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6.1 Ratios, Proportions and Geometric Mean

6.1 Ratios, Proportions and Geometric Mean. Objectives. Write ratios Use properties of proportions Find the geometric mean between two #s. Ratios and Proportions.

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6.1 Ratios, Proportions and Geometric Mean

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  1. 6.1 Ratios, Proportions and Geometric Mean

  2. Objectives • Write ratios • Use properties of proportions • 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 asa to b, a : b, or as a fraction awhere b ≠ 0. • Also, recall that a proportionis 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 = 14x7 b

  4. 0.3 can be written as Example 1: 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. To find this ratio, divide the number of athletes by the total number of students. Answer: The athlete-to-student ratio is 0.3.

  5. Multiple-Choice Test ItemIn 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 Example 2: Answer: C

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

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

  8. Example 4: 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.

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

  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 numberxsuch that a : x= x : bWe can also write these as fractions, a= x x bor as cross products, x 2= ab.

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

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

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

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

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