ichha148 wrote:
Tanks X and Y contain 500 and 200 gallons of water respectively. If water is being pumped out of tank X at a rate of K gallons per minute and water is being added to tank Y at a rate of M gallons per minute, how many hours will elapse before the two tanks contain equal amounts of water?
A. \(\frac{5}{M + K}\text{ hours}\)
B. \(6(M + K)\text{ hours}\)
C. \(\frac{300}{M + K}\text{ hours}\)
D. \(\frac{300}{M - K}\text{ hours}\)
E. \(\frac{60}{M - K}\text{ hours}\)
m22 q17
TL;DR
500 - Kx = 200 + Mx
300 = x (K + M)
x = 300/(K + M) minutes
x = 300/(K + M) minutes * (1 hr/60 min)
x = 5/(K + M) hours
ANSWER: A
Veritas Prep Official Solution
There are two tanks with different water levels. Note that the rate of pumping is given as K gallons per min and M gallons per min i.e. they are different. So we cannot say that they both will have equal amount of water when they have 350 gallons. They could very well have equal amount of water at 300 gallons or 400 gallons etc. So when one expects that water in both tanks will be at 350 gallon level, one is making a mistake. The two tanks are working for the same time to get their level equal but their rates are different. So the work done is different. Note here that equal level does not imply equal work done. The equal level could be achieved at 300 gallons when work done would be different – 200 gallons removed from tank X and 100 gallons added to tank Y. The equal level could be achieved at 400 gallons when work done would be different again – 100 gallons removed from tank X and 200 gallons added to tank Y.
To achieve the ‘equal level,’ tank Y needs to gain water and tank X needs to lose water. Total 300 gallons (500 gallons – 200 gallons) of work needs to be done. Which tank will do how much depends on their respective rates.
Work to be done together = 300 gallons
Relative rate of work = (K + M) gallons/minute
The rates get added because they are working in opposite directions – one is removing water and the other is adding water. So we get relative rate (which is same as relative speed) by adding the individual rates.
Note here that rate is given in gallons per minute. But the options have hours so we must convert the rate to gallons per hour.
Relative rate of work = (K + M) gallons/minute = (K + M) gallons/(1/60) hour = 60*(K + M) gallons/hour
Time taken to complete the work = 300/(60(K + M)) hours = 5/(K + M) hours