One characteristic of these work problems is that some job (washing windows in this case, but it doesn't really matter what) takes one person (or combination of people) some length of time, and another combination of people working together some other length of time, and the question is always to say what length of time it takes for yet another combination of people to do the same job.

The interesting thing about all these measurements is that they are given in terms of the time needed for various groups of people to do a job, and you can't add up those times, because they're not rates.  A rate is a unit of work (jobs, say) that are done per unit of time (hour, for example).  Once you express everything in terms of rates, then everything gets a whole lot easier.  If Alex does a job at rate a, and Betty does the job at rate b, then the two of them working together will do the job at rate a+b.

Example 1

A typical "work" problem goes like this:

Henry and Irene working together can wash all the windows of their house in 1 hr 48 minutes.
Working alone, it takes Henry 1 and 1/2 hr more than Irene to do the job.
how long does it take each person working alone to wash all the windows?

Here, we see that the problem is given in terms of hours per job -- Henry and Irene take 1.8 hours to do one job.  The first order of business is to express it in terms of a rate -- jobs per hour, which is just the reciprocal of hours per job.  So the first sentence becomes:

Henry and Irene working together can complete 1/1.8 jobs per hour, which is 5/9 jobs per hour.

Let's assign a variable, x, to Henry's rate.  So then we know Irene's rate is 5/9 - x.

x is Henry's rate -- Henry completes x jobs per hour.
5/9 - x is Irene's rate -- Irene completes 5/9 - x jobs per hour.

Now looking at the second sentence, we can make an equation relating the two rates.  1/x is the time it takes Henry to do a job, and 1/(5/9-x), or 9/(5-9x) is the time it takes Irene to do a job.  1/x is 3/2 longer than 9/(5-9x):

1/x = 3/2 + 9/(5 - 9x)

Multiplying through by the common denominator 2x(5-9x),

2(5-9x) = 3x(5-9x) + 18x
10-18x = 15x - 27x^2 + 18x
27x^2 - 51x + 10 = 0
(9x-2)(3x-5) = 0

Now we see Henry's rate is 2/9 or else it's 5/3.  If it's 2/9, then Irene's rate, 5/9 - x, is 3/9, or 1/3.  But what if Henry's rate is 5/3?  In that case, Irene's rate is 5/9 - 15/9 = -10/9, a negative number, which is impossible (or is it?  more about that later) so there is just one answer:

2/9 is Henry's rate, so he can do a job in 9/2, or 4.5 hours.
1/3 is Irene's rate, so she can do a job in 3 hours.

Now, what about that strange second solution to the quadratic equation?  Henry's rate was an astonishingly swift 5/3, and Irene's rate was -10/9.  First, what does it mean to do a job at a negative rate?  Quite simply, it means the job is undone at that rate.  In other words, Irene dirties the windows of 10/9 houses per hour.  Let's see how this all looks:

5/3 is Henry's rate, so he can do a job in 3/5 hours.
-10/9 is Irene's rate, so she can do a job in -9/10 hours.

In a crazy way, this solution is valid.  It takes Henry 1.5 hours longer than Irene to do a job -- 0.6 hours for Henry, and -0.9 hours for Irene.  And working "together" with Henry washing the windows, and Irene dirtying them, but not quite as fast, each job will eventually get done in 1.8 hours.

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The webmaster and author of the Math Help site is Graeme McRae.
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