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PRESENTATION 9 Ratios and Proportions

Learn the fundamentals of ratios and proportions, with explanations, examples, and practical problems. Understand direct and inverse proportions through step-by-step solutions.

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PRESENTATION 9 Ratios and Proportions

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  1. PRESENTATION 9Ratios and Proportions

  2. 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

  3. 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

  4. 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

  5. RATIOS • Example: Express 10 to as a ratio in lowest terms • Divide • The ratio is 12 : 1

  6. 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

  7. 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)

  8. PROPORTIONS • Example: Solve the proportion below for F: • Cross multiply: 21.7F = 6.2(9.8) • Divide both sides by 21.7: • Therefore F = 2.8

  9. 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

  10. 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

  11. 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)

  12. DIRECT PROPORTIONS • Solve for t: • It will take 9 hours to produce 720 pieces

  13. 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

  14. 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

  15. 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

  16. 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

  17. INVERSE PROPORTIONS • Solve for x: • It will take 3 hours

  18. 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

  19. 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

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

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