Bunuel wrote:
A tank is emptied everyday at a fixed time point. Immediately thereafter, either pump A or pump B or both start working until the tank is full. On Monday, A alone completed filling the tank at 8 pm. On Tuesday, B alone completed filling the tank at 6 pm. On Wednesday, A alone worked till 5 pm, and then B worked alone from 5 pm to 7 pm, to fill the tank. At what time was the tank filled on Thursday if both pumps were used simultaneously all along?
A. 4 : 12 PM
B. 4 : 24 PM
C. 4 : 36 PM
D. 4 : 46 PM
E. 4 : 48 PM
Are You Up For the Challenge: 700 Level Questions Solution:If we let a be the number of hours it takes pump A to fill the tank by itself, then 1/a is the rate of pump A. Since pump B alone can fill the tank in 2 hours faster (6 pm vs. 8 pm), the number of hours it takes pump B to fill the tank by itself is (a - 2), and thus its rate is 1/(a - 2). Since on Wednesday, A alone worked till 5 pm, and then B worked alone from 5 pm to 7 pm to fill the tank, we can create the equation:
(a - 3) x 1/a + 2 x 1/(a - 2) = 1
(a - 3)/a + 2/(a - 2) = 1
Multiplying the equation by a(a - 2), we have:
(a - 3)(a - 2) + 2a = a(a - 2)
a^2 - 5a + 6 + 2a = a^2 - 2a
-3a + 6 = -2a
6 = a
Therefore, it takes pump A 6 hours to fill the tank by itself and pump B 4 hours to fill the tank by itself. Furthermore, the tank is filled starting at 2 pm. So on Thursday, if both pumps were used simultaneously at 2 pm, it will take 1/(⅙ + ¼) = 1/(4/24 + 6/24) = 1/(10/24) = 24/10 = 2.4 hours to fill the tank. Since 2.4 hours = 2 hours and 24 minutes, the tank will be filled at 4:24 pm.
Alternate Solution:
Let t be the number of hours it takes for pump B to fill the tank. Since pump A alone finishes filling the tank 2 hours later, compared pump B alone (6 pm vs. 8 pm), it takes pump A (t + 2) hours to fill the tank.
On Wednesday at 5 pm, pump A had 3 more hours to fill the entire tank; but pump B completed this job in only 2 hours. Thus, pump B works 3/2 faster compared to pump A or, equivalently, the time for pump A to fill the tank alone must be 3/2 of the time it takes for pump B to fill the tank. Since the time of pump A was t + 2 hours and the time of pump B was t hours, we can create the following equation:
t + 2 = 3t/2
2t + 4 = 3t
t = 4
Therefore, it takes pump B 4 hours to fill the tank by itself and pump A 4 + 2 = 6 hours to fill the tank by itself. Furthermore, the tank is filled starting at 2 pm. So on Thursday, if both pumps were used simultaneously at 2 pm, it took 1/(⅙ + ¼) = 1/(4/24 + 6/24) = 1/(10/24) = 24/10 = 2.4 hours to fill the tank. Since 2.4 hours = 2 hours and 24 minutes, the tank was filled at 4:24 pm.
Answer: B _________________
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