Fill pipe P is 21 times faster than fill pipe Q. If Q can fill a cistern in 110 minutes, find the time it takes to fill the cistern when both fill pipes are opened together.

Correct — B. Work in rates per minute, never in times.

Q fills the cistern in 110 minutes, so Q's rate = 1⁄110 of the cistern per minute.

P is 21 times faster, so P's rate = 21 × 1⁄110 = 21⁄110 per minute.

Together = 1⁄110 + 21⁄110 = 22⁄110 = 1⁄5 of the cistern per minute.

Time = 1 ÷ 1⁄5 = 5 minutes → option (b).

Why the others are wrong

• (a)
  3 minutes — 3 minutes needs a combined rate of 1⁄3. Q supplies only 3⁄110 of the tank in that time, leaving 107⁄110 to P — about 36 times Q's rate, not 21.
• (c)
  4 minutes — 4 minutes is where the stricter reading of 'faster' half-lands: taking P as 22 times Q gives 23⁄110 per minute and 110⁄23 ≈ 4.78 minutes, which is not 4 either.
• (d)
  6 minutes — P alone at 21⁄110 per minute finishes in 110⁄21 ≈ 5.24 minutes, so the pair cannot possibly need 6. Opening a second pipe can only shorten the time.

Pipe-and-cistern sums are work sums in disguise: never add or average times, add rates.

A pipe that fills in t minutes contributes 1⁄t per minute, and pipes running together simply have their rates summed.

The phrase 21 times faster is the live ambiguity. Taken as 21 times the rate, P gives 21⁄110 and the pair gives 22⁄110 = 1⁄5, landing exactly on 5 minutes.

Key facts

• A pipe that fills a cistern in t minutes works at 1⁄t of the cistern per minute.
• Rates of pipes opened together add, so 1⁄110 + 21⁄110 = 1⁄5 and the tank fills in 5 minutes.
• An emptying pipe enters the same sum with a minus sign.

Common traps

• Averaging the two filling times instead of adding the rates
• Reading 21 times faster as 22 times the rate and then truncating 4.78 to 4

No directly related past PYQ was found.