Solving Applications of Proportions

Transcription

1 5.5 Solving Applications of Proportions 5.5 OBJECTIVE. Solve an application that involves a proportion Now that you have learned how to find an unknown value in a proportion, let s see how this can be used in the solution of applications. Step by Step: Solving Applications of Proportions Step Step Step 3 Read the problem carefully to determine the given information. Write the proportion necessary to solve the problem. Use a letter to represent the unknown quantity. Be sure to include the units in writing the proportion. Solve, answer the question of the original problem, and check the proportion as before. Example Using a Proportion to Find an Unknown Value In a shipment of 00 parts, are found to be defective. How many defective parts should be expected in a shipment of 000? Assume that the ratio of defective parts to the total number remains the same. defective 00 total x defective 000 total We have decided to let x be the unknown number of defective parts. Multiply: 00x,000 Divide by the coefficient, 00. x defective parts should be expected in the shipment. Checking the original proportion, we get McGraw-Hill Companies,000,000 CHECK YOURSELF An investment of $3000 earned $330 for year. How much will an investment of $0,000 earn at the same rate for year? Let s look at an application involving fractions in the proportion. 3

2 CHAPTER 5 RATIOS AND PROPORTIONS Example Using Proportions to Find an Unknown Value The scale on a map is given as inch (in.) = 3 miles (mi). The distance between two towns is in. on the map. How far apart are the towns in miles? NOTE We could divide both sides by : x 3 x 3 x then invert and multiply x 8 For this solution we use the fact that the ratio of inches (on the map) to miles remains the same. in. in. 3 mi x mi x 3 x 3 x 3 x 8 (mi) CHECK YOURSELF Jack drives 5 mi in hours (h). At the same rate, how far will he be able to travel in h? (Hint: Write as an improper fraction.) We may also find decimals in the solution of an application. Using Proportions to Find an Unknown Value Jill works. h and receives $. How much will she get if she works 0 h? The ratio of hours worked to the amount of pay remains the same.. h $ Example 3 0 h $a Let a be the unknown amount of pay. 00 McGraw-Hill Companies

3 SOLVING APPLICATIONS OF PROPORTIONS SECTION a 0.a. 0. a $50 Divide both sides by.. CHECK YOURSELF 3 A piece of cable 8.5 centimeters (cm) long weighs 68 grams (g). What will a 0-cm length of the same cable weigh? In Example we must convert the units stated in the problem. Example Using Proportions to Find an Unknown Value A machine can produce 5 tin cans in minutes (min). At this rate how many cans can it make in an 8-h period? In writing a proportion for this problem, we must write the times involved in terms of the same units. 5 cans min x cans 80 min x 5 80 or x 700 x 3600 cans Because h is 60 min, convert 8 h to 80 min. CHECK YOURSELF Instructions on a can of film developer call for ounces (oz) of concentrate to quart (qt) of water. How much of the concentrate is needed to mix with gallon (gal) of water? ( qt gal.) An important use of proportions is in solving problems involving similar geometric figures. These are figures that have the same shape and whose corresponding sides are proportional. For instance, in the similar triangles shown below, 00 McGraw-Hill Companies a proportion involving corresponding sides is

4 6 CHAPTER 5 RATIOS AND PROPORTIONS Example 5 Solving an Application Using Similar Triangles If a 6-foot-tall man casts a shadow that is 0 feet long, how tall is a tree that casts a shadow that is 0 feet long? Let s look at a picture of the two triangles involved. NOTE Connect the top of the tree to the end of the shadow to create a triangle. Connecting the top of the man to the end of his shadow creates a similar triangle. h ft. 6 ft 0 ft. From the similar triangles, we have the proportion 6 0 h 0 Using the proportion rule, we have h 0 h 80 0 h h 8 The tree is 8 feet tall. CHECK YOURSELF 5 If a woman who is 5 feet tall casts a shadow that is 3 feet long, how tall is a building that casts a shadow that is 90 feet long? CHECK YOURSELF ANSWERS. $00. 5 mi 5 h x mi h 5 x 500 x 00 mi g. 8 oz feet tall Divide both sides by 5 00 McGraw-Hill Companies

