Work Problems
Work problems are another staple of algebra courses. A work problem is normally stated as two workers (two people, machines, hoses, drains, etc.) working together and working separately to complete a task. Often one worker performs faster than the other. Sometimes the problem states how fast each can complete the task alone and you are asked to find how long it takes for them to complete the task together. At other times, you are told how long one worker takes to complete the task alone and how long it takes for both to work together to complete it; you are asked how long the second worker would take to complete the task alone. The formula is quantity (work done—usually ‘‘1’’) ¼ rate times time: Q = rt. The method outlined below will help you solve most, if not all, work problems. The following chart is useful in solving these problems.
There are four equations in this chart. One
of them will be the one you will use to solve for the unknown. Each horizontal
line in the chart represents the equation Q = rt for that particular line. The
fourth equation comes from the sum of each worker’s rate set equal to the
together rate. Often, the fourth equation is the one you will need to solve.
Remember, as in all word problems, that all units of measure must be
consistent.
Examples
Joe takes 45
minutes to mow a lawn. His older brother Jerry takes 30 minutes to mow the
lawn. If they work together, how long will it take for them to mow the lawn?
The quantity in each of the three cases is
1—there is one yard to be mowed. Use the formula Q = rt and the data given in
the problem to fill in all nine boxes. Because we are looking for the time (in
minutes) it takes for them to mow the lawn together, let t represent the number
of minutes needed to mow the lawn together.
Because Q = rt,
r = Q/t. But Q = 1, so r = 1/t. This makes Joe’s rate 1/45 and Jerry’s rate
1/30. The together rate is 1/t.
Of the four
equations on the chart, only ‘‘Joe’s rate + Jerry’s rate = Together rate’’ has
enough information in it to solve for t.
The equation
to solve is 1/45 + 1/30 = 1/t. The LCD is 90t.
They can mow
the yard in 18 minutes.
Tammy can
wash a car in 40 minutes. When working with Jim, they can wash the same car in
15 minutes. How long would Jim need to wash the car by himself?
Let t represent the number of minutes Jim
needs to wash the car alone.
The equation
to solve is 1/40 + 1/t = 1/15. The LCD is 120t.
Jim needs 24
minutes to wash the car alone.
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