Applications of Multiplication of Fractions

Transcription

1 .6 Applications of Multiplication of Fractions.6 OBJECTIVES. Solve applications involving multiplication of fractions. Use appropriate units analysis when solving applications Units Analysis When dividing two denominate numbers, the units are also divided. This yields a unit in fraction form. Examples 0 mi mi 0 mi 0 gal mi/gal (read miles per gallon ) 0 gal gal 60 ft 60 ft 0 s ft/s ( feet per second ) 0 s When we multiply denominate numbers that have these units in fraction form, they behave just like fractions. Examples mi/gal gal mi gal 00 mi gal (If we look at the units, we see that the gallons essentially cancel when one is in the numerator and the other in the denominator) ft ft/s 60 s/min 60 s 70 ft 70 ft/min s min min (again, the seconds cancel, leaving feet in the numerator and minutes in the denominator) Let s look at some applications of our work with the multiplication of fractions. In solving these word problems, we will use the same approach we used earlier with whole numbers. Let s review the four-step process introduced in Section.. 00 McGraw-Hill Companies Step by Step: Step Step Step Step Solving Applications Involving the Multiplication of Fractions Read the problem carefully to determine the given information and what you are asked to find. Decide upon the operation or operations to be used. Write down the complete statement necessary to solve the problem and do the calculations. Check to make sure that you have answered the question of the problem and that your answer seems reasonable. 79

2 80 CHAPTER MULTIPLYING AND DIVIDING FRACTIONS Let s work through some examples, using these steps. Example An Application Involving Multiplication Lisa worked 0 hours per day h for days. How many hours did she work? day Step Step Step Step We are looking for the total hours Lisa worked. We will multiply the hours per day by the days. 0 h day days h 0 days day h h Note the days cancel, leaving only the unit hours. The units should always be compared to the desired units from step. The answer also seems reasonable. An answer like hours or 00 hours would not seem reasonable. CHECK YOURSELF Carlos gets 0 mi/gal in his Miata. How far should he be able to drive with an -gal tank of gas? In the next example, we will follow the four steps for solving applications, but we won t label the steps. You should still think about these steps as we solve the problem. Example An Application Involving the Multiplication of Mixed Numbers A sheet of notepaper is 6 inches (in.) wide by 8 in. long. Find the area of the paper. Multiply the given length by the width. This will give the desired area. First, we will estimate the area 7 in. 9 in. 6 in. Now, we will find the exact area. 8 in. 6 in. 6 in. 7 in. NOTE Recall that the area of a rectangle is the product of its length and its width. 7 in. 8 in. The units (square inches) are units of area. Note from our estimate that the result is reasonable. CHECK YOURSELF A window is feet (ft) high by ft wide. What is its area? 00 McGraw-Hill Companies

3 APPLICATIONS OF MULTIPLICATION OF FRACTIONS SECTION.6 8 The next example reminds us that an abstract number multiplied by a denominate number will yield the units of the denominate number. NOTE The word of indicates multiplication. Example An Application Involving the Multiplication of a Mixed Number and a Fraction A state park contains 8 acres. According to the plan for the park, of the park is to be left as a wildlife preserve. How many acres will this be? We want to find of 8 acres. We then multiply as shown: acres 9 acres CHECK YOURSELF A backyard has square yards (yd ) of open space. If Patrick wants to build a vegetable garden covering of the open space, how many square yards will this be? We have mentioned the word of indicates multiplication. You should also note that it indicates that the fraction preceding it is an abstract number (it has no units attached). There are even occasions, as in the next example, when we are looking at the product of two abstract numbers. Example An Application Involving the Multiplication of Fractions A grocery store survey shows that of the customers will buy meat. Of these, will buy at least one package of beef. What portion of the store s customers will buy beef? Step We know that of the customers will buy meat and that of these customers will buy beef. 00 McGraw-Hill Companies NOTE Remember, in this problem, of means to multiply. Step We wish to know of The operation here is multiplication.. Step Multiplying, we have Step From step we have the result: of the store s customers will buy beef.

