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7-1 Ratios and Proportions You solved problems by writing and solving equations. β’ Write ratios. β’ Write and solve proportions. Ratio β’ A ratio is a comparison of two quantities using division. β’ The ratio of quantities π and π can be expresses π as π to π, π: π, or where π β 0. π β’ Ratios are usually expressed in simplest form. β’ 32:18 = 16:9 π€πππ‘β ππ π πππππ βπππβπ‘ ππ π πππππ = 32 ππ 18 ππ = 32÷2 18÷2 = 16 9 SCHOOL The number of students who participate in sports programs at Central High School is 520. The total number of students in the school is 1850. Find the athlete-to-student ratio to the nearest tenth. To find this ratio, divide the number of athletes by the total number of students. 0.3 can be written as Answer: The athlete-to-student ratio is 0.3. The country with the longest school year is China, with 251 days. Find the ratio of school days to total days in a year for China to the nearest tenth. (Use 365 as the number of days in a year.) A. 0.3 B. 0.5 C. 0.7 D. 0.8 Extended Ratios Extended ratios can be used to compare three or more quantities. The expression π: π: π means β’ the ratio of the first two quantities is π: π, β’ the ratio of the last two quantities is π: π, β’ the ratio of the first and last quantities is π: π. In ΞEFG, the ratio of the measures of the angles is 5:12:13. Find the measures of the angles. 5 or 5:12 is equivalent to ______ 5x Just as the ratio___ 12x 12 or 5x:12x, the extended ratio 5:12:13 can be written as 5x:12x:13x. Write and solve an equation to find the value of x. 5x + 12x + 13x = 180 30x = 180 x =6 Triangle Sum Theorem Combine like terms. Divide each side by 30. Answer: So, the measures of the angles are 5(6) or 30, 12(6) or 72, and 13(6) or 78. The ratios of the angles in ΞABC is 3:5:7. Find the measure of the angles. A. 30, 50, 70 B. 36, 60, 84 C. 45, 60, 75 D. 54, 90, 126 Definition β’ Proportion--an equation stating that two ratios are equal. 8 2 ο½ 12 3 300mi x mi ο½ 10gal 1gal Properties of Proportions Proportion β an equation stating that two ratios are equal. π π In the proportion = , π π the numbers π and π are called the extremes of the proportion. the numbers π and π are called the means of the proportion. π π mean = extreme mean π π Cross products - the products of the extremes ππ and the product of the means ππ extreme Page 462 A. Original proportion Cross Products Property Multiply. Divide each side by 6. y = 27.3 Answer: y = 27.3 B. Original proportion Cross Products Property Simplify. Add 30 to each side. Divide each side by 24. Answer: x = β2 A. A. b = 0.65 B. b = 4.5 C. b = β14.5 D. b = 147 Use Proportions to Make Predictions PETS Monique randomly surveyed 30 students from her class and found that 18 had a dog or a cat for a pet. If there are 870 students in Moniqueβs school, predict the total number of students with a dog or a cat. Write and solve a proportion that compares the number of students who have a pet to the number of students in the school. 18 β 870 = 30x 15,660 = 30x β Students who have a pet β total number of students Cross Products Property Simplify. 522 = x Divide each side by 30. Answer: Based on Monique's survey, about 522 students at her school have a dog or a cat for a pet. Brittany randomly surveyed 50 students and found that 20 had a part-time job. If there are 810 students in Brittany's school, predict the total number of students with a part-time job. A. 324 students B. 344 students C. 405 students D. 486 students Page 463 When setting up proportions, the trick is to have all of the same information in the numerators and all of the same information in the denominators. miles gallon or gallons mile 7-1 Assignment Page 464, 10-28 even, 29-30 Show your workβ¦