Problem Solving

VJesus12 wrote:Tap A can fill a tank in 10hrs, Tap B can fill it in 20hrs, and an outlet, Tap C can empty the tank in 30hrs. Each tap is opened, one by one, for exactly one hr and then closed. If its 7AM, the tank is 1/4th full and taps work in alphabetic order(A then B and then C),
at what time will the tank begin to overflow?

1. 1:30 AM next day
2. 12PM
3. 8AM
4. 10AM
5. 5PM

We can see that if taps A, B and C each work 1 hour consecutively in that order, then 1/10 + 1/20 - 1/30 = 6/60 + 3/60 - 2/60 = 7/60 of the tank would be filled. Since the tank is already 1/4 full, then the taps need to fill the remaining 3/4 of the tank. Since (3/4)/(7/60) = 3/4 x 60/7 = 45/7, it would appear that it will take 45/7 x 3 = 135/7 hours, or 19 2/7 hours, to fill the tank. However, remember that the problem asks for the time the tank will begin to overflow. That is, at certain point, if the tank is filled to its capacity by either tap A or B, then there won't be a tap C emptying it.

So let's see what happen after 18 hours (notice that, at this time, tap C just finishes emptying the pool and tap A takes over the filling):

1/4 + 6 x 7/60 = 1/4 + 7/10 = 15/60 + 42/60 = 57/60 = 19/20

We see that the leftover portion of the tank is 1/20, and it will take tap A (1/20)/(1/10) = 1/2 hour to fill the tank to its full capacity. Therefore, it will take 18 and ½ hours to fill the tank before it overflows.

7 AM + 18:30 hours = 25:30 --->25:30 - 24 = 1:30 AM the next day

Alternate Solution:

Let's assume that the tank has a capacity of 60 gallons. Since it takes 10 and 20 hours for A and B to fill the tank, respectively; the rate of A and B are 6 gal/hr and 3 gal/hr, respectively. Similarly, since it takes C 30 hours to empty the tank, the rate of C is 2 gal/hr.

Since 1/4 of the tank is full, there are 60/4 = 15 gallons of water in the tank and thus 60 - 15 = 45 gallons more should be filled before the tank overflows.

In a three hour period, 6 + 3 - 2 = 7 gallons of water accumulates in the tank. After six such cycles (or, after 18 hours), 6 x 7 = 42 gallons of water is in the tank and 45 - 42 = 3 more gallons is needed for the tank to overflow. Since it is tap A's turn to operate, it will take tap A 3/6 = 1/2 hours to fill 3 gallons; therefore the tank will overflow after 18.5 hours of operation.

7 AM + 18:30 hours = 25:30 ---> 25:30 - 24 = 1:30 AM the next day

Answer: 1/A