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Question 203502This question is from textbook Elementary & Intermediate Algebra
: Please help with word problem:
Filling a tank. A water tank has an inlet pipe and a drain pipe. A full tank can be emptied in 30 minutes if the drain is opened and an empty tank can be filled in 45 minutes with the inlet pipe opened. If both pipes are accidentally opened when the tank is full, then how long will it take to empty the tank?
This question is from textbook Elementary & Intermediate Algebra
Found 3 solutions by Earlsdon, Theo, ikleyn:
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! If the empty tank can be filled in 45 minutes, then 1/45 of the tank can be filled in 1 minute.
If the full tank can be emptied in 30 minutes, then 1/30 of the tank can be emptied in 1 minute.
With both pipes open, then (1/45 - 1/30) of the full tank can be emptied in 1 minute.
 of the tank can be emptied (negative) in 1 minute.
So this means it will take 90 minutes to empty the tank completely.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! Problem:
Filling a tank. A water tank has an inlet pipe and a drain pipe. A full tank can be emptied in 30 minutes if the drain is opened and an empty tank can be filled in 45 minutes with the inlet pipe opened. If both pipes are accidentally opened when the tank is full, then how long will it take to empty the tank?
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empty tank in 30 minutes.
fill tank in 45 minutes.
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let T = amount of liquid in the tank when it's full.
T can be anything, i.e. gallons, liters, etc. It's not specified.
you do know that when the tank is full it has T amount of liquid in it, whatever T represents (gallons, liters, quarts, whatever)
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let e = rate of empty = T/30
let f = rate of fill = T/45
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the question was:
if both drains are open, how long does it take to empty the tank.
what you have is both drains open so water is pouring out of both at the same time.
the assumption is that the rate of drainage from the fill pipe would be the same as the fill rate.
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water is draining out of the drain pipe at the rate of e = T/30.
water is draining out of the fill pipe at the rate of f = T/45.
the total water to be drained is T.
the equation should be:
(f+e)*m = T
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since f = T/45 and e = T/30, this equation becomes:
((T/30) + (T/45)) * m = T
where m represents the number of minutes it takes to drain the tank.
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multiply both sides of this equation by (30)*(45) to get
45*T + 30*T * m = 45*30*T
which becomes:
75*T*m = 45*30*T
divide both sides of this equation by (75*T) to get:
m = (45*30*T)/(75*T)
which becomes:
m = 18
the tank would drain in 18 minutes.
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(T/45)*18 = .4T
(T/30)*18 = .6T
((T/45)+(T/30))*18 = T
answer looks good. tank will drain in 18 minuters if both fill pipe and drain pipe are left open.
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Answer by ikleyn(52957)  (Show Source):
You can put this solution on YOUR website! .
A water tank has an inlet pipe and a drain pipe. A full tank can be emptied in 30 minutes
if the drain is opened and an empty tank can be filled in 45 minutes with the inlet pipe opened.
If both pipes are accidentally opened when the tank is full, then how long will it take to empty the tank?
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The solution by @Theo, giving the answer 18 minutes to empty the tank
is (a) incorrect and (b) contradicts to common sense.
I will give here short, clear and correct solution.
The drain opening removes water from the tank at the rate of  of the tank volume per minute.
The inlet pipe adds water top the tank at the rate of  of the tank volume per minute.
The combined rate when both drain and inlet pipe works simultaneously is the difference
- = use the common denominator 90 =  -  =  .
In other words, when the full tank remained with both the drain and the inlet opened,
then of the tank volume is removed each minute from the tank.
From it, we conclude that the time to empty the tank is 90 minutes, or 1 hour and 30 minutes. <<<---=== ANSWER
Solved.
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Since the @Theo' solution is incorrect, ignore his post for the peace in your mind.
His error is that he uses the sum of rates of works as an effective/combined rate,
while in this problem the difference of rates should be used.
It is a typical error for those who write wordy: when a person writes wordy, it opens the space for many errors.
Use my solution, instead. It is short, clear and instructive.
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