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Word Problems solved using Ratios and Proportions

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- 1. Using Proportions to solve Problems<br />Many application problems can be easily solved using proportions.<br />
- 2. Ratio and Proportion<br />A ratio is a comparison of two values expressed as a quotient<br />Example: A class has 12 girls and 18 boys. The ratio of girls to boys is<br />This ratio can also be expressed as an equivalent fraction<br />A proportion is an equation stating that two ratios are equal.<br />Example: = <br />
- 3. Cross Products<br />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:<br /> 2 • 18 = 12 • 3<br /> 36 = 36<br />In this case, the cross products are equal, so we have a true proportion<br />This feature of proportions is very useful in solving problems.<br />
- 4. Example Proportion Problems<br />Change 42 inches to feet <br />FIND: number of feet<br />FACTS: there are 12 inches in one foot<br />FORMULA: we’ll set up a proportion<br />(label the values to ensure the ratio is<br /> accurate on bothsides of the proportion)<br />SOLVE: 12x = 42 (find cross products)<br /> x = 3.5 (solve for x)<br />ANSWER: Result: 42 inches = 3.5 feet. Substitute that value in the original proportion and take cross products:<br /> (12)(3.5) = (42)(1)<br /> 42 = 42<br />
- 5. More Proportion Problems<br />If 12 gallons of gas cost $26.68, how much will 15 gallons cost?<br />FIND: cost of 15 gallons<br />FACTS: 12 gallons cost $26.68<br />FORMULA: set up a proportion and label the parts<br />SOLVE: 12 x = 15 •26.68 (multiply cross products)<br /> 12 x = 400.20<br /> x = 33.35 (solve for x)<br />ANSWER: 15 gallons will cost $33.35. Substitute and check cross products: (15)(26.68) = (12)(33.35)<br /> 400.20 = 400.20<br />
- 6. More Proportion Problems<br />Yogurt is on sale at 5 cartons for $3.00, how much will it cost for 12 cartons?<br />FIND: the cost of 12 cartons of yogurt<br />FACTS: 5 cartons cost $3<br />FORMULA: create a proportion and label the parts<br />SOLVE: 5x = 36 (multiply cross products)<br /> x = 7.20 (solve for x)<br />ANSWER: 5 cartons will cost $7.20. To check, substitute and verify cross products: (5)(7.20) = (12)(3)<br /> 36 = 36<br />
- 7. More Proportion Problems<br />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?<br />FIND: number of successful free throws expected<br />FACTS: he normally makes 5 out of 8<br />FORMULA: create a proportion and label the parts<br />SOLVE: 8x = 1000 (multiply cross products)<br /> x = 125 (solve for x)<br />ANSWER: John would be expected to make about 125 out 200 free throws. Check by substituting and checking cross products.<br />
- 8. Similar Triangles<br />Two triangles are similar if their corresponding angles have the same measure. Corresponding sides of similar triangles are proportional.<br />Thus, the following is true about the triangles pictured: <br />FIND: the measure of side x<br />FACTS: corresponding sides are proportional<br />FORMULA: we can use either of the other two corresponding sides to create our proportion to solve<br />SOLVE: 4x = 48<br /> x = 12<br />ANSWER: The measure of the missing side is 12. We can substitute it in the original proportion to check.<br />

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