Two taps A and B can fill a tank with water in 20 minutes and 15 minutes, respectively. A third tap C can empty the full tank in 6 minutes. After the taps A and B fill the tank for 5 minutes, the tap C is opened to empty while A and B continue to fill it. In how many minutes after the start shall the tank get empty?

Correct — D, 11 2/3 minutes. Take the tank as 60 units, the lowest common multiple of 20, 15 and 6. Then A fills 3 units a minute, B fills 4 and C drains 10. In the first five minutes A and B alone put in (3 + 4) × 5 = 35 units. Once C opens, the three taps together change the level by 3 + 4 − 10 = −3 units a minute, so the 35 units drain away in 35 ÷ 3 = 11 2/3 minutes. That is the figure the paper offers, and no other option matches any step of the working.

• (a)10 1/3 minutes — 10 1/3 corresponds to 31 units draining at 3 a minute. The tank holds 35 units when C opens, not 31.
• (b)10 2/5 minutes — This would need the net drain rate to be 3.365 units a minute. The three taps give exactly 3 units a minute against a 35-unit head.
• (c)10 4/5 minutes — 10.8 minutes at 3 units a minute empties 32.4 units. It answers a tank filled for four and a half minutes rather than five.

A pipes-and-cisterns question is a time-and-work question with one sign reversed: a filling tap is a positive rate and an emptying tap a negative one. Fixing the tank as the lowest common multiple of the individual times turns every rate into a whole number, after which the phases of the problem can simply be added up.

Read the stem's last line with care. The working gives 11 2/3 minutes measured from the moment tap C opens, and 16 2/3 minutes measured from the first minute of filling; the second figure is not among the options, so the paper is counting from the opening of C even though it says 'after the start'. The booklet page was rendered to check this, and the printed stem and the four printed options are exactly as reproduced here, so the looseness belongs to the paper and not to the transcription. Work the physical quantity first — 35 units at a net drain of 3 a minute — and then match it to the option list.

• On a 60-unit tank, A fills 3 units a minute, B fills 4, and C drains 10.
• Five minutes of A and B together put in 35 units.
• With all three open the net rate is 3 + 4 − 10 = −3 units a minute.
• 35 ÷ 3 = 11 2/3 minutes for the tank to empty once C is opened.
• An emptying pipe enters the sum as a negative rate; if the net rate were positive the tank would never empty.

• Forgetting that A and B keep running after C opens, and using a drain rate of 10 instead of 3.
• Computing the 35 units and then dividing by 10 rather than by the net rate.
• Assuming the tank is full when C opens; five minutes of filling leaves it a little over half full.

A cistern item in two phases, with an emptying pipe joining a filling pair — the standard test of whether rates are being added with the right signs.

• practice — not a real PYQ

  A tap fills a tank in 12 minutes and a drain empties it in 20 minutes. With both open, the tank fills in

  • (a)24 minutes
  • (b)28 minutes
  • (c)30 minutes
  • (d)32 minutes

  Answer(c) 30 minutes — on a 60-unit tank the rates are +5 and −3, so the net is +2 and 60 ÷ 2 = 30.

• practice — not a real PYQ

  Two taps fill a tank in 10 and 15 minutes. Both are opened together. How long does the tank take to fill?

  • (a)5 minutes
  • (b)6 minutes
  • (c)8 minutes
  • (d)12.5 minutes

  Answer(b) 6 minutes — on a 30-unit tank the rates are 3 and 2, so together 5 a minute and 30 ÷ 5 = 6.