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Question 892114: Please help me solve this problem:
A larger pipe can fill a tank in 32 minutes less time than a smaller pipe. If they both are turned on at the same time, they can fill the tank in 30 minutes. How long does it take each pipe alone to fill the tank?
Thank you.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! rate * time = quantity
quantity is 1 full tank = 1
time for the small pipe is x.
time for the large pipe is x - 32.
the rate for the small pipe is calculated as follows:
rate * time = quantity
rate * x = 1
rate = 1/x
the rate for the large pipe is calculated as follows:
rate * time = quantity
rate * (x-32) = 1
rate = 1 / (x-32)
rate for the small pipe is 1/x
rate for the large pipe is 1/(x-32)
when they work together their rates are additive.
when they work together they can fill the tank in 30 minutes.
rate * time = quantity.
(1/x + 1/(x-32)) * 30 = 1
you need to solve for x.
when you combine the denominators into one common denominator, the equation becomes:
((2x-32)/(x(x-32))*30 = 1
multiply both sides of this equation by (x(x-32) to get:
(2x-32)*30 = x(x-32)
simplify to get 60x - 960 = x^2 - 32x
subtract (60x - 960) from both sides of the equation to get:
0 = x^2 - 32x - (60x - 960)
simplify to get:
x^2 - 92x + 960 = 0
factor this equation to get (x-12) * (x-80) = 0
solve for x to get x = 12 or x = 80
x can't be equal to 12 because x - 32 would then be negative.
your solution for x is that x = 80.
the small pipe fills the tank in 80 minutes.
the large pipe fills the tank in 80 - 32 = 48 minutes.
the rate of the small pipe is 1/80.
the rate of the large pipe is 1/48
this is derived from rate * time = quantity
if the small pipe fills the tank in 80 minutes, then the rate of the small pipe is 1/80 of the tank in 1 minutes.
if the big pipe fills the tank in 48 minutes, then the rate of the big pipe is 1/48 of the tank in 1 minute.
when the pipes work together, their rates are additive.
you get rate * time = quantity becomes:
(1/80 + 1/48) * 30 = 1
solve this equation to get 1 = 1 which confirms the rates are calculated correctly.
the solution to the problem is:
the small pipe can fill the tank in 80 minutes.
the large pipe can fill the tank in 48 minutes.
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