Question 896484
let A equal the rate of tap A.
let B equal the rate of tap B.


when they work together their rates are additive.


the formula to use is R*T = Q


Q = 1 which represents 1 full pool.


T = 15


R = A+B


formula becomes:


(A+B)*15 = 1


you can solve for (A+B) to get:


(A+B) = 1/15


that's the combined rate when both pipes are open.


now you run both pipes for 12 hours.


since R*T = Q, your formula becomes:


(A+B)*12 = x


Q is equal to x because we don't know what it is yet.


but we do know that (A+B) = 1/15, so we get:


1/15 * 12 = x which results in x = 12/15 which can be simplified to x = 4/5.


the both pipes can fill 4/5 of the pool in 12 hours.


that leaves 1/5 of the pool still needing to be filled.


since pipe B is closed, pipe A has to finish the job.


the formula is still R*T = Q


this time Q = 1/5 and T = 8 and R = A only because B is closed.


the formula becomes:


8*A = 1/5


solve for A to get A = 1/40


that's the rate of pipe A.


pipe A can fill 1/40 of the pool in one hour.


now we want to know the rate of pipe B.


since we know that A+B = 1/15 and we know that A = 1/40, we can substitute in this equation to find B.


the formula becomes:


1/40 + B = 1/15


subtract 1/40 from both sides of this equation to get:


B = 1/15 - 1/40 which is equivalent to:


B = 8/120 - 3/120 which is equal to 5/120 which is equal to 1/24.


we now have the individual rates.


A = 1/40
B = 1/24


pipe A will take 40 hours to fill the pool by itself.
pipe B will take 24 hours to fill the pool by itself.


working together, the formula becomes (A+B)*15 = 1


replace A and B with their respective values and you get:


(1/40 + 1/24) * 15 = 1


solve to get:


1 = 1 which confirms the solutions are good.


12 hours working together fills 4/5 of the pool and then pipe A takes over for the remaining 1/5 of  the pool working at 1/40 of the pool per hour.


1/40 * 8 = 8/40 = 1/5.


solution is confirmed.


rates are:


A = 1/40 of the pool per hour.
B = 1/24 of the pool per hour.