What are the steps to solve the following problem? I know the answer, but not how you get it.?

Question

Nine large pipes will drain a pond in 8 hrs and 6 small pipes will drain the same pond in 16 hrs. How long will it take 3 large pipes and 5 small pipes to drain the pond???

Answer 1

suppose your pool is G gallons.

9 large pipes drain it in 8 hours
so 9 large pipes put G gallons of water through them in 8 hours
so 1 large pipe drains G/9 gallons in 8 hours
so 1 large pipe drains G/72 gallons an hour

6 small pipes, 16 hours, G gallons, so (similarly)
1 small pipe drain G/96 gallons an hour

so now you have 3 large ones and 5 small ones. Together, they drain

3 * G / 72 + 5 * G / 96 = 4G/96 + 5G/96 = 9G/96 = 3G/32 = (3/32) * G gallons an hour.

How long will it take to drain G gallons? Let's call this time X.

X = G / ((3/32) * G) = 1 / (3/32) = 32/3 = 10.6666... hours.
cheers!

Answer 2

Let R be the rate of drainage by a large pipe, and r be the rate of drainage by a small one. Let t be the unknown time it takes for 3 large and 5 small pipes to drain the pond. We have, since the volume the pond is the same in all 3 cases:
R*9*8 = r*6*16 = R*3*t + r*5*t
We solve for t in this equation:
R*9*8 = R*3*t + r*5*t
t = R*9*8 / (R*3+ r*5)
But we also solve for r in this equation:
R*9*8 = r*6*16
r = R*(9*8) / (6*16)
So, we plug this into the other equation, and we have:
t = R*9*8 / (R*3+ (R*(9*8*5) / (6*16)))
R drops out of this equation, and so we work out the numbers:
t = (9*8) / (3+ (9*8*5) / (6*16)) = 32/3 hours

Answer 3

9/8=3/x

6/16=5/y

you can solve these two proportions to find x and y.