A pipe can fill an empty tank in 5 minutes and another pipe can empty it in 6 minutes. If both pipes are opened simultaneously, how long (in minutes) will it take to fill the empty tank?
- (a)33 min
- (b)25 min
- (c)35 min
- (d)30 min
Answer
Why
Correct — D. Turn each time into a rate, filling positive and emptying negative.
Filling pipe: 1/5 of the tank per minute
Emptying pipe: −1/6 of the tank per minute
Net rate = 1/5 − 1/6
= 6/30 − 5/30 = 1/30 of the tank per minute
The net rate is positive, so the tank does fill.
Time = 1 ÷ 1/30 = 30 minutes → option (d)
Why the others are wrong
- (a)33 min — 33 min needs a net rate of 1/33. The only rates in play are 1/5 and 1/6, and their difference is 1/30 — nothing in the question produces a thirty-third.
- (b)25 min — 25 min is 30 − 5: times subtracted instead of rates. Rates add and subtract, times do not, so 1/5 − 1/6 has to be formed first and inverted only at the end.
- (c)35 min — 35 min needs a net rate of 1/35. The LCM of 5 and 6 is 30, so every fraction in this working carries a denominator built from 2, 3 and 5 — a thirty-fifth is unreachable.
Concept
Pipe-and-cistern questions are work questions in disguise. A pipe that fills in t minutes does 1/t of the job each minute, and an outlet that empties in t minutes contributes −1/t.
Rates add. Times do not. So the move is always the same: convert each time to a rate, combine the rates with their signs, then invert the net rate to recover a time.
Here 1/5 − 1/6 = 1/30, a positive net rate, so the tank fills — in 30 minutes.
The stem image on the response sheet ends at the word 'simultaneously' and the closing clause was not captured with it. All four options are times in minutes and 30 min is exactly the net fill time, so this card answers the standard reading: how long the tank takes to fill.
Key facts
- A pipe that fills a tank in t minutes works at 1/t of the tank per minute.
- For consecutive integers n and n+1, 1/n − 1/(n+1) = 1/(n(n+1)), which is why 1/5 − 1/6 = 1/30.
- If the emptying rate is the larger of the two, the net rate is negative and the tank never fills.
Study next
Common traps
- Subtracting the two times, 6 minus 5, and answering 1 minute
- Adding the rates instead of subtracting, treating the outlet as a second inlet
- Quoting 30 without first checking that the net rate came out positive
The same two-pipe setup is asked at 25 Sep 2024, 16:00, Quant Q.3, with a tap filling in 10 minutes against one emptying in 12, and a three-pipe version at 17 Sep 2024, 16:00, Quant Q.9.
Related PYQs
No directly related past PYQ was found.