Answer: If the a and b of your formula mean how many hours it takes each
person, like a = 2 and b = 1, then the formula gives 3/2 or 1 hour and
30 minutes. This is not reasonable, since one of the painters alone
could do it in 1 hour.
Actually, your formula is UPSIDE DOWN. It should be (a*b)/(a+b).
Here's why.
Let h be the number of hours it takes for the two of them to do it
together. The first person could do it all in a hours, but can paint
only a fraction of the wall working h hours. Assuming a constant rate
of painting, that fraction is h/a. That is, the first painter will
paint h/a of the wall in h hours.
Similarly, the second painter will paint h/b of the wall in that time.
So, when will they be done? When the whole wall is painted, which
happens whenever those 2 fractions add up to 1.
You need to solve the following equation:
(h/a) + (h/b) = 1
h*b + h*a = a*b Multiplying through by a*b
h*(b+a) = a*b Factoring out h
h = (a*b)/(a+b) Dividing both sides by b+a
Okay, now say 3 painters could do it in a hours, b hours or c hours,
respectively. Let h be the number of hours they all must work to get
it all painted. The first painter finishes h/a of the wall in that
amount of time. The other 2 painters manage to paint h/b of the wall
and h/c of the wall in that time. Since h is the time to finish the
job, all 3 of these fractions must add up to the whole wall, or 1.
So you have the equation:
h h h
--- + --- + --- = 1
a b c
Solve this for h (start by multiplying through by a*b*c) and you will
get the generalized formula for 3 painters. It's not the formula you
wrote above but I think you can finish it off.
(I hope this helps. I'm sorry if it's wrong)
