The ratio of the efficiencies of two workers, P and Q, is 5 : 2. If P can complete a project in 12 days, how many days will Q take to complete the same project alone?

Correct — B. For the same job, time is inversely proportional to efficiency.

Efficiency P : Q = 5 : 2
Invert for time: P : Q = 2 : 5
P's 2 parts = 12 days → 1 part = 6 days
Q's 5 parts = 5 × 6 = 30 days → option (b)

Why the others are wrong

• (a)
  24 days — 24 days is 12 × 2, the time for a worker half as efficient as P. Q's efficiency is 2⁄5 of P's, so Q takes 5⁄2 as long: 30 days.
• (c)
  35 days — 35 days would make the time ratio 12 : 35, so the efficiency ratio would be 35 : 12, not 5 : 2.
• (d)
  40 days — 40 days gives a time ratio of 12 : 40 = 3 : 10, which means efficiencies of 10 : 3, not 5 : 2.

Efficiency is work done per day. For a fixed job, doubling efficiency halves the time, so times run in the reverse ratio of efficiencies.

A units check gives the same answer: call P's daily work 5 units, so the job is 5 × 12 = 60 units. Q does 2 units a day and needs 60 ÷ 2 = 30 days.

Key facts

• Work = efficiency × time, so for a fixed job efficiency × time is constant.
• An efficiency ratio a : b means a time ratio b : a.
• Here 5 × 12 = 2 × t, so t = 60 ÷ 2 = 30 days.

Common traps

• Multiplying 12 by 2⁄5 instead of 5⁄2 gives 4.8 days: the less efficient worker would then finish faster, which cannot be right.

No directly related past PYQ was found.