Another "typical" work problem is the "one guy did part of the job" or "the number of workers changed at some point during the job" type. We'll still need to do the computations for how much each guy does per unit time (usually hours or days), but we may need to use the fact that "a completed task" is represented by "1". Then we'll figure out the number of time units is needed for each combination of workers, in order to end up with one fully-completed task.
A typical exercise of this type might run something like this:
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Since the first mechanic can do the job in 6 hours, then she does of the job per hour. The new kid needs 8 hours for the same task, so he only does of the job per hour. Together, they can do:
per hour
Taking the reciprocal, this means that it would take them hours to complete the job, if they both worked together for the whole job. Putting this info into the usual set-up, I have:
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hours to complete job:
her: 6
him: 8
together:
completed per hour:
her:
him:
together:
But how do I account for the fact that they didn't work together for the whole time? I need to use the fact that they're working toward one completed task, which is represented by 1.
First, they worked together for two hours. Since they can do of the task per hour when they work together, and since they worked together for two hours, then, in those two hours, they completed:
of the job
I calculated this by using the fact that (fraction of work) = (rate per unit time) × (number of units of time). That is, I multiplied how much they can do per hour by the number of hours they worked, to find the fraction of the entire job that they had completed together.
This leaves the rest of the task for the new kid. That is, he needs to complete the remainder of the job, which is:
of the job
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(Note: This is just under half of the job. Since the new kid can do the whole job in eight hours, I should expect an answer of something under four hours for him to complete the remainder of this task.)
The kid can do of the job in an hour. So how many hours will it take him to complete the remaining of the job? Well, how many 's fit into ? To find out, I divide:
...or three and one-third hours. (This is a good match to my expectation; namely, that the time would be somewhat under four hours.) One-third of sixty minutes is twenty minutes, so:
It takes the new kid another three hours and twenty minutes to finish fixing the car.
By the way, this means that they took a total of [two hours with two mechanics] plus [three hours and twenty minutes with one mechanic] equals [four man-hours] plus [three-and-a-third man-hours] equals [seven-and-a-third man-hours] to fix your car. Yes, you may get billed for more when the mechanic is less qualified.
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I'll start by converting their times to unit rates, making sure to convert "hours" to "minutes", so the units match:
minutes to complete job:
Maria: 100
Shaniqua: 120
Liu: t
Inverting, I get their per-minute rates:
completed per minute:
Maria:
Shaniqua:
Liu:
They weren't all working together for the first half hour; only Maria and Shaniqua were working. To figure how much those two completed during that thirty minutes, I'll need to remember that (amount completed per unit of time) times (number of units of time) equals (fraction of work completed). In other words:
Maria and Shaniqua together:
So they do of the job per minute. Since they worked together for thirty minutes:
...they completed of the job. This leaves of the job still to be completed.
Liu joined them for the remainder of the job. She does per minute. Then the total done, per minute, when all three are working together, is:
They worked at this rate for another twenty minutes:
Since they were finished at the end of the twenty minutes, then this is equal to the remainder of the job, which is of the job. Since these both represent "the rest of the job", then these are equal; I can set them equal and create an equation:
Solving, I get:
I chose the variable t to stand for the number of minutes, so this answer represents 240 minutes. There are sixty minutes to an hour, so this is equal to four hours.
Liu takes four hours to complete the task alone.
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