Assistant Provident Fund Commissioner and Enforcement Officer / Accounts Officer (APFC and EO/AO) 2025 recruitment test. Transcribed from the Commission's own booklet (series A, PENA-A-PTS); answer worked out by our faculty — UPSC has published no answer key for this sitting. This question is in Part B. Answer last reviewed 25 August 2026.

If five persons take five hours to paint five walls of equal area, how many hours will 7 persons take to paint 7 walls of equal area, provided both the sets of walls are identical ?

(a)8
(b)7
(c)6
(d)5

Answer

Why

Correct — D, (d) 5. The number of hours does not change, because the number of painters and the number of walls change together.

The quickest route — one person, one wall. Five persons paint five walls in five hours. Since the walls are identical and of equal area, share the work out: each of the five painters takes one wall, and finishes it in five hours. So ONE person paints ONE wall in FIVE hours, and that is the fact the whole question rests on. Now give seven persons seven walls: each takes one wall and finishes it in five hours. They all work at the same time, so the job is done in five hours.

The formal route — person-hours. Work done is persons multiplied by hours. Five persons for five hours is 25 person-hours, and that produced five walls, so one wall costs 25 ÷ 5 = 5 person-hours. Seven walls therefore need 35 person-hours, and with seven persons working together the time is 35 ÷ 7 = 5 hours. The same answer, arrived at without the shortcut.

Why the answer feels wrong at first. The instinct is to reason 'five, five, five — so seven, seven, seven'. But the two sevens pull in OPPOSITE directions: more walls means more work, more painters means more capacity, and here they increase in exactly the same proportion, so they cancel. The rate per person is what is constant, and it stays five hours per wall no matter how many people are working. Any question of the form 'n workers do n units in n hours; how long do m workers take for m units' has the same answer as the original time, n hours.

The final clause 'provided both the sets of walls are identical' is there to close the only loophole — it guarantees that a wall in the second job is the same amount of work as a wall in the first.

Why the others are wrong

(a)8 — Eight hours suggests that the work grew faster than the workforce. There is no basis for that here: the walls and the painters both rise from five to seven, which is the same factor of seven-fifths on each side. A figure above five could only arise if the extra walls were larger or the extra painters slower, and the stem rules both out by saying the walls are of equal area and the two sets identical.
(b)7 — The trap the question is built around, and by far the most popular wrong answer. It comes from carrying the pattern of the sentence rather than its arithmetic: five, five, five becomes seven, seven, seven. Test it against the rate — seven persons painting for seven hours is 49 person-hours, but seven walls need only 35, so the job would be finished long before the seventh hour.
(c)6 — A compromise between five and seven, and compromises have no place in a problem where the rate is exactly determined. Six hours would mean seven painters producing 42 person-hours of work for a job requiring 35 — two walls' worth of effort wasted. When two changes cancel exactly, the answer is unchanged; there is nothing to split the difference between.

Concept

Time-and-work problems are solved by fixing the RATE. If one person completes a unit of work in T hours, then n persons working together complete n units in T hours, or one unit in T/n hours. The bookkeeping unit is the person-hour: total work equals persons multiplied by hours, so persons and time are inversely proportional for a fixed amount of work, and time is directly proportional to the amount of work for a fixed number of persons. Where both the workforce and the workload change, the two effects multiply, and where they change by the same factor they cancel exactly. The same reasoning covers pipes filling a tank, machines producing items, and workers of different efficiencies, which are handled by converting each worker into a rate and adding.

This is one of the best-known reasoning puzzles in circulation and it appears in recruitment papers precisely because the intuitive answer is wrong. The examiner has also mixed spelt-out numbers and figures in the same sentence, which does nothing to the mathematics but does encourage the pattern-matching that produces the wrong answer. Slowing down for one sentence — how long does ONE person take for ONE wall — defeats the whole construction.

This is one of the best-known puzzles in circulation, and it is set here because the intuitive answer is not merely wrong but confidently wrong. The paper reinforces the effect by mixing spelt-out numbers with figures in one sentence, which encourages pattern-matching over arithmetic. The remedy is a single question asked deliberately: how long does ONE person take for ONE wall? Once that is answered, the rest of the problem is trivial, and the same question opens every problem in the time-and-work family, including pipes filling tanks and machines producing items.

Key facts

Total work = number of persons × time, measured in person-hours.
Five persons painting five walls in five hours means one person paints one wall in five hours.
One wall costs 5 person-hours; seven walls cost 35 person-hours; seven persons take 35 ÷ 7 = 5 hours.
For a fixed amount of work, persons and time are inversely proportional.
For a fixed workforce, time is directly proportional to the amount of work.
When workload and workforce change by the same factor, the time is unchanged.
In general, n workers doing n units in n hours implies m workers do m units in n hours.
The equal-area and identical-walls conditions are what make the unit of work the same in both cases.

Study next

Time and work with workers of unequal efficiencyPipes and cisterns, including outlets working against inletsChain rule problems combining several proportional changesWork done in alternating shifts or with workers joining part-wayDirect and inverse proportion, and how to tell which applies

Common traps

Repeating the pattern of the numbers in the sentence instead of computing the rate.
Treating more workers and more work as pulling in the same direction.
Forgetting to reduce the data to one worker and one unit before scaling up.
Ignoring the clause that guarantees the two jobs are of equal size.

Time-and-work items are asked in every paper of this family, and this classic form appears regularly. Reduce everything to person-hours as the first step, and the whole family — including pipes, machines and mixed-efficiency teams — becomes a single technique. These items are usually placed early in the quantitative run and look trivial, which is exactly why they should be worked rather than read — the whole difficulty of the family is that its answers contradict a first impression.

Related PYQs

EPFO_EOAO_2020_Q82

If the ratio of speeds of ‘A’ and ‘B’ is 5 : 6 and ‘B’ allows ‘A’ a start of 70 metres in a 1·2 km race, who will win the race and by what distance ?

(a) ‘A’ wins by 30 m.
(b) ‘B’ wins by 200 m.
(c) ‘B’ wins by 130 m.
(d) The race finishes in a dead heat.

Answer(c) ‘B’ wins by 130 m.

A rate problem from the earlier paper worked on the same principle — a race in which two runners' speeds are in a given ratio and one is allowed a start, decided by comparing rates rather than totals.

Practice

practice — not a real PYQ
If 8 machines make 8 articles in 8 minutes, how many minutes will 20 machines take to make 20 articles ?
- (a)8
- (b)12
- (c)20
- (d)25
Answer(a) 8
practice — not a real PYQ
Six workers build a wall in 12 days. How many days will nine workers, working at the same rate, take to build a wall of the same size ?
- (a)6
- (b)8
- (c)9
- (d)18
Answer(b) 8

Why