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**PRESENTATION 9 Ratios and Proportions**

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**RATIOS A ratio is the comparison of two like quantities**

The terms of a ratio must be compared in the order in which they are given Terms must be expressed in the same units The first term is the numerator of a fraction, and the second term is the denominator A ratio should be expressed in lowest fractional terms

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**RATIOS Ratios are expressed in two ways:**

With a colon between the terms, such as 4 : 9 This is read as “4 to 9” With a division sign separating the two numbers, such as 4 ÷ 9 or

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**RATIOS Example: Express 5 to 15 as a ratio in lowest terms**

Write the ratio as a fraction and reduce The ratio is 1 : 3

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**RATIOS Example: Express 10 to as a ratio in lowest terms Divide**

The ratio is 12 : 1

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PROPORTIONS A proportion is an expression that states the equality of two ratios Proportions are expressed in two ways As 3 : 4 = 6 : 8, which is read as “3 is to 4 as 6 is to 8” As , which is the equation form

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**PROPORTIONS A proportion consists of four terms**

The first and fourth terms are called extremes The second and third terms are called means In the proportion 3 : 4 = 6 : 8, 3 and 8 are the extremes and 4 and 6 are the means The product of the means equals the product of the extremes (if the terms are cross-multiplied, their products are equal)

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**PROPORTIONS Example: Solve the proportion below for F:**

Cross multiply: F = 6.2(9.8) Divide both sides by 21.7: Therefore F = 2.8

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DIRECT PROPORTIONS Two quantities are directly proportional if a change in one produces a change in the other in the same direction When setting up a direct proportion in fractional form: Numerator of the first ratio must correspond to the numerator of the second ratio Denominator of the first ratio must correspond to the denominator of the second ratio

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DIRECT PROPORTIONS Example: A machine produces 280 pieces in 3.5 hours. How long does it take to produce 720 pieces? Analyze: An increase in the number of pieces produced (from 280 to 720) requires an increase in time Time increases as production increases; therefore, the proportion is direct

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DIRECT PROPORTIONS Set up the proportion and let t represent the time required to produce 720 pieces The numerator of the first ratio corresponds to the numerator of the second ratio (280 pieces to 3.5 hours) The denominator of the first ratio corresponds to the denominator of the second ratio (720 pieces to t)

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**DIRECT PROPORTIONS Solve for t:**

It will take 9 hours to produce 720 pieces

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INVERSE PROPORTIONS Two quantities are inversely or indirectly proportional if a change in one produces a change in the other in the opposite direction Two quantities are inversely proportional if An increase in one produces a decrease in the other A decrease in one produces an increase in the other

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INVERSE PROPORTIONS When setting up an inverse proportion in fractional form: The numerator of the first ratio must correspond to the denominator of the second ratio The denominator of the first ratio must correspond to the numerator of the second ratio

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INVERSE PROPORTIONS Example: Five identical machines produce the same parts at the same rate. The 5 machines complete the required number of parts in 1.8 hours. How many hours does it take 3 machines to produce the same number of parts? Analyze: A decrease in the number of machines (from 5 to 3) requires an increase in time Time increases as the number of machines decrease and this is an inverse proportion

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INVERSE PROPORTIONS Let x represent the time required by 3 machines to produce the parts The numerator of the first ratio corresponds to the denominator of the second ratio; 5 machines corresponds to 1.8 hours The denominator of the first ratio corresponds to the numerator of the second ratio; 3 machines corresponds to x

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INVERSE PROPORTIONS Solve for x: It will take 3 hours

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PRACTICAL PROBLEMS A piece of lumber 2.8 meters long weighs 24.5 kilograms A piece 0.8 meters long is cut from the 2.8-meter length Determine the weight of the 0.8- meter piece

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PRACTICAL PROBLEMS Analyze: Since the weight of 0.8 meters is less than the total weight of the piece of lumber, this is a direct proportion Set up the proportion and let x represent the weight of the 0.8-meter piece

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PRACTICAL PROBLEMS Solve for x: The piece of lumber weighs 7 kilograms

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