Dhaval and Dhiraj together can do a piece of work in 12 days. Dhaval alone can do it in 20 days. In how many days can Dhiraj alone do it?

Correct — A. Turn each time into a one-day rate, then subtract.

Together, per day = 1⁄12 of the work
Dhaval alone, per day = 1⁄20
Subtract: 1⁄12 − 1⁄20 = 5⁄60 − 3⁄60 = 2⁄60 = 1⁄30

Dhiraj does 1⁄30 of the work a day, so he takes 30 days → option (a)

Why the others are wrong

• (b)
  28 — 1⁄20 + 1⁄28 = 3⁄35, so with a 28-day Dhiraj the pair would finish in 35⁄3 ≈ 11.7 days, faster than the 12 in the stem.
• (c)
  29 — 1⁄20 + 1⁄29 = 49⁄580, a joint time of 580⁄49 ≈ 11.8 days. Only a 30-day Dhiraj makes the pair take exactly 12.
• (d)
  27 — 1⁄20 + 1⁄27 = 47⁄540, a joint time of about 11.5 days. A faster Dhiraj pulls the pair below the stated 12.

Days do not add or subtract; rates do. Someone who finishes a job in n days does 1⁄n of it each day.

Combined rate − known rate = unknown rate, then flip the rate back into days.

The LCM method does the same with whole numbers. Take the work as LCM(12, 20) = 60 units: together they do 5 units a day, Dhaval 3, so Dhiraj does 2, and 60 ÷ 2 = 30 days.

Key facts

• Rate = 1 ⁄ time: a job done in 20 days means 1⁄20 of it per day.
• Rates of people working together add, so an unknown rate = combined rate − known rate.
• B alone = (A × T) ⁄ (A − T), where A is A's time alone and T the joint time: 20 × 12 ⁄ 8 = 30.

Common traps

• Subtracting the days: 20 − 12 = 8 is neither a rate nor the answer.
• Adding the rates instead of subtracting: 1⁄12 + 1⁄20 = 2⁄15 gives 7.5 days, faster than the pair.

No directly related past PYQ was found.