Applications of Algebraic Fractions

Transcription

1 5.6 Applications of Algebraic Fractions 5.6 OBJECTIVES. Solve a word problem that leads to a fractional equation 2. Apply proportions to the solution of a word problem Many word problems will lead to fractional equations that must be solved by using the methods of the previous section. The five steps in solving word problems are, of course, the same as you saw earlier. Example Solving a Numerical Application If one-third of a number is added to three-fourths of that same number, the sum is 26. Find the number. Step Step 3 Read the problem carefully. You want to find the unknown number. Choose a letter to represent the unknown. Let x be the unknown number. Form an equation. NOTE The equation expresses the relationship between the two parts of the number. 3 x 3 x 26 4 One-third of number Three-fourths of number Step 4 Solve the equation. Multiply each side (every term) of the equation by 2, the LCD. 2 3 x 2 3 x Simplifying yields 4x 9x 32 3x 32 x McGraw-Hill Companies NOTE Be sure to answer the question raised in the problem. The number is 24. Step 5 Check your solution by returning to the original problem. If the number is 24, we have and the solution is verified. 447

2 448 CHAPTER 5 ALGEBRAIC FRACTIONS CHECK YOURSELF The sum of two-fifths of a number and one-half of that number is 8. Find the number. Number problems that involve reciprocals can be solved by using fractional equations. Example 2 illustrates this approach. Example 2 Solving a Numerical Application 3 One number is twice another number. If the sum of their reciprocals is, what are the two numbers? 0 Step You want to find the two numbers. Let x be one number. Then 2x is the other number. Step 3 Twice the first NOTE The reciprocal of a fraction is the fraction obtained by switching the numerator and denominator. x The reciprocal of the first number, x 2x 3 0 The reciprocal of the second number, 2x Step 4 The LCD of the fractions is 0x, and so we multiply by 0x. NOTE x was one number, and 2x was the other. 0x x 0x 2x 0x 0 3 Simplifying, we have 0 5 3x 5 3x 5 x The numbers are 5 and 0. Step 5 Again check the result by returning to the original problem. If the numbers are 5 and 0, we have The sum of the reciprocals is. 0 CHECK YOURSELF 2 2 One number is 3 times another. If the sum of their reciprocals is, find the two numbers McGraw-Hill Companies

3 APPLICATIONS OF ALGEBRAIC FRACTIONS SECTION The solution of many motion problems will also involve fractional equations. Remember that the key equation for solving all motion problems relates the distance traveled, the speed or rate, and the time: Definitions: Motion Problem Relationships d r t Often we will use this equation in different forms by solving for r or for t. So r d t or t d r Example 3 Solving an Application Involving r d t Vince took 2 hours (h) longer to drive 225 miles (mi) than he did on a trip of 35 mi. If his speed was the same both times, how long did each trip take? 225 miles 35 miles Step You want to find the times taken for the 225-mi trip and for the 35-mi trip. NOTE It is often helpful to choose your variable to suggest the unknown quantity here t for time. 2 h longer Let t be the time for the 35-mi trip (in hours). Then t 2 is the time for the 225-mi trip. It is often helpful to arrange the information in tabular form such as that shown. 200 McGraw-Hill Companies NOTE Remember that rate is distance divided by time. The rightmost column is formed by using that relationship. 35-mi trip 35 t Distance Time Rate 225-mi trip 225 t 2 35 t 225 t 2 Step 3 In forming the equation, remember that the speed (or rate) for each trip was the same. That is the key idea. We can equate the rates for the two trips that were found in step 2.