5 Name 5.5 Exercises Section Date Solve the following applications.. Book purchases. If books are purchased for $0, how much will you pay for 8 books at the same rate?. Construction. If an 8-foot (ft) two-by-four costs 96, what should a -ft two-byfour cost? 3. Consumer affairs. A box of 8 tea bags is marked 90. At that price, what should a box of 8 tea bags cost?. Consumer affairs. Cans of orange juice are marked for 93. What would the price of a case of cans be? 5. Workload. A worker can complete the assembly of 5 tape players in 6 hours (h). At this rate, how many can the worker complete in a 0-h workweek? 6. Consumer affairs. If 3 pounds (lb) of apples cost 90, what will 0 lb cost? ANSWERS Elections. The ratio of yes to no votes in an election was 3 to. How many no votes were cast if there were 880 yes votes?. 8. College enrollment. The ratio of men to women at a college is 7 to 5. How many women students are there if there are 3500 men? 9. Photography. A photograph 5 inches (in.) wide by 6 in. high is to be enlarged so that the new width is 5 in. What will be the height of the enlargement? 0. Shift work. Meg s job is assembling lawn chairs. She can put together 55 chairs in h. At this rate, how many chairs can she assemble in an 8-h shift?. Distance. Christy can travel 0 miles (mi) in her new car on 5 gallons (gal) of gas. How far can she travel on a full tank, which has usable gal? 00 McGraw-Hill Companies. Property taxes. The Changs purchased an $80,000 home, and the property taxes were $00. If they make improvements and the house is now valued at $0,000, what will the new property tax be? 7

6 ANSWERS Distance. A car travels 65 mi in 3 h. How far will it travel in 8 h if it continues at the same speed?. Consumer affairs. A battery pack is on sale at for $3. At this rate, how much will 7 packs cost? 5. Manufacturing. The ratio of teeth on a smaller gear to those on a larger gear is 3 to 7. If the smaller gear has 5 teeth, how many teeth does the larger gear have? 6. Consumer affairs. A store has T-shirts on sale at for $5.50. At this rate, what will five shirts cost? Using the given map, find the distances between the cities named in exercises 7 to 0. L a k e E r i e Erie NEW YORK Bradford 7 6 Warren Allegheny Mansfield Towanda 8 Pymatuning Reservior Reservior Meadville 6 Carbondale OHIO 6 5 Franklin Scranton Oil City Sharon Williamsport Wilkes-Barre 80 Youngstown 80 New Castle Du Bois Milton Hazleton NEW N Butler Punxsutawney 8 JERSEY Kittanning State College Lewisburg Mars Indiana Bethlehem Allentown Alquippa Altoona 78 Pittsburgh Lebanon Reading McKeesport Latrobe Johnstown Pottstown 5 Hershey Washington Carlisle Norristown 70 8 Lancaster Philadelphia Waynesburg FALLINGWATER Uniontown 5 Chambersburg York Kennett Chester Camden Gettysburg Square Wilmington Morgantown Cumberland Hagerstown Vineland Frederick Aberdeen Fairmont Martinsburg MARYLAND Millville WEST VIRGINIA Charles Town Atlantic City Baltimore Clarksburg VA DE queha nna PENNSYLVANIA River Ohio 90 Jamestown MAGELLAN Geographix SM Santa Barbara, CA (800) 99-MAP Olean Harrisburg Elmira Ithaca Sus River Binghamton Railroad 0 0 mi Delaware River Trenton 7. Find the distance from Harrisburg to Philadelphia. 8. Find the distance from Punxsutawney (home of the groundhog) to State College (home of the Nittany Lions). 9. Find the distance from Gettysburg to Meadville. 0. Find the distance from Scranton to Waynesburg.. Manufacturing. An inspection reveals 30 defective parts in a shipment of 500. How many defective parts should be expected in a shipment of 00?. Investments. You invest $000 in a stock that pays a $80 dividend in year. At the same rate, how much will you need to invest to earn $70? 00 McGraw-Hill Companies 8

7 ANSWERS 3. Football. A football back ran yards (yd) in the first two games of the season. If he continues at the same pace, how many yards should he gain in the -game season? Cooking. A 6-lb roast will serve people. What size roast is needed to serve people? Lawn care. A -lb box of grass seed is supposed to cover 500 square feet (ft ) of lawn. How much seed will you need for 8750 ft of lawn? 6. Fencing. A 6-ft fence post casts a 9-ft shadow. How tall is a nearby pole that casts a 5-ft shadow? 7. Lighting. A 9-ft light pole casts a 5-ft shadow. Find the height of a nearby tree that is casting a 0-ft shadow. h ft 8. Construction. On the blueprint of the Wilsons new home, the scale is 5 in. equals 7 ft. What will be the actual length of a bedroom if it measures 0 in. long on the blueprint? 9. Distance. The scale on a map is in. 50 mi. If the distance between two towns on the map is 6 in., how far apart are they in miles? 00 McGraw-Hill Companies 9

