The ratio of the time taken by A and B to complete a task is 3 : 2. If B can complete the task alone in 18 hours, how many hours will they take to complete the task if they work together?

Correct — B.
Time ratio A : B = 3 : 2, and B takes 18 hours.
A's time = 18 × 3⁄2 = 27 hours

One hour's work together = 1⁄27 + 1⁄18
= 2⁄54 + 3⁄54 = 5⁄54

Time together = 54 ÷ 5 = 10.8 hours → option (b)

Why the others are wrong

• (a)
  12 hours — 12 hours is too long: at 5⁄54 of the task an hour, 12 hours would get through 60⁄54, more than the whole task.
• (c)
  9.8 hours — 9.8 hours is too short: 9.8 × 5⁄54 = 49⁄54 of the task, so 5⁄54 of it is still undone.
• (d)
  10 hours — 10 hours covers 10 × 5⁄54 = 50⁄54 of the task. The last 4⁄54 takes another 0.8 hour, which makes 10.8.

Time and rate are inverses. A takes 3 parts of time to B's 2, so A works at 2 parts of rate to B's 3.

Rates add, times do not. Turn each time into a one-hour rate, add the rates, then invert the sum: 1⁄27 + 1⁄18 = 5⁄54, so the pair needs 54⁄5 hours.

Shortcut for two workers: time together = ab ÷ (a + b) = 27 × 18 ÷ 45 = 10.8.

Key facts

• If two times are in the ratio a : b, the rates are in the ratio b : a.
• Two workers who need a and b hours alone need ab ÷ (a + b) hours together.
• Here A needs 27 hours alone and the pair 10.8 hours.

Common traps

• Reading the 3 : 2 time ratio as a rate ratio, which makes A the faster worker and gives A 12 hours instead of 27.
• Averaging the two times instead of adding the two rates.

No directly related past PYQ was found.