4 450 CHAPTER 5 ALGEBRAIC FRACTIONS The two rates are shown in the rightmost column of the table. Thus we can write 35 t 225 t 2 NOTE Notice that the equation is in proportion form. So we could solve by setting the product of the means equal to the product of the extremes. Step 4 To solve the above equation, multiply each side by t(t 2), the LCD of the fractions. t(t 2) 35 t t(t 2) 225 t 2 Simplifying, we have 35(t 2) 225t 35t t t t 3h The time for the 35-mi trip was 3 h, and the time for the 225-mi trip was 5 h. We ll leave the check to you. CHECK YOURSELF 3 Cynthia took 2 h longer to bicycle 75 mi than she did on a trip of 45 mi. If her speed was the same each time, find the time for each trip. Example 4 uses the d r t relationship to find the speed. Example 4 Solving an Application Involving d r t A train makes a trip of 300 mi in the same time that a bus can travel 250 mi. If the speed of the train is 0 mi/h faster than the speed of the bus, find the speed of each. Step You want to find the speeds of the train and of the bus. Let r be the speed (or rate) of the bus (in miles per hour). NOTE Remember that time is distance divided by rate. Here the rightmost column is found by using that relationship. Then r 0 is the rate of the train. 0 mi/h faster Again let s form a table of the information. Distance Rate Time Train 300 r 0 Bus 250 r 300 r r 200 McGraw-Hill Companies

5 APPLICATIONS OF ALGEBRAIC FRACTIONS SECTION Step 3 To form an equation, remember that the times for the train and bus are the same. We can equate the expressions for time found in step 2. Again, working from the rightmost column, we have 250 r 300 r 0 Step 4 We multiply each term by r(r 0), the LCD of the fractions. 250 r (r 0) r r(r 0) 300 r 0 Simplifying, we have NOTE Remember to find the rates of both vehicles. 250(r 0) 300r 250r r r r 50 mi/h The rate of the bus is 50 mi/h, and the rate of the train is 60 mi/h. You can check this result. CHECK YOURSELF 4 A car makes a trip of 280 mi in the same time that a truck travels 245 mi. If the speed of the truck is 5 mi/h slower than that of the car, find the speed of each. The next example involves fractions in decimal form. Mixture problems often use percentages, and those percentages can be written as decimals. Example 5 illustrates this method. Example 5 Solving an Application Involving Solutions A solution of antifreeze is 20% alcohol. How much pure alcohol must be added to 2 quarts (qt) of the solution to make a 40% solution? Step You want to find the number of quarts of pure alcohol that must be added. Let x be the number of quarts of pure alcohol to be added. 200 McGraw-Hill Companies Step 3 To form our equation, note that the amount of alcohol present before mixing must be the same as the amount in the combined solution. A picture will help. 2 qt 20% x qt 00% 2 x qt 40%

6 452 CHAPTER 5 ALGEBRAIC FRACTIONS NOTE Express the percentages as decimals in the equation. So 2(0.20) x(.00) (2 x)(0.40) The amount of The amount of The amount of alcohol in the pure alcohol alcohol in the first solution ( pure is 00%, mixture (20% is 0.20) or.00) Step 4 Most students prefer to clear the decimals at this stage. It s easy here multiplying by 00 will move the decimal point two places to the right. We then have 2(20) x(00) (2 x)(40) x x 60x 240 x 4 qt CHECK YOURSELF 5 How much pure alcohol must be added to 500 cubic centimeters (cm 3 ) of a 40% alcohol mixture to make a solution that is 80% alcohol? There are many types of applications that lead to proportions in their solution. Typically these applications will involve a common ratio, such as miles to gallons or miles to hours, and they can be solved with three basic steps. Step by Step: To Solve an Application by Using Proportions Step Step 3 Assign a variable to represent the unknown quantity. Write a proportion, using the known and unknown quantities. Be sure each ratio involves the same units. Solve the proportion written in step 2 for the unknown quantity. Example 6 illustrates this approach. Example 6 Solving an Application Using Proportions A car uses 3 gallons (gal) of gas to travel 05 miles (mi). At that mileage rate, how many gallons will be used on a trip of 385 mi? Step Assign a variable to represent the unknown quantity. Let x be the number of gallons of gas that will be used on the 385-mi trip. Miles Gallons Write a proportion. Note that the ratio of miles to gallons must stay the same x Miles Gallons 200 McGraw-Hill Companies

