114 CHAPTER 1 Equations and Inequalities EXAMPLE 2 Solving a Motion Problem Maria and Eduardo are traveling to a business conference. The trip takes 2 hr for Maria and 2.5 hr for Eduardo because he lives 40 mi farther away. Eduardo travels 5 mph faster than Maria. Find their average rates. SOLUTION Step 1 Read the problem. We must find Maria’s and Eduardo’s average rates. Step 2 Assign a variable. Because average rates are to be found, we let the variable represent one of these rates. Let x = Maria’s rate. Because Eduardo travels 5 mph faster than Maria, we can express his average rate using the same variable. Then x + 5 = Eduardo’s rate. Make a table. The expressions in the last column were found by multiplying the corresponding rates and times. Step3 Write an equation. Eduardo’s distance traveled exceeds Maria’s distance by 40 mi. Translate this into an equation. r t d Maria x 2 2x Eduardo x + 5 2.5 2.51x + 52 Summarize the given information in a table. Use d = rt. Eduardo’s 40 more distance is than Maria’s. (11)11* 5 (1111)1111* 2.51x + 52 = 2x + 40 Step 4 Solve. 2.5x + 12.5 = 2x + 40 Distributive property 0.5x = 27.5 Subtract 2x and 12.5. x = 55 Divide by 0.5. Step 5 State the answer. Maria’s rate of travel is 55 mph, and Eduardo’s rate is 55 + 5 = 60 mph. Step 6 Check. The conditions of the problem are satisfied, as shown below. Distance traveled by Maria: 21552 = 110 mi 150 - 110 = 40, as required. Distance traveled by Eduardo: 2.51602 = 150 mi S Now Try Exercise 19. Mixture Problems Problems involving mixtures of two types of the same substance (salt solution, candy, and so on) often involve percentages. PROBLEM-SOLVING HINT In mixture problems involving solutions, rate (percent) amount of pure of concentration # quantity = substance present. The concentration of the final mixture must be between the concentrations of the two solutions making up the mixture. George Polya (1887–1985) Polya, a native of Budapest, Hungary, wrote more than 250 papers and a number of books. He proposed a general outline for solving applied problems in his classic book How to Solve It.
RkJQdWJsaXNoZXIy NjM5ODQ=