Three pipes, P, Q and R, together take four hours to fill a tank. All the three pipes were opened at the same time. After three hours, P was closed, and Q and R filled the remaining tank in two hours. How many hours will P alone take to fill the tank?

Correct — C. Work with rates per hour, which add.

P + Q + R fill the tank in 4 hours → combined rate = 1⁄4 tank per hour
First 3 hours, all three open: 3 × 1⁄4 = 3⁄4 of the tank filled

Remaining = 1 − 3⁄4 = 1⁄4, and Q + R take 2 hours over it
Q + R = (1⁄4) ⁄ 2 = 1⁄8 per hour

P alone = 1⁄4 − 1⁄8 = 1⁄8 per hour
So P fills the tank in 8 hours → option (c).

Why the others are wrong

• (a)
  10 — 10 hours makes P's rate 1⁄10 and Q + R equal 3⁄20. They would clear the last quarter in 5⁄3 hours, not the 2 hours stated.
• (b)
  12 — 12 hours leaves Q + R at 1⁄6 per hour, which finishes the remaining quarter in 1.5 hours — faster than the question allows.
• (d)
  9 — 9 hours puts Q + R at 5⁄36 per hour and the last quarter takes 1.8 hours. Only a rate of 1⁄8 stretches it to exactly 2 hours.

Every pipe question turns on one idea: rates add, times do not. A pipe that fills a tank in t hours contributes 1⁄t of the tank each hour.

The three-hour stretch gives 3⁄4 of the tank, so the residue is 1⁄4. That residue and the 2 hours spent on it give the combined rate of Q and R, which is all the question needs.

Subtracting that from the rate of all three isolates P: 1⁄4 − 1⁄8 = 1⁄8, so 8 hours.

Key facts

• Pipes running together have a combined rate equal to the sum of their individual rates.
• Filling a tank in 4 hours is a rate of 1⁄4 of the tank per hour.
• Q and R fill the last 1⁄4 of the tank in 2 hours, a rate of 1⁄8 per hour.

Common traps

• Subtracting times rather than rates, and reading 4 − 2 = 2 as the answer.
• Taking the remaining 1⁄4 as a share of the remaining time instead of of the tank.
• Assuming P did one third of the work because three pipes were running.

No directly related past PYQ was found.