Stephanie M. answered • 07/07/15
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If 4 houses take 3 men 9 days to paint, then 4 houses take a single man 27 days to paint.
Thus, a single man paints 4/27 of a house per day.
Thus, x men paint (4/27)x of a house per day.
Thus, x men paint 3(4/27)x = (4/9)x of a house in 3 days.
If 4 houses take 8 women 9 days, then 4 houses take a single woman 72 days.
Thus, a single woman paints 4/72 = 1/18 of a house per day.
Thus, y women paint (1/18)y of a house per day.
Thus, y women paint 3(1/18)y = (1/6)y of a house in 3 days.
Working together, then, x men and y women paint (4/9)x + (1/6)y of a house in 3 days. We'd like that quantity to equal 6, since there are 6 houses being painted. So:
(4/9)x + (1/6)y = 6
Now, we'll have to do a bit of guessing and checking. We need both x and y to be whole numbers, since you can't have fractional men or women. If the answer is (a), then there are a total of x + y = 15 people. Solve that system of equations for x and y:
(4/9)x + (1/6)y = 6
x + y = 15
Solve the second equation for x, then plug that into the first equation to solve for y:
x + y = 15
x = 15 - y
(4/9)x + (1/6)y = 6
(4/9)(15 - y) + (1/6)y = 6
20/3 - (4/9)y + (1/6)y = 6
(-5/18)y = -2/3
y = 2.4 women
That can't be true, so let's try x + y = 16:
(4/9)(16 - y) + (1/6)y = 6
64/9 - (4/9)y + (1/6)y = 6
(-5/18)y = -10/9
y = 4 women
With 16 people total, that means the builder could have 4 women and 12 men paint 6 houses in exactly 3 days. This is the smallest of the given numbers that will work, though others will work as well (check 20 women and 6 men for a total of 26 members, for example).
