Lesson 8.1. Ratio: a ratio is a quotient of two numbers. a:ba to ba÷b Always given in lowest terms. Slope of a line is a ratio between two points. (rise.

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Presentation transcript:

Lesson 8.1

Ratio: a ratio is a quotient of two numbers. a:ba to ba÷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.)

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.

 Example:

Arithmetic Mean Arithmetic means are just averages. Given two numbers a and b, find the arithmetic mean. a + b 2

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

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: = 15 Geometric mean: = x 2 = 81 x =  9

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

Find the mean proportional(s) between 4 and 16. = x 2 = 64 x =  8 Notice that the principal or positive root is taken although in algebra it could be either. Since we will be dealing in measurement the only one we are interested in is the positive. Find the geometric mean between 12 and 6 Remember to simplify radicals. One of the rules for radicals is that no factor of the radicand can be a perfect square.

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