7 APPLICATIONS OF ALGEBRAIC FRACTIONS SECTION Step 3 means. Solve the proportion. The product of the extremes is equal to the product of the NOTE To verify your solution, return to the original problem and check that the two ratios are equivalent. 05x x 55 05x x gal So gal of gas will be used for the 385-mi trip. CHECK YOURSELF 6 A car uses 8 liters (L) of gasoline in traveling 00 kilometers (km). At that rate, how many liters of gas will be used on a trip of 250 km? Proportions can also be used to solve problems in which a quantity is divided by using a specific ratio. Example 7 shows how. Example 7 Solving an Application Using Proportions A piece of wire 60 inches (in.) long is to be cut into two pieces whose lengths have the ratio 5 to 7. Find the length of each piece. Step Let x represent the length of the shorter piece. Then 60 x is the length of the longer piece. NOTE A picture of the problem always helps. Shorter x Longer 60 x McGraw-Hill Companies NOTE On the left and right, we have the ratio of the length of the shorter piece to that of the longer piece. x 60 x 5 7 Step 3 5 The two pieces have the ratio, so 7 Solving as before, we get 7x (60 x)5 7x 300 5x 2x 300 x 25 (Shorter piece) 60 x 35 (Longer piece) The pieces have lengths 25 in. and 35 in.

8 454 CHAPTER 5 ALGEBRAIC FRACTIONS CHECK YOURSELF 7 A board 2 feet (ft) long is to be cut into two pieces so that the ratio of their lengths is 3 to 4. Find the lengths of the two pieces. CHECK YOURSELF ANSWERS. The number is The numbers are 6 and mi trip: 5 h; 45-mi trip: 3 h 4. Car: 40 mi/h; truck: 35 mi/h cm L 7. 9 ft; 2 ft 200 McGraw-Hill Companies

9 Name 5.6 Exercises Section Date Solve the following word problems.. Adding numbers. If two-thirds of a number is added to one-half of that number, the sum is 35. Find the number. 2. Subtracting numbers. If one-third of a number is subtracted from three-fourths of that number, the difference is 5. What is the number? 3. Subtracting numbers. If one-fourth of a number is subtracted from two-fifths of a number, the difference is 3. Find the number. 4. Adding numbers. If five-sixths of a number is added to one-fifth of the number, the sum is 3. What is the number? 5. Consecutive integers. If one-third of an integer is added to one-half of the next consecutive integer, the sum is 3. What are the two integers? 6. Consecutive integers. If one-half of one integer is subtracted from three-fifths of the next consecutive integer, the difference is 3. What are the two integers? 7. Reciprocals. One number is twice another number. If the sum of their reciprocals is, find the two numbers Reciprocals. One number is 3 times another. If the sum of their reciprocals is, find the two numbers Reciprocals. One number is 4 times another. If the sum of their reciprocals is, find the two numbers Reciprocals. One number is 3 times another. If the sum of their reciprocals is, what are the two numbers? 5 ANSWERS Reciprocals. One number is 5 times another number. If the sum of their reciprocals 6 is, what are the two numbers? Reciprocals. One number is 4 times another. The sum of their reciprocals is. What are the two numbers? McGraw-Hill Companies 3. Reciprocals. If the reciprocal of 5 times a number is subtracted from the reciprocal 4 of that number, the result is. What is the number? Reciprocals. If the reciprocal of a number is added to 4 times the reciprocal of that 5 number, the result is. Find the number Driving rate. Lee can ride his bicycle 50 miles (mi) in the same time it takes him to drive 25 mi. If his driving rate is 30 mi/h faster than his rate bicycling, find each rate. 455