4 8 CHAPTER MULTIPLYING AND DIVIDING FRACTIONS CHECK YOURSELF A supermarket survey shows that of the customers will buy lunch meat. Of these, will buy boiled ham. What portion of the store s customers will buy boiled ham? Example An Application Involving the Multiplication of Mixed Numbers Shirley drives at an average speed of miles per hour (mi/h) for How far has she traveled at the end of h. h? NOTE Remember, distance is the product of speed and time. mi h h Speed Time mi h h mi 69 mi CHECK YOURSELF A. The scale on a map is inch (in.) 60 miles (mi). What is the distance in miles between two towns that are in. apart on the map? B. Maria is ordering concrete for a new sidewalk that is to be yd thick, yd long, and 9 yd wide. How much concrete should she order if she must order a whole number of cubic yards? CHECK YOURSELF ANSWERS. 0 mi ft 6 yd 0. a. 0 mi b. The answer, yd, is rounded up to yd. 00 McGraw-Hill Companies

5 Name.6 Exercises Section Date Evaluate the following. Be sure to use the proper units.. 6 mi/h h. 80 cal/g g. joules/s s. lb/ft ft. 88 ft/s mi/80 ft 600 s/h 6. h/day 600 s/h 6 days/yr Solve the following applications. 7. Maria earns $7 per hour. Last week, she worked 9 hours per day for 6 days. What was her gross pay? 8. The gas tank in Luigi s Toyota Camry holds 7 gal when full. The car gets mi gal. How far can he travel on three full tanks? 9. Recipes. A recipe calls for cup of sugar for each serving. How much sugar is needed for six servings? 0. Recipes. Mom-Mom s French toast requires cup of batter for each serving. If five people are expected for breakfast, how much batter is needed? ANSWERS Gardening. A patch of dirt needs square feet (ft ) of sod to cover it. If Nick 6 decides to cover only of the dirt, how much sod does he need?. Construction. A driveway requires cubic yards (yd ) of concrete to cover it. If 6 Sheila wants to enlarge her driveway to times its current size, how much concrete will she need? 00 McGraw-Hill Companies 8

6 ANSWERS.... Map scales. The scale on a map is inch (in.) 00 miles (mi). What actual distance, in miles, does in. represent? Salary. You make $90 a day on a job. What will you receive for working of a day?. Size. A lumberyard has a stack of 80 sheets of plywood. If each sheet is in. thick, how high will the stack be? 6. Family budget. A family uses of its monthly income for housing and utilities on average. If the family s monthly income is $70, what is spent for housing and utilities? What amount remains? 7. Elections. Of the eligible voters in an election, were registered. Of those registered, actually voted. What fraction of those people who were eligible voted? Surveys. A survey has found that of the people in a city own pets. Of those who 0 own pets, have dogs. What fraction of those surveyed own dogs? 9. Area. A kitchen has dimensions yards (yd) by yd. How many square yards (yd ) of linoleum must be bought to cover the floor? 00 McGraw-Hill Companies 8

7 ANSWERS 0. Distance. If you drive at an average speed of miles per hour (mi/h) for how h, far will you travel? 0... Distance. A jet flew at an average speed of 0 mi/h on a -h flight. What was the distance flown? Area. A piece of land that has acres is being subdivided for home lots. It is estimated that of the area will be used for roads. What amount remains to be used 7 for lots?. Circumference. To find the approximate circumference or distance around a circle, we multiply its diameter by What is the circumference of a circle with a diameter 7. of in.? 6. Area. The length of a rectangle is and its width is What is its area in 7 yd, 6 yd. square yards?. Volume. Find the volume of a box that measures by by in. 7 8 in. 6 in. 6. Topsoil. Nico wishes to purchase mulch to cover his garden. The garden measures 77 feet (ft) by 0 He wants the mulch to be ft deep. How much mulch should 8 8 ft. Nico order if he must order a whole number of cubic feet? 00 McGraw-Hill Companies 7 78 ft 0 8 ft 8

8 ANSWERS 7. The formula for the area of a triangle is 8. h A h b 9. b in which h is the height of the triangle and b is the base Find the area of a triangle with a height of in. and a base of in. 8. Find the area of a triangle with a height of 7 and a base of 8 in. in. 9. A recipe calls for the following ingredients: 7 cup of flour, cup of sugar, cup of milk, and teaspoon of salt. This recipe 8 6 makes eight servings. What amount of each quantity would you use if you wanted to serve two people? 0. Obtain a map of your state and, using the legend provided, determine the distance between your state capital and any other city. Would this be the actual distance you would travel by car if you made the journey? Why or why not? Answers. mi. 60 joules. 60 mi/h 7. $78 9. cups mi. 60 in ft yd. 0 mi. 66 in.. 9 in. 7. in McGraw-Hill Companies 86