8 ANSWERS Science. A metal bar expands in. for each F rise in temperature. How much will it expand if the temperature rises 8 F? 3. Car maintenance. Your car burns quarts (qt) of oil on a trip of 5000 mi. How many quarts should you expect to use when driving 700 mi? 3. Lighting. A 6-ft person casts a 7 -ft shadow. If the shadow of a nearby pole is 30 ft long, how tall is the pole? 33. Manufacturing. A piece of tubing 0.5 centimeters (cm) long weighs 35 grams (g). What is the weight of a piece of the same tubing that is 5 cm long? 3. Salary. Jane works 7.75 h and receives $38.75 pay. What will she receive at the same rate if she works h? 35. Sales tax. The sales tax on an item costing $80 is $5.0. What will the tax be for an item costing $50? 36. Conversion. If 8 kilometers (km) is approximately.8 mi, how many kilometers will equal mi? 37. Timing. You find that your watch gains minutes (min) in 6 h. How much will it gain in 3 days? 38. Painting. If qt of paint will cover 5 ft, how many square feet will gal cover? ( gal qt) 39. Construction. Directions on a box of cups of wallpaper paste are to mix the contents with 5 qt of water. To mix a smaller batch using cup of paste, how much water (in ounces) should be added? ( qt 3 oz) 0. Film processing. A film processing machine can develop three rolls of film every 8 min. At this rate, how many rolls can be developed in a -h period? 00 McGraw-Hill Companies 50

9 ANSWERS. Carpooling. Approximately 7 out of every 0 people in the U.S. workforce drive to work alone. During morning rush hour there are 5,000 cars on the streets of a medium-sized city. How many of these cars have one person in them?. Carpooling. Approximately 5 out of every 00 people in the U.S. workforce carpool to work. There are an estimated 30,000 people in the workforce of a given city. How many of these people are in car pools? Use a proportion to find the unknown side, labeled x, in each of the following pairs of similar figures x x 6 5. x 6. 3 x 8 7. A recipe for servings lists the following ingredients: cups ziti 7 cups spaghetti sauce cups ricotta cheese cup parsley teaspoon garlic powder teaspoon pepper cups mozzarella tablespoons parmesan cheese cheese Determine the amount of ingredients necessary to serve 5 people. 00 McGraw-Hill Companies Answers 8 tea bags 8 tea bags. $60 3. ; 8x 30; x 0, or $.0 90 x 5 gal gal players no votes 9. 8 in.. 0 mi x mi ; 5x 30; x 6 mi 3. 0 mi teeth 7. 0 mi yd x yd 9. 5 mi. 7 defective parts 3. ; x 66 yd games games 5

10 5. 7 lb 7. ft mi 3. 5 Write as 5 x qt 5000 mi 700 mi ; 5000x 8,000; x 3.6 qt g 35. $9.75 min x min 37. (Write 3 days as 7 h) ; x min 6 h 7 h oz. 80,500 cars with one person McGraw-Hill Companies 5

11 Using Your Calculator to Solve Proportions When the numbers involved in a proportion are large, your calculator can be very useful. Consider Example. Example Solving a Proportion for an Unknown Value Solve the proportion for n. n Start the solution as before. 505n n Divide both terms by the coefficient, 505. Find n, using the calculator Display 3 n has the value 3. To check, replace n with 3 and use your calculator to multiply. CHECK YOURSELF Solve the proportion for n. n In practical applications, you may have to round the result after using your calculator in the solution of a proportion. Example shows such a situation. Example Using Proportions to Find an Unknown Value 00 McGraw-Hill Companies Micki drives 78 miles (mi) on 3.6 gallons (gal) of gas. If the gas tank of her car holds gal, how far can she travel on a full tank of gas? We can write the proportion 78 mi x mi 3.6 gal gal Multiply. 3.6 x 78 53

12 5 CHAPTER 5 RATIOS AND PROPORTIONS Now divide both terms by x Now to find x, we must multiply 78 by and then divide by 3.6. On the calculator, Let s round the result to the nearest mile; Micki can drive about 9 mi on a full tank of gas. CHECK YOURSELF Life insurance costs $.37 for each $000 of insurance. How much does a $5,000 policy cost? Your calculator can be very handy for comparing prices at the grocery store. As we said in Section 5., to find the unit price, just divide the cost of the item by the number of units. Example 3 Using Proportions to Find an Unknown Value A dishwashing liquid comes in three sizes: (a) ounces (oz) for 77 (b) oz for $.33 (c) 3 oz for $.85 Which is the best buy? For each size, let s find the unit price in cents per ounce. (a) For the first size (77 for oz), using your calculator, divide oz per ounce We have chosen to round to the nearest tenth of a cent. (b) To find the unit price for the second size, divide again: NOTE Notice that we consider $.33 as 33 to find the ratio cents per ounce. 33 $.33 oz per ounce (c) For the third size, $ per ounce 3 oz Treat $.33 as 33. Again round to the nearest tenth of a cent. 00 McGraw-Hill Companies

