1. 6.1 Ratios, Proportions and Geometric Mean

2. Objectives WWWWrite ratios UUUUse properties of proportions FFFFind the geometric mean between two #s

3. Ratios and Proportions  Recall that a ratio is simply a comparison of two quantities. It can be expressed as a to b, a : b, or as a fraction a where b ≠ 0.  Also, recall that a proportion is an equation stating two ratios are equivalent (i.e. 2/3 = 4/6).  Finally, to solve a proportion for a variable, we multiply the cross products, or the means and the extremes. 10 = 5  5x = 70  x = 14 x 7 b

4. The total number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. Answer: The athlete-to-student ratio is 0.3. To find this ratio, divide the number of athletes by the total number of students. 0.3 can be written as Example 1:

5. Multiple- Choice Test Item In a triangle, the ratio of the measures of three sides is 3:4:5, and the perimeter is 42 feet. Find the measure of the longest side of the triangle. A 10.5 ft B 14 ft C 17.5 ft D 37 ft Answer: C Example 2:

6. Original proportion Cross products Multiply. Answer: 27.3 Solve Divide each side by 6. Example 3a:

7. Original proportion Cross products Simplify. Answer: –2 Add 30 to each side. Divide each side by 24. Solve Example 3b:

8. A boxcar on a train has a length of 40 feet and a width of 9 feet. A scale model is made with a length of 16 inches. Find the width of the model. Because the scale model of the boxcar and the boxcar are in proportion, you can write a proportion to show the relationship between their measures. Since both ratios compare feet to inches, you need not convert all the lengths to the same unit of measure. Example 4:

9. Substitution Cross products Multiply. Divide each side by 40. Answer: The width of the model is 3.6 inches. Example 4:

10. Geometric Mean  The geometric mean between two numbers is the positive square root of their products.  In other words, given two positive numbers such as a and b, the geometric mean is the positive number x such that a : x = x : b We can also write these as fractions, a = x x b or as cross products, x 2 = ab. x b or as cross products, x 2 = ab.

11. Find the geometric mean between 2 and 50. Definition of geometric mean Let x represent the geometric mean. Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is 10. Example 5a:

12. Find the geometric mean between 25 and 7. Definition of geometric mean Let x represent the geometric mean. Cross products Take the positive square root of each side. Simplify. Answer: The geometric mean is about 13.2. Use a calculator. Example 5b:

13. a. Find the geometric mean between 3 and 12. b. Find the geometric mean between 4 and 20. Answer: 6 Answer: 8.9 Your Turn:

14. Assignment Geometry: Workbook Pg. 103 – 105 #1 – 41 odds