How to Teach Math Ratios
You are teaching ratio and proportion to a classroom of students. Alternatively, you may be tutoring a friend or relative. In any event, you want to help the students learn and retain the information for future mathematics, business and science courses. They will also apply ratios to everyday activities such as reading maps, calculating unit costs and manipulating recipes in the kitchen. Teach the students how to properly set up and reduce ratios and solve proportions.
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Instructions
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Define ratio. A ratio is a comparison of two or more numbers. For example, if a bag contains 3 red marbles, 5 blue marbles and 9 green marbles, you can use the three-term ratio 3:5:9 to represent all the marbles. The colon represents the word "to." You can also set up the following two-term ratios: 3:5, 5:9 and 3:9. Reduce the ratio 3:9 to 1:3. You can use fraction notation to set up all two-term ratios. In this example, set up 3/5, 5/9 and 1/3.
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Define proportion. A proportion is a statement that two ratios are equal. For example, 7:8 = 14:16. Solve simple proportion problems by inspection. In the proportion, 3:4 = 15:x, solve for x by examining the relationship between the first two terms of each ratio. To get 15, multiply 3 by 5. Apply this relationship to the 4 in the first ratio. Therefore, 4 x 5 = 20. The missing number in this proportion is 20.
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Use cross-multiplication to solve more difficult proportions. In cross-multiplication, the product of the means (the inside numbers) equals the product of the extremes (the outside numbers). Solve the following proportion: 2:x = 5:12. The product of the inside numbers is 5x and the product of the outside numbers is 2 x 12 or 24. Solve the equation 5x = 24. Therefore, x = 4.8.
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Practice using cross-multiplication with fraction notation. Solve 4/x = 9/30. When you cross-multiply over the equal sign, 9x = 4 x 30 or 9x = 120. Solve to obtain x = 13.3.
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Apply cross-multiplication to solving proportions with three or more terms. Solve 9:x:5 = y:8:7. Set up the proportion in fraction notation: 9/y = x/8 = 5:7. Note that one ratio has no variables or unknowns. Use this ratio to form the following equations: 9/y = 5/7 and x/8 = 5:7. Cross-multiply in the first equation to obtain 5y = 63. Therefore, y = 12.6. Cross-multiply in the second equation to obtain 7x = 40. Therefore, x = 5.7.
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Solve the following word problem using cross-multiplication. A plane travels 2,000 miles in 4.5 hours. How long would it take to travel 3,000 miles at that same rate of speed? For the solution, let x represent the length of time it takes to travel 3,000 miles. Set up the equation 4.5/2,000 = x/3,000. Cross multiply to obtain 2000x = 13,500. Therefore, x = 6.75. The plane takes 6.75 hours to travel 3,000 miles.
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Solve the following word problem by using ratios to form a simple equation. If the ratio of girls to boys in an elementary school is 7:5, how many girls are in the school if the student population is 624 students? Let 7x and 5x represent the number of girls and boys in the school. Set up the equation 7x + 5x = 624. Solve 12x = 624 to obtain x = 52. Multiply 7 by 52 to find out the number of girls in the school. Therefore, there are 364 girls in the school.
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Tips & Warnings
Order matters when writing ratios. A ratio of 3:5 is not the same as a ratio of 5:3.
Express all ratios in lowest terms. Reduce 12:18:36 to 2:3:6.
When teaching ratio and proportion for the first time, help the students build up confidence by using inspection to solve a few easy proportions.
Encourage students to alternate between using fraction and colon notation.
References
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