Problem Solving

A tank is fitted with an inlet which takes 1 hour to completely fill the empty tank. The tap was turned on when the tank was empty. When the tank was half filled , it developed a leak and as a result it took totally 65 minutes to fill the tank. As soon as the tank got filled completely, the tap was turned off but the leakage continued. How long will it take for the leakage to empty the tank completely.

Answer is 7hrs

According to me-

Rate is 1tank/hr
Rate of leak is x tank/hr

Due to the leak it has taken 65minutes to fill the tank.Since the leaking spot is in the middle of the tank,its safe to say that it has taken (65-30)=35minutes to fill the top half of the tank
So we can formulate an equation like this one below:

(1-x)*35=(1/2)

35-35x=1/2
34.5=35x => x=(34.5/35)

Now we have the rate of leakage equaling (34.5/35)

Now (34.5/35) * Time =0.5(Because only half the tank could possibly leak) Therefore
Time=(35/69)hrs==========This answer is wrong

However if Ive made a mistake of considering the leak to be happening just at the half of the tank,
the main equation can be reformulated as

(1-x)*65=1
65-65x=1

x=64/65

(64/65)*Time = 1

Time =65/64 hrs======This too is a wrong answer

Please tell me where my understanding is misleading me even though there might be a totally different approach to this problem.

The above method I used was influenced to an extent by this problem below

Two inlet taps can fill a tank in 12 hours and 15 hours respectively. There is an out let tap at the middle of the tank which can empty the full tank in 60 hours. If all the taps are opened together, after how much time, the tank will be full?
1) 7 hrs 5 mins 2) 7 hrs 30 mins 3) 9 hrs 12 mins 4) 12 hrs

Thanks

Dan

dddanny2006 wrote:A tank is fitted with an inlet which takes 1 hour to completely fill the empty tank. The tap was turned on when the tank was empty. When the tank was half filled , it developed a leak and as a result it took totally 65 minutes to fill the tank. As soon as the tank got filled completely, the tap was turned off but the leakage continued. How long will it take for the leakage to empty the tank completely.

Inlet alone:
Time to fill the whole tank = 60 minutes.
Thus, the time for the inlet to fill half the tank = 30 minutes.

Inlet and leak together:
Since the leak develops when the tank is half full, the leak develops after 30 minutes have passed.
Since the tank is full after 65 minutes, the time for the inlet and leak to fill half the tank = 35 minutes.
Thus, the time for the inlet and leak to fill the WHOLE tank = 70 minutes.

Let the tank = the LCM of 60 and 70 = 420 liters.
Rate for the inlet alone = w/t = 420/60 = 7 liters per minute.
Rate for the inlet and leak together = w/t = 420/70 = 6 liters per minute.
Since the leak reduces the rate by 1 liter per minute, the rate for the leak alone = 1 liter per minute.
Time for the leak to empty the tank = w/r = 420/1 = 420 minutes = 7 hours.

Hi Rich
Here's my doubt-When the tank is half filled it springs a leak,so a part of the initial half filled tank gets laekd.So how can we say that the last 35 minutes filled only the top half of the tank?It could have also filled the initial portion that leaked plus the top half of the tank right?Please clarify.

[email protected] wrote:Hi dddanny2006,

This question isn't written in "GMAT style", so you should be a bit suspicious of its source. That having been said, you should have included the 5 answer choices for reference; sometimes the answers are written in such a way as to help you figure out how to do the "math" or how to avoid it altogether.

Since the prompt does NOT give us much information to work with, we can TEST Values to get to the solution.

We're told that a tank can be filled in 1 hour. When the tank is half-full, it springs a leak, so filling the remaining half talk takes 35 minutes (instead of 30 minutes). The question asks how long the leak would take to empty the tank completely.

Let's say that we have a 60 gallon tank.

Under normal circumstance, it takes 1 hour to fill; this means that we fill the tank at a rate of 1 gallon/minute.

The first 30 gallons takes 30 minutes
The last 30 gallons takes 35 minutes

This means that 5 more minutes were needed to input 30 "net" gallons.
In those 35 minutes, 35 gallons went "in" and 5 gallons went "out"

So 5 gallons are lost every 35 minutes.

Since we have a 60 gallon tank, we have 12 "sets" of 5 gallons...

12(35 minutes) = 420 minutes = 7 hours.

GMAT assassins aren't born, they're made,
Rich

Hi Mitch,
This question is a little ambiguous.When the leak happens,it happens once the tank is half-filled.So the water from the initial half itself could have leaked thereby making then next 35 minutes fill the later half as well as the leaked portion of the bottom half.
Please clear this doubt.I have also attached the image below

Thanks

Dan

GMATGuruNY wrote:

dddanny2006 wrote:A tank is fitted with an inlet which takes 1 hour to completely fill the empty tank. The tap was turned on when the tank was empty. When the tank was half filled , it developed a leak and as a result it took totally 65 minutes to fill the tank. As soon as the tank got filled completely, the tap was turned off but the leakage continued. How long will it take for the leakage to empty the tank completely.

Inlet alone:
Time to fill the whole tank = 60 minutes.
Thus, the time for the inlet to fill half the tank = 30 minutes.

Inlet and leak together:
Since the leak develops when the tank is half full, the leak develops after 30 minutes have passed.
Since the tank is full after 65 minutes, the time for the inlet and leak to fill half the tank = 35 minutes.
Thus, the time for the inlet and leak to fill the WHOLE tank = 70 minutes.

Let the tank = the LCM of 60 and 70 = 420 liters.
Rate for the inlet alone = w/t = 420/60 = 7 liters per minute.
Rate for the inlet and leak together = w/t = 420/70 = 6 liters per minute.
Since the leak reduces the rate by 1 liter per minute, the rate for the leak alone = 1 liter per minute.
Time for the leak to empty the tank = w/r = 420/1 = 420 minutes = 7 hours.

dddanny2006 wrote:Hi Mitch,
This question is a little ambiguous.When the leak happens,it happens once the tank is half-filled.So the water from the initial half itself could have leaked thereby making then next 35 minutes fill the later half as well as the leaked portion of the bottom half.
Please clear this doubt.I have also attached the image below

Thanks

Dan

The problem indicates that the tap is shut off only after the tank is full.
Implication:
When the leak develops at the 30-minute mark, the tap continues to pump in water.

There is only one way for the volume in the tank to decrease:
Water would have to LEAK OUT faster than water is PUMPED IN.
In this case, there would be a NET LOSS every minute, with the result that the tank would never fill.
Since the tank fills after 35 minutes, we know that there must be a NET GAIN every minute -- even AFTER the leak develops.

Since there is no loss of volume, these 35 minutes must represent the time it takes for the inlet and the leak together to fill the remaining half of the tank.