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Jack can paint a house in 5 days, and Richard can paint the same house in 7 days. Working together, how long will it take them to finish the job?

In 7*5 = 35 days, Jack can paint 7 houses.

In 5*7 = 35 days, Richard can paint 5 houses.

So in 35 days, the two of them can paint 12 houses. To paint just one house, they'll need 1/12 the time, or 35/12 = 2 11/12 days.

There's a general formula for solving this kind of problem.

If we let the time it takes the first person to do a job equal a, and the time it takes the second person to do the job equal b, the following formula will find the time it will take them to do the job together:

    a * b    <--- time to do (a+b) jobs
    a + b    <-------------- (a+b) jobs
    5 * 7     35
    -----  =  --  =  2 11/12
    5 + 7     12

But wait - isn't there a common way of explaining these problems that involves adding fractions?

Here's a similar problem:

There are two small holes in a tub filled with water. If opened, one hole will empty the tub in 3 hours; the other will empty it in 6 hours. If both holes are opened at the same time, how long will it take to empty the tub?

One hole will empty the tub in 3 hours; so in 6*3 hours, it would empty 6 tubs.

The other hole will empty the tub in 6 hours, so in 3*6 hours it would empty 3 tubs.

Together, in 6*3 hours, the two holes would empty 6+3 tubs. To empty one tub would take 1/9 of that time, or 18/9 = 2 hours.

    a * b    <--- time to empty (a+b) tubs
    a + b    <----------------- (a+b) tubs
    3 * 6     18
    -----  =  --  =  2 
    3 + 6      9

Notice that one rate is a multiple of the other. The faster hole is like two of the slower ones, so the whole thing is like having three holes, any one of which would empty the tub in 6 hours. Three times more flow means three times less time, or 2 hours.

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