four terms: First and fourth terms are the extremes. Second and third terms are the means. Theorem 59: In a proportion, the product of the means is equal to the product of the extremes. (means-extremes product theorem.)

=  ad = bc If they aren’t equal, then the ratios aren’t in proportion. Theorem 60: If the product of a pair of non-zero numbers is equal to the product of another pair of non-zero numbers, then either pair of numbers may be made the extremes, and the other pair the means, of a proportion. (means-extremes ratio theorem.)

Given: pq = rs Then: = = = pq = rs pq = rs pq = rs This theorem is harder to state than to use! Given: pq = rs Then: = = = pq = rs pq = rs pq = rs These proportions are all equivalent since their cross products are equivalent equations.

Geometric Mean: = = x is the geometric mean 4 is the geometric mean In a mean proportion, the means are the same. = = x is the geometric mean 4 is the geometric mean

Find the arithmetic & geometry means between 3 and 27. Definition: If the means in a proportion are equal, either mean is called a geometric mean or mean proportional between the extremes. Find the arithmetic & geometry means between 3 and 27. Arithmetic mean: Geometric mean: = x2 = 81 x =  9 = 15

Solve: Find the fourth term (sometimes called the fourth proportional) of a proportion if the first three terms are 2, 3, and 4. = You might want to reduce the fraction first. = 7x = 42 x = 6 2x = 12 x = 6

= x2 = 64 x =  8 Find the mean proportional(s) between 4 and 16. If we are looking for the length of a segment, then only the positive number works.

If 3x = 4y, find the ratio of x to y. Make x and 3 the extremes and y and 4 the means. 3x = 4y =

Is = ? equal to = b(x-2y) = y(a-2b) bx-2by = ay-2by bx = ay ay = bx Cross multiply and simplify both sets. b(x-2y) = y(a-2b) bx-2by = ay-2by bx = ay ay = bx Yes, they are equal.