Using Proportions to solve Problems

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Presentation transcript:

Using Proportions to solve Problems

Ratio and Proportion A ratio is a comparison of two values expressed as a quotient Example: A class has 12 girls and 18 boys. The ratio of girls to boys is This ratio can also be expressed as an equivalent fraction A proportion is an equation stating that two ratios are equal. Example: =

Cross Products A test to verify that we have a true proportion is to check the cross products. In a true proportion, the cross products are equal: 2 • 18 = 12 • 3 36 = 36 In this case, the cross products are equal, so we have a true proportion This feature of proportions is very useful in solving problems.

Example Proportion Problems Change 42 inches to feet FIND: number of feet FACTS: there are 12 inches in one foot FORMULA: we’ll set up a proportion (label the values to ensure the ratio is accurate on both sides of the proportion) SOLVE: 12x = 42 (find cross products) x = 3.5 (solve for x) ANSWER: Result: 42 inches = 3.5 feet. Substitute that value in the original proportion and take cross products: (12)(3.5) = (42)(1) 42 = 42

More Proportion Problems If 12 gallons of gas cost $26.68, how much will 15 gallons cost? FIND: cost of 15 gallons FACTS: 12 gallons cost $26.68 FORMULA: set up a proportion and label the parts SOLVE: 12 x = 15 •26.68 (multiply cross products) 12 x = 400.20 x = 33.35 (solve for x) ANSWER: 15 gallons will cost $33.35. Substitute and check cross products: (15)(26.68) = (12)(33.35) 400.20 = 400.20

More Proportion Problems Yogurt is on sale at 5 cartons for $3.00, how much will it cost for 12 cartons? FIND: the cost of 12 cartons of yogurt FACTS: 5 cartons cost $3 FORMULA: create a proportion and label the parts SOLVE: 5x = 36 (multiply cross products) x = 7.20 (solve for x) ANSWER: 5 cartons will cost $7.20. To check, substitute and verify cross products: (5)(7.20) = (12)(3) 36 = 36

More Proportion Problems John has a free throw average of 5 out of 8 attempts. At this rate, how many successful free throws would he be expected to make out of 200 attempts? FIND: number of successful free throws expected FACTS: he normally makes 5 out of 8 FORMULA: create a proportion and label the parts SOLVE: 8x = 1000 (multiply cross products) x = 125 (solve for x) ANSWER: John would be expected to make about 125 out 200 free throws. Check by substituting and checking cross products.

Similar Triangles Two triangles are similar if their corresponding angles have the same measure. Corresponding sides of similar triangles are proportional. Thus, the following is true about the triangles pictured: FIND: the measure of side x FACTS: corresponding sides are proportional FORMULA: we can use either of the other two corresponding sides to create our proportion to solve SOLVE: 4x = 48 x = 12 ANSWER: The measure of the missing side is 12. We can substitute it in the original proportion to check.