A and B can complete a task together in 12 days, while B and C can finish it in 16 days. After A works on it for 5 days and B for 7 days, C takes the remaining 13 days to finish the work. How many days would it take for C to complete the work alone?

Correct — A. Regroup the individual days into days of the two given pairs.

Rates: a + b = 1⁄12 and b + c = 1⁄16. The job: 5a + 7b + 13c = 1.
Regroup: 5a + 7b + 13c = 5(a + b) + 2(b + c) + 11c

Substitute: 5⁄12 + 2⁄16 + 11c = 13⁄24 + 11c = 1
Solve: 11c = 11⁄24, so c = 1⁄24
C alone takes 24 days → option (a).

Why the others are wrong

• (b)
  16 days — 16 days is B and C together. If C alone took 16 days, B's rate would be 1⁄16 − 1⁄16 = 0, yet B works 7 days on the job.
• (c)
  20 days — If c = 1⁄20, then b = 1⁄80 and a = 17⁄240, and the stated days add up to 262⁄240 of the job. That is more than the whole, so C must take longer than 20 days.
• (d)
  18 days — If c = 1⁄18, then b = 1⁄144 and a = 11⁄144, and the stated days add up to 166⁄144 of the job — again more than the whole, so C must take longer than 18 days.

When only pair rates are known, rewrite each person's days so they line up with the pairs.

A's 5 days join 5 of B's days to make 5 days of A + B. B's other 2 days join 2 of C's days to make 2 days of B + C. What is left is 11 days of C alone.

The rest follows: b = 1⁄16 − 1⁄24 = 1⁄48 and a = 1⁄12 − 1⁄48 = 1⁄16. Check: 5⁄16 + 7⁄48 + 13⁄24 = 1.

Key facts

• A + B do 1⁄12 of the job a day and B + C do 1⁄16.
• The one-day rates are A = 1⁄16, B = 1⁄48 and C = 1⁄24.
• Alone, A takes 16 days, B 48 days and C 24 days.

Common traps

• Taking 16 days, the time for B and C together, as C's own time.
• Pairing A's days with C's — neither given pair contains both A and C.

No directly related past PYQ was found.