# Math

Bryan and Kim are painting a house. Working together they can paint a house in three-fifths the time it takes Kim to paint the house working alone. Bryan takes 15 hours to paint the house alone. How long does it take Kim to paint the house working alone?

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1. let Kim's time working alone be x hrs

then Kim's rate = job/x
Brian's rate = job/15
combined rate = job/x + job/15 = job(x+15)/(15x)

time taken working together = job รท [job(x+15)/(15x)]
= 15x/(x+15) hrs

but 15x/(x+15) = (3/5)x
3x^2 + 45x = 75x
3x^2 - 30x = 0
3x(x - 10) = 0
x = 0 or x = 10, but clearly x > 0
so x = 10

It would take kim 10 hours to work alone.

check: combined rate = job(1/10 + 1/15) = job(1/6)
so the combined time = 6/1 = 6 hours
what is 3/5 of 10 ?
yes, 6 hours.

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