A and B together can finish a job in 20 days. B and C together can finish the same job in 30 days. If A and C together can finish it in 24 days, in how many days can A alone finish the job?

Correct — C, 34 2/7 days. Work with rates rather than days. If A, B and C do 1/A, 1/B and 1/C of the job per day, the three statements read 1/A + 1/B = 1/20, 1/B + 1/C = 1/30 and 1/A + 1/C = 1/24. Add all three: the left side becomes twice the combined rate, and the right side is 6/120 + 4/120 + 5/120 = 15/120 = 1/8. So 1/A + 1/B + 1/C = 1/16 — all three together would finish in 16 days. A's own rate is the combined rate minus the pair that excludes him, that is 1/16 − 1/30 = (15 − 8)/240 = 7/240. Inverting, A alone needs 240/7 days, which is 34 and 2/7 days.

• (a)35 2/7 days — 240/7 is 34 2/7, and 35 2/7 would be 247/7. It is the answer you write after a slip of one whole day in the final division.
• (b)37 1/7 days — This is 260/7, which corresponds to a rate of 7/260 — no combination of the three given pair-rates produces it.
• (d)33 2/7 days — 233/7, again a whole-day slip from the correct 240/7. Both this and option (a) sit one day either side of the answer, so an approximate calculation cannot separate them.

Time-and-work problems become linear the moment you switch from days to work per day. Rates add: if two people work together their rates sum. That converts a set of awkward statements about days into a small system of linear equations, and the standard three-pair system is solved by adding all three equations to get twice the total rate.

The bridge value here is the sixteen-day figure for all three together, and it is worth writing down even though the question never asks for it, because every individual rate is then one subtraction away. B alone would need 1/16 − 1/24 = 1/48, that is 48 days, and C alone 1/16 − 1/20 = 1/80, that is 80 days. Check: 1/34·29 + 1/48 + 1/80 does return 1/16. Fractional answers with a common denominator of seven are a signal in themselves — they tell you the arithmetic is meant to be exact, so a decimal approximation will not distinguish the options.

• Convert days to rates: n days means 1/n of the job per day, and rates of people working together add.
• Adding the three pair equations gives twice the combined rate; here 1/20 + 1/30 + 1/24 = 1/8, so the combined rate is 1/16.
• A's rate = combined rate − (B + C) rate = 1/16 − 1/30 = 7/240.
• A alone takes 240/7 = 34 2/7 days; B alone 48 days and C alone 80 days.
• The three of them working together would finish in 16 days.

• Adding or averaging the days themselves instead of the rates.
• Subtracting the wrong pair — A's rate needs the total minus B and C, not minus A and B.
• Rounding to a decimal when the four options differ by a single unit of one seventh.

The standard three-pair time-and-work item. Adding all three equations is the move that unlocks every version of it.

• practice — not a real PYQ

  A and B together finish a job in 12 days, B and C in 15 days and A and C in 20 days. How long would all three together take?

  • (a)8 days
  • (b)10 days
  • (c)12 days
  • (d)16 days

  Answer(b) 10 days — the three rates sum to 1/12 + 1/15 + 1/20 = 1/5, which is twice the combined rate, so the combined rate is 1/10.

• practice — not a real PYQ

  A alone can do a piece of work in 24 days and B alone in 12 days. Working together they will finish it in

  • (a)6 days
  • (b)8 days
  • (c)9 days
  • (d)18 days

  Answer(b) 8 days — the rates 1/24 and 1/12 add to 1/8.