Suppose A, B and C are three taps fixed to the bottom of a tank with draining capacity 1 : 2 : 3. When all three of them are on, it takes 1 hour to drain out the full tank. If A and C are on but B is off, then how much time, in minutes, will it take to empty out a full tank of water ?

Correct — B, 90. Draining capacity 1 : 2 : 3 means the three taps move water at rates in that proportion, so let them be 1, 2 and 3 units per minute. All three together move 6 units per minute and empty the tank in 60 minutes, which fixes the tank at 6 × 60 = 360 units. With B shut, A and C carry 1 + 3 = 4 units per minute, so the time needed is 360 ÷ 4 = 90 minutes. The same result follows without ever naming the tank size: closing B removes 2 of the 6 parts of the flow, leaving four-sixths of it, and time varies inversely with rate, so the new time is 60 × 6/4 = 90 minutes. The whole item is one instruction — add rates, never add times.

• (a)75 — This answers to A and C together carrying four-fifths of the total flow, since 60 × 5/4 is 75. No reading of the ratio 1 : 2 : 3 makes the parts add to five; they add to six.
• (c)105 — It corresponds to a total of seven parts rather than six, as 60 × 7/4 gives 105. The arithmetic slip is in the sum of the ratio terms, not in the method.
• (d)120 — The instinctive answer — one tap of three is shut, so double the time. Closing B removes only two of the six parts of the flow, not three, so the time rises by half rather than doubling. Doubling would be right only if B alone carried as much as A and C together.

Rates add; times do not. If one agent completes a job in t units of time its rate is 1/t of the job per unit, and several agents working simultaneously have a combined rate equal to the sum of their individual rates. That is why the standard technique is to convert every time into a rate, add or subtract the rates as the situation requires, and only then invert the result back into a time. A filling tap counts as a positive rate and a draining tap as a negative one when both are open at once.

Assigning the tank a convenient size is what makes this kind of item quick. Taking the total as the least common multiple of the numbers in play — or here simply as 6 units per minute times 60 minutes, giving 360 — replaces fractions with whole numbers throughout. A ratio question is even easier, because the ratio itself does the work: the answer depends only on what share of the total flow remains open, so the tank size cancels out. Confirm the direction of the answer before writing it down. Shutting a tap must make the tank take longer to empty, so any option below 60 minutes could have been rejected on sight.

• Rates of work add; times taken do not.
• A ratio of capacities 1 : 2 : 3 means six parts in total, so each tap carries one-sixth, one-third and one-half of the flow.
• Taking the tank as 360 units makes the three rates 1, 2 and 3 units per minute over the 60-minute full-flow time.
• With B closed the remaining rate is 4 of 6 parts, so the time becomes 60 × 6/4 = 90 minutes.
• Time is inversely proportional to rate, so a smaller open flow must always give a longer emptying time.

• Averaging or adding the times instead of the rates.
• Assuming that closing one of three taps must double the time, without checking that tap's share.
• Reading a capacity ratio as a ratio of times, which reverses the order of the taps.

Asked as a ratio-of-rates item where the tank's actual size is never given, because it cancels out and is not needed.

• practice — not a real PYQ

  Two taps can fill a tank in 12 minutes and 24 minutes respectively. Opened together, they fill it in

  • (a)6 minutes
  • (b)8 minutes
  • (c)16 minutes
  • (d)18 minutes

  Answer(b) 8 minutes — the rates are one-twelfth and one-twenty-fourth of the tank per minute, which add to one-eighth, so the tank fills in 8 minutes.

• practice — not a real PYQ

  A tap fills a tank in 20 minutes while a leak at the bottom empties it in 30 minutes. With both open, the tank fills in

  • (a)10 minutes
  • (b)25 minutes
  • (c)50 minutes
  • (d)60 minutes

  Answer(d) 60 minutes — the net rate is one-twentieth minus one-thirtieth, which is one-sixtieth of the tank per minute.