A tank can be filled by an inlet pipe in 8 hours when the drain pipe
is open and in 5 hours if the drain pipe is closed. How long will it
take to empty a full tank when the inlet pipe is closed.
I'm afraid your method is wrong. Only 3 1/3 hours is much too fast
for the drain pipe to be able to empty the pool if it only slows
the fill pipe down from filling it in 5 hours to filling it in 8
hours. If it drained that fast, the inlet pipe would never be able
to fill the pool at all when the drain pipe was open for it would
be draining water out much faster than the inlet pipe was putting
water in!
Here's a method very much like distance-rate-time problems that I
think you may understand better.
Let's make this DRT-chart, except here the D stands for "Deeds Done"
rather than "Distance". So D represents the number of deeds, or
poolfuls it caused.
D R T
No. of Rate in No. of hrs.
poolfuls pool/hr required
Inlet pipe alone
Drain pipe alone
both open
Let x = the length of time required for the drain pipe to empty the
pool when the inlet pipe is open.
First fill in the three times, 5 hours for the inlet pipe, x hours
for the drain pipe, and 8 hours when they are both open.
D R T
No. of Rate in No. of hrs.
poolfuls pool/hr required
Inlet pipe alone 5
Drain pipe alone x
both open 8
Next we will fill in the number of poolfuls.
The inlet pipe caused the GAIN of 1 poolful in 5 hours. So we put
1 for the number of poolfuls it caused in the 5 hours it was open
by itself.
D R T
No. of Rate in No. of hrs.
poolfuls pool/hr required
Inlet pipe alone 1 5
Drain pipe alone x
both open 8
Now the drain pipe caused a LOSS of 1 poolful in x hours. So we
put -1 for the number of poolfuls it caused in the x hours it was
open by itself. (That -1 is the tricky part!)
D R T
No. of Rate in No. of hrs.
poolfuls pool/hr required
Inlet pipe alone 1 5
Drain pipe alone -1 x
both open 8
Now when both were open that caused a GAIN of 1 poolful in 8 hours.
So we put 1 for the number of poolfuls they caused in the 8 hours
they both were open. So we fill in 1 for the no. of poolfuls caused
by both of them being open.
D R T
No. of Rate in No. of hrs.
poolfuls pool/hr required
Inlet pipe alone 1 5
Drain pipe alone -1 x
both open 1 8
Now we fill in the rates by using R = D/T.
D R T
No. of Rate in No. of hrs.
poolfuls pool/hr required
Inlet pipe alone 1 1/5 5
Drain pipe alone -1 -1/x x
both open 1 1/8 8
Now the logic in setting up the equation is just like
when a boat is in a river going downstream, we add the
rate of the boat in still water and the rate of the
stream to get the combined rate. So we use the same
idea here:
Rate of inlet pipe + Rate of drain pipe = their combined rate, or
1/5 + (-1/x) = 1/8
1/5 - 1/x = 1/8
Solve that and you'll get 13 1/3 hours for the drain pipe to
drain 1 full pool. That answer makes a lot more sense.
Edwin