13 SOLVING APPLICATIONS OF PROPORTIONS SECTION Comparing the three unit prices, we see that the 3-oz size of dishwashing liquid, at 5.8 per ounce, is the best buy. Note: All the ratios used must be in terms of the same units. If quantities involve different units, they must be converted. CHECK YOURSELF 3 A floor cleaner comes in three sizes: 3 oz for $.89 8 oz for $ oz for $.96 Which is the best buy? Applying Proportions to Find an Unknown Value Vegetable oil is sold in the following quantities: (a) 6 oz for $.7 (b) pint (pt) 8 oz for $.79 (c) quart (qt) 6 oz for $.89 Which is the best buy? (a) (b) Because pt is 6 oz, pt 8 oz is (6 8) oz, or oz. So we write pt 8 oz as oz. $.79 oz (c) Because qt is 3 oz, qt 6 oz is (3 6) oz, or 38 oz. Write qt 6 oz as 38 oz. $ oz Example $.7 6 oz 7.9 per ounce 7.5 per ounce 7.6 per ounce In this case, by comparing the unit prices we see that the -pt 8-oz size is the best buy. CHECK YOURSELF Ketchup is sold in the following quantities: 00 McGraw-Hill Companies oz for $0.68 pt 5 oz for $.05 qt 7 oz for $.89 Which is the best buy? CHECK YOURSELF ANSWERS. 5. $ oz size at 7.09 per oz. qt 7 oz for.8 per oz

14 Name Section ANSWERS Date Calculator Exercises Solve for the unknown a x (to nearest tenth). (to nearest hundredth) 8. n (to nearest tenth) 6. (to nearest hundredth) 0.0 x 3.8 n 0.08 Solve the following applications n 7 7. Salary. Bill earns $8.0 for working 3.5 hours (h). How much will he receive if he works at the same pay rate for 3.75 h? 8. Construction. Construction-grade lumber costs $38.50 per 000 board-feet. What will be the cost of 686 board-feet?. 9. Speed of sound. A speed of 88 feet per second (ft/s) is equal to a speed of 60 miles per hour (mi/h). If the speed of sound is 750 mi/h, what is the speed of sound in feet per second? 0. Manufacturing. A shipment of 75 parts is inspected, and 6 are found to be faulty. At the same rate, how many defective parts should be found in a shipment of 39? Round your result to the nearest whole number.. Taxes. The property tax on a $67,50 home is $35. At the same rate, what will be the tax on a home valued at $87,65? Round your result to the nearest dollar.. Production. A machine produces 58 items in minutes (min). At the same rate, how many items will it produce in 8 h? 00 McGraw-Hill Companies 56

15 ANSWERS 3. Gas mileage. Sally travels 50 mi, using.6 gallons (gal) of gas. How many gallons of gas will she need for a trip of 800 mi? Give your answer to the nearest tenth of a gallon.. Tom and Jerry operate a food concession stand at a local amusement park. They sell the most food when the attendance at the park is at maximum capacity. When this happens, they sell an average of 50 pork roll sandwiches and 550 cheese steak sandwiches. The company that owns the park is going to expand; they will increase the capacity of the park from 6000 to 9000 people next season. Tom and Jerry plan to expand their concession stand so they can sell more sandwiches. (a) Using the same ratio of attendance to sandwiches, how many additional sandwiches of each kind would Tom and Jerry expect to sell? (b) The following costs are associated with the anticipated expansion: Item Construction Supplies Employee costs Cost per Unit $70/square foot $0.65/pork roll sandwich $.5/steak sandwich $8/hour Currently, pork roll sandwiches sell for $.50, and steak sandwiches sell for $.75. Based on these prices and the information in part (a), how would you plan the expansion, and what would you charge for a sandwich? Find the best buy in each of the following exercises. 5. Dishwashing liquid: 6. Canned corn: (a) oz for 79 (a) 0 oz for (b) oz for $.9 (b) 7 oz for McGraw-Hill Companies 7. Syrup: 8. Shampoo: (a) oz for 99 (a) oz for $.6 (b) oz for $.59 (b) 7 oz for $.5 (c) 36 oz for $.9 (c) 5 oz for $ Salad oil ( qt is 3 oz): 0. Tomato juice ( pt is 6 oz): (a) 8 oz for 89 (a) 8 oz for 37 (b) qt for $.39 (b) pt 0 oz for $.9 (c) qt 6 oz for $.9 (c) qt oz for $.99 57

16 ANSWERS... Peanut butter ( lb is 6 oz):. Laundry detergent: (a) oz for $.5 (a) lb oz for $.99 (b) 8 oz for $.7 (b) lb oz for $.89 (c) lb oz for $.59 (c) lb 8 oz for $.9 (d) lb 8 oz for $3.76 (d) 5 lb for $7.99 Answers $ ft/s. $ gal 5. b 7. c 9. b. c 00 McGraw-Hill Companies 58