10 ANSWERS Running rate. Tina can run 2 mi in the same time it takes her to bicycle 72 mi. If her bicycling rate is 20 mi/h faster than her running rate, find each rate Driving rate. An express bus can travel 275 mi in the same time that it takes a local bus to travel 225 mi. If the rate of the express bus is 0 mi/h faster than that of the local bus, find the rate for each bus. 8. Flying time. A light plane took hour (h) longer to travel 450 mi on the first portion of a trip than it took to fly 300 mi on the second. If the speed was the same for each portion, what was the flying time for each part of the trip? 9. Train speed. A passenger train can travel 325 mi in the same time a freight train takes to travel 200 mi. If the speed of the passenger train is 25 mi/h faster than the speed of the freight, find the speed of each. 20. Flying time. A small business jet took h longer to fly 80 mi on the first part of a flight than to fly 540 mi on the second portion. If the jet s rate was the same for each leg of the flight, what was the flying time for each leg? 2. Driving time. Charles took 2 h longer to drive 240 mi on the first day of a vacation trip than to drive 44 mi on the second day. If his average driving rate was the same on both days, what was his driving time for each of the days? 22. Driving time. Ariana took 2 h longer to drive 360 mi on the first day of a trip than she took to drive 270 mi on the second day. If her speed was the same on both days, what was the driving time each day? 23. Flying time. An airplane took 3 h longer to fly 200 mi than it took for a flight of 480 mi. If the plane s rate was the same on each trip, what was the time of each flight? 24. Traveling time. A train travels 80 mi in the same time that a light plane can travel 280 mi. If the speed of the plane is 00 mi/h faster than that of the train, find each of the rates. 200 McGraw-Hill Companies 456

11 ANSWERS 25. Canoeing time. Jan and Tariq took a canoeing trip, traveling 6 mi upstream against a 2 mi/h current. They then returned to the same point downstream. If their entire trip took 4 h, how fast can they paddle in still water? [Hint: If r is their rate (in miles per hour) in still water, their rate upstream is r 2 and their rate downstream is r 2.]. 26. Flying speed. A plane flies 720 mi against a steady 30 mi/h headwind and then returns to the same point with the wind. If the entire trip takes 0 h, what is the plane s speed in still air? 27. Alcohol solution. How much pure alcohol must be added to 40 ounces (oz) of a 25% solution to produce a mixture that is 40% alcohol? 28. Mixtures. How many centiliters (cl) of pure acid must be added to 200 cl of a 40% acid solution to produce a 50% solution? 29. Speed conversion. A speed of 60 miles per hour (mi/h) corresponds to 88 feet per second (ft/s). If a light plane s speed is 50 mi/h, what is its speed in feet per second? 30. Cost. If 342 cups of coffee can be made from 9 pounds (lb) of coffee, how many cups can be made from 6 lb of coffee? Fuel consumption. A car uses 5 gallons (gal) of gasoline on a trip of 60 mi. At the same mileage rate, how much gasoline will a 384-mi trip require? 32. Fuel consumption. A car uses 2 liters (L) of gasoline in traveling 50 kilometers (km). At that rate, how many liters of gasoline will be used in a trip of 400 km? 200 McGraw-Hill Companies 33. Yearly earnings. Sveta earns $3,500 commission in 20 weeks in her new sales position. At that rate, how much will she earn in year (52 weeks)? 34. Investment earning. Kevin earned $65 interest for year on an investment of $500. At the same rate, what amount of interest would be earned by an investment of $2500? 35. Insect control. A company is selling a natural insect control that mixes ladybug beetles and praying mantises in the ratio of 7 to 4. If there are a total of 0 insects per package, how many of each type of insect is in a package? 457

12 ANSWERS Individual height. A woman casts a shadow of 4 ft. At the same time, a 72-ft building casts a shadow of 48 ft. How tall is the woman? Consumer affairs. A brother and sister are to divide an inheritance of $2,000 in the ratio of 2 to 3. What amount will each receive? 38. Taxes. In Bucks County, the property tax rate is $25.32 per $000 of assessed value. If a house and property have a value of $28,000, find the tax the owner will have to pay. Answers , , , 2. 7, mi/h bicycling, 50 mi/h driving 7. Express 55 mi/h, local 45 mi/h 9. Freight 40 mi/h, passenger 65 mi/h 2. 5 h, 3 h h, 2 h mi/h oz ft/s 3. 2 gal 33. $35, ladybugs, 40 praying mantises 37. Brother $4800, sister $ McGraw-Hill Companies 458