Overview:
Motion problems are special types of rational expressions that are based on the relationship between distance, rate, and time (d=rt). In order to solve them, it is important to read the story problem, recognize the essential information, and organize the facts in an equation.
Reading the Story Problem
Story problems usually contain all the information that is necessary, and sometimes additional information that is not needed. If the problem involves the speed of something traveling, people or vehicles either going in opposite directions or overtaking each other, and a rate of speed that may change or differ, it will follow the distance = rate X time pattern. Suppose two drivers start out at the same time travelling in opposite directions. One driver was driving 6 miles slower than the other driver. After 10 hours they were 940 miles apart. Find the speed each driver was driving.
Recognizing the Essential Information
In the sample story problem, there are several things that are known. First, the total distance, 940 miles, is known. Second, the time for each driver is known (10 hours). If Driver A is going r miles per hour, Driver B is going r-6 miles per hour. We know the rate is miles per hour rather than feet per second or kilometers per hour, because the time traveled is measured in hours and the distance measured is measured in miles.
Organizing the Data
The data can then be organized in an equation. If d=rt, the value for d, 940 miles, is already known. That is the same value for both Driver A and Driver B. In addition, we also know that the time for both drivers is the same (10 hours). The rate for Driver A is r and the rate for Driver B is r-6. In order to set up the equation so that both Driver A and Driver B are included, the equation would then be 10r (the rate of Driver A) + 10(r-6) (the rate of Driver B) = 940.
Solving and Checking the Equation
The equation, 10r + 10(r-6) = 940, can be expanded to 10r +10r -60 =940. Then like terms can be combined as 20r – 60 = 940. Then 60 can be added to each side so that 20r = 940 + 60 or 20r = 1000. Dividing both sides by 20, r equals 50 and r – 6 = 44. Also, 20(50) – 60 = 940.
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