Two pipes X and Y can fill a tank in 14 hours and 21 hours, respectively. Both pipes are opened simultaneously to fill the tank. In how many hours will the empty tank be filled?

Correct — C. Add the rates, then invert the sum.

X fills 1⁄14 of the tank per hour
Y fills 1⁄21 of the tank per hour

Together: 1⁄14 + 1⁄21 = 3⁄42 + 2⁄42 = 5⁄42 per hour

Time = 42⁄5 = 8.4 hours = 8 2⁄5 hours → option (c), the image reading 8 2⁄5 hours

Why the others are wrong

• (a)
  Option (a) is the image reading 6 2⁄5 hours, that is 32⁄5. At the joint rate of 5⁄42 tank per hour that fills only 32⁄42 of the tank, so it stops short.
• (b)
  Option (b) is the image reading 7 2⁄5 hours, that is 37⁄5 — exactly one hour short of 42⁄5. That missing hour is the last 5⁄42 of the tank.
• (d)
  Option (d) is the image reading 5 2⁄5 hours. Even two pipes as fast as X would need 7 hours, so no time under 7 is reachable when Y is the slower pipe.

Work problems run on rates, not times. A pipe that fills a tank in t hours does 1⁄t of the tank each hour, and rates add.

1⁄14 + 1⁄21 = 5⁄42, so the pair fills 5 units of a 42-unit tank each hour and finishes in 42⁄5 hours.

The LCM shortcut says the same without fractions: call the tank LCM(14, 21) = 42 units, so X does 3 units an hour and Y does 2, and 42 ÷ 5 = 8.4.

A quick sanity bound: two filling pipes always finish faster than the quicker one alone, but never in less than half its time.

Key facts

• A pipe filling a tank in t hours does 1⁄t of the job each hour.
• Two filling pipes taking a and b hours together take ab ⁄ (a + b) hours.
• For 14 and 21 that is (14 × 21) ⁄ 35 = 294⁄35 = 42⁄5 = 8.4 hours.

Common traps

• Adding the two times to get 35 hours, or averaging them to get 17.5
• Adding the rates correctly and then forgetting to invert 5⁄42 back into a time
• Writing 8.4 hours as 8 hours 40 minutes instead of 8 hours 24 minutes

No directly related past PYQ was found.