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7-1 Ratios and Proportions 1. PETS Out of a survey of 1000 households, 460 had at least one dog or cat as a pet. What is the ratio of pet owners to households? SOLUTION: The ratio of per owners to households is 23:50. 3. The ratio of the measures of three sides of a triangle is 2:5:4, and its perimeter is 165 units. Find the measure of each side of the triangle. SOLUTION: Just as the ratio or 2:5 is equivalent to or 2x:5x , the extended ratio can be written as 2x:5x:4x. The perimeter is 165 units, so the sum of the lengths of the sides is 165. Solve for x. So the measures of the three sides are 2(15) or 30, 5(15) or 75, and 4(15) or 60. Solve each proportion. 5. SOLUTION: Cross multiply. Solve for x. 48 = 3x x = 16 7. SOLUTION: Cross multiply. eSolutions Manual - Powered by Cognero Solve for x. Page 1 Solve for x. 48 = 3x 7-1 Ratios x = 16and Proportions 7. SOLUTION: Cross multiply. Solve for x. 9. CCSS MODELING Ella is baking apple muffins for the Student Council bake sale. The recipe that she is using calls for 2 eggs per dozen muffins, and she needs to make 108 muffins. How many eggs will she need? SOLUTION: Let the unknown number be x. Form a proportion for the given information. Cross multiply. Solve for x. MOVIES For Exercises 10 and 11, refer to the graphic below. 11. Which film listed had the lowest ratio of awards to nominations? eSolutions Manual - Powered by Cognero SOLUTION: Page 2 Solve for x. 7-1 Ratios and Proportions MOVIES For Exercises 10 and 11, refer to the graphic below. 11. Which film listed had the lowest ratio of awards to nominations? SOLUTION: We can write them in decimal form and then compare their values on the same scale: Movie A has the lowest ratio, that is, 1:2. 13. The ratio of the measures of the three sides of a triangle is 9: 7: 5. Its perimeter is 191.1 inches. Find the measure of each side eSolutions Manual - Powered by Cognero Page 3 SOLUTION: Just as the ratio or 9:7 is equivalent to or 9x:7x , the extended ratio can be written as 9x:7x:5x. 7-1 Ratios Movie and A hasProportions the lowest ratio, that is, 1:2. 13. The ratio of the measures of the three sides of a triangle is 9: 7: 5. Its perimeter is 191.1 inches. Find the measure of each side SOLUTION: Just as the ratio or 9:7 is equivalent to or 9x:7x , the extended ratio can be written as 9x:7x:5x. The perimeter is 191.1 inches so the sum of the lengths of the sides is 191.1. Solve for x. So the measures of the three sides are 9(9.1) or 81.9 in., 7(9.1) or 63.7 in., and 5(9.1) or 45.5 in.. 15. The ratio of the measures of the three sides of a triangle is Its perimeter is 4.75 feet. Find the length of the longest side. SOLUTION: The given ratio is equivalent to The perimeter is 4.75 feet, so the sum of the lengths of the sides is 4.75. Solve for x. So the measures of the three sides are or 2.2 ft, or 1.1 ft, and or 1.47 ft. The length of the longest side is 2.2 ft. Find the measures of the angles of each triangle. 17. The ratio of the measures of the three angles is 3:6:1. SOLUTION: Just as the ratio or 3:6 is equivalent to or 3x:6x , the extended ratio can be written as 3x:6x:1x. We know that sum of the measures of all interior angles in a triangle is 180 degrees. Set the sum of the extended ratios equal to 180 and solve for x. So the measures of the three angles are 3(18) or 54, 6(18) or 108, and 1(18) or 18. eSolutions by Cognero ratio of- Powered the measures of the 19. The Manual SOLUTION: three angles is 10:8:6. Page 4 7-1 Ratios and Proportions So the measures of the three angles are 3(18) or 54, 6(18) or 108, and 1(18) or 18. 19. The ratio of the measures of the three angles is 10:8:6. SOLUTION: Just as the ratio or 10:8 is equivalent to or 10x:8x , the extended ratio can be written as 10x:8x:6x. We know that sum of the measures of all interior angles in a triangle is 180. Set the sum of the extended ratio equal to 180 and solve for x. So the measures of the three angles are 10(7.5) or 75, 8(7.5) or 60, and 6(7.5) or 45. Solve each proportion. 21. SOLUTION: Cross multiply. Solve for y. 23. SOLUTION: Cross multiply. Solve for x. eSolutions Manual - Powered by Cognero 25. Page 5 7-1 Ratios and Proportions 23. SOLUTION: Cross multiply. Solve for x. 25. SOLUTION: Cross multiply. Solve for x. 27. SOLUTION: Cross multiply. Solve for x. eSolutions Manual - Powered by Cognero Page 6 29. NUTRITION According to a recent study, 7 out of every 500 Americans aged 13 to 17 years are vegetarian. In a 7-1 Ratios and Proportions 27. SOLUTION: Cross multiply. Solve for x. 29. NUTRITION According to a recent study, 7 out of every 500 Americans aged 13 to 17 years are vegetarian. In a group of 350 13- to 17-year-olds, about how many would you expect to be vegetarian? SOLUTION: Let the unknown number be x. Form a proportion for the given information. Cross multiply. Solve for x. ALGEBRA Solve each proportion. Round to the nearest tenth. 31. SOLUTION: eSolutions Manual - Powered by Cognero Page 7 Solve for x. 7-1 Ratios and Proportions ALGEBRA Solve each proportion. Round to the nearest tenth. 31. SOLUTION: 33. SOLUTION: Use the quadratic formula. a = 9, b = –114, c = –24 eSolutions Manual - Powered by Cognero Page 8 7-1 Ratios and Proportions 33. SOLUTION: Use the quadratic formula. a = 9, b = –114, c = –24 35. The perimeter of a rectangle is 220 inches. The ratio of its length to its width is 7: 3. Find the area of the rectangle. SOLUTION: The ratio or 5:2 is equivalent to or 7x:3x . The perimeter is 220 feet, so the sum of the lengths of the sides is 220. Solve for x. So the length and width of the rectangle are 7(11) or 77 in. and 3(11) or 33 in. respectively. Thus the area of the rectangle is 2541 square inches. 37. The ratio of the measures of the angles of a quadrilateral is 2:4:6:3. Find the measures of the angles of the eSolutions Manual - Powered by Cognero quadrilateral. SOLUTION: Page 9 7-1 Ratios and Proportions Thus the area of the rectangle is 2541 square inches. 37. The ratio of the measures of the angles of a quadrilateral is 2:4:6:3. Find the measures of the angles of the quadrilateral. SOLUTION: Just as the ratio or 2:4 is equivalent to or 2x:4x , the extended ratio can be written as 2x:4x:6x:3x. The sum of the measures of all interior angles in a quadrilateral is 360.Set the sum of the extended ratio equal to 360 and solve for x. So the measures of the four angles are 2(24) or 48, 4(24) or 96, 6(24) or 144, and 3(24) or 72. eSolutions Manual - Powered by Cognero Page 10