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Fractions as Ratios and Proportions

Fractions as Ratios and Proportions. A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group. Ratio. Comparison of two numbers

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Fractions as Ratios and Proportions

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  1. Fractions as Ratios and Proportions A "ratio" is just a comparison between two different things. For instance, someone can look at a group of people, count noses, and refer to the "ratio of men to women" in the group.

  2. Ratio • Comparison of two numbers • Expresses the relative size of two quantities as the quotient of one divided by the other • Written in 3 ways: a:b or a/b or a to b

  3. Example • Suppose there are 240 people in a class, 78 are women and 162 are men. Numerically 78 : 162 78 to 162 78/162 Could reduce to 13/27 or decimal of 0.481481 • Would be the opposite putting the value for men first.

  4. The order in which the ratio is written is important because it defines the comparison • Ratios should be left in their original form to represent the size of the sample compared • In our example • Ratio of women to men is 78 to 162 • Notice that, in the expression "the ratio of women to men", “women" came first. • This order is very important, and must be respected:  whichever word came first, its number must come first. • If the expression had been "the ratio of men to women", then the ratio would have been “162 to 78"

  5. Reducing Ratios Let's return to the 162 men and 78 women in our original group. 162 e had expressed the ratio as a fraction, namely, 15/20. This fraction reduces to 3/4. This means that you can also express the ratio of men to women as 3/4, 3 : 4, or "3 to 4".

  6. However… • This points out something important about ratios: the numbers used in the ratio might not be the absolute references. • The ratio “78 women to 162 men" refers to the absolute numbers of women and men, respectively. • But “13 to 27" just tells you that, for every 13 women, there are 27 men. • This also tells you that, in any representative set of 40 people (13 + 27 = 40) from this group, 13 will be women and 27 men.

  7. Using Ratios to Solve Word Problems In a certain class, the ratio of passing grades to failing grades is 7 to 5. How many of the 36 students failed the course? • The ratio, "7 to 5" (or 7 : 5 or 7/5), tells you that, of every 7 + 5 = 12 students, five failed. • That is, 5/12 of the class flunked.

  8. So in a class of 36 students – 5 X 36 = 180 = 15 12 1 12 = 15 students failed.

  9. Units in Ratios • Ratios may or may not have units – it depends on what you are comparing • In some cases units may cancel out Express the ratio in simplest form: $10 to $45 This means that you should write the ratio as a fraction, and you should then reduce the fraction: 10/45 = 2/9 Note that the units "canceled" on the fraction, since the units, "$", were the same on both values. So there is no unit on the answer

  10. Ratios and Units • Express the ratio in simplest form: 240 miles to 8 gallons • In this case, you would have (240 miles)/(8 gallons) = (30 miles)/(1 gallon) In more common language, 30 miles per gallon. • Properly, this answer should have units on it, since the units, "miles" and "gallons", do not cancel out.

  11. 11:19 Write two equivalent ratios for each ratio 3 17 54 to 24

  12. Write each ratio in simplest form. 32:20 15:33 149 21 48

  13. Ratios are said to be in proportion when their corresponding fractions are equal 78/162 = 13/27 OR 78:162 = 13:27

  14. A statement that two ratios are equal. A comparison of one fraction to another For example: What is a Proportion? • = X • 162 193

  15. Solve the Problem • Cross Multiply and set up an equation • women = X women • 162 men 193 men (78) (193) = (162) X 15054 = 162 X 15054 = X 162 X = 92.9259 women X = 93 women

  16. Check your answer to see if the equations are equal 78 = 92.93 162 193 78/162 = 0.48 92.93/193 = 0.48 The Proportion is true if the both fractions reduce to the same value.

  17. Check your answer to see if the equations are equal 78 X 193 = 15054 92.9259 X 162 = 15054 15054 15054 • = 92.9259 • 162 193 = 1

  18. State whether the ratios are proportional. yes or no • = 2 7 28 2 = 6 11 33 7 = 30 10 21 40 = 4 50 5

  19. Practice • If 18 plums weigh 54 ounces, then 27 plums weigh _____ ounces. • If 40 nails hold 5 rafters, then 96 nails hold ______ rafters. • If 60 sliced mushrooms are on 4 pizzas, them ______ sliced mushrooms are on 15 pizzas.

  20. Making Pancakes This many? OR This many??

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