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 x percent of a is same as y percent of b, then what percent of a is z percent of b ?

(a)zx/y
(b)xy/z
(c)yz/x
(d)x/yz

Answer

Why

Correct — A, (a) zx/y. That is, z per cent of b is (zx/y) per cent of a.

Translate the given condition into an equation. 'x percent of a is same as y percent of b' means (x/100) × a = (y/100) × b. The hundreds cancel, leaving xa = yb, and therefore b = xa/y.

Now work out what is asked. 'z percent of b' is (z/100) × b = (z/100) × (xa/y) = (zx/y) × (a/100).

Read the last expression carefully. It says that the quantity equals (zx/y) hundredths of a — and 'p hundredths of a' is exactly what 'p per cent of a' means. So z per cent of b is (zx/y) per cent of a, which is option (a).

Check it with numbers, which is the fastest way to be sure in an algebraic option set. Take x = 10, y = 20 and a = 100. Then 10 per cent of 100 is 10, and for 20 per cent of b to equal 10 we need b = 50. Now take z = 40: forty per cent of b is 40 per cent of 50 = 20, and 20 is 20 per cent of a = 100. The formula gives zx/y = 40 × 10 ÷ 20 = 20. It matches.

That substitution method is worth adopting for every algebraic option set. Choose small, distinct numbers — never equal ones, which would make several options agree — compute the required quantity directly, then test the four expressions and keep the one that reproduces it.

A second substitution guards against a lucky first one. Take x = 20, y = 5 and a = 50: then 20 per cent of 50 is 10, so 5 per cent of b must be 10 and b = 200. With z = 3, three per cent of b is 6, and 6 is 12 per cent of a = 50. The formula gives zx/y = 3 × 20 ÷ 5 = 12, matching again — while xy/z gives 33·3, yz/x gives 0·75 and x/yz gives 1·33, all wrong. Two independent substitutions leave no doubt.

Why the others are wrong

(b)xy/z — This puts z in the denominator, when z is the percentage being APPLIED and must appear on top: doubling z doubles the quantity in question, so the answer must be directly proportional to z, not inversely. Under the numerical test — x = 10, y = 20, z = 40 — it gives 10 × 20 ÷ 40 = 5, where the true answer is 20.
(c)yz/x — Has x and y the wrong way round. The relation b = xa/y makes b LARGER when x is larger and SMALLER when y is larger, so the final expression must carry x on top and y underneath. This option inverts that, and the same numbers give 20 × 40 ÷ 10 = 80 instead of 20. It is the likeliest slip for a candidate who forms the equation correctly and then transposes in haste.
(d)x/yz — Printed as a single fraction with the PRODUCT yz underneath, so it means x divided by the product of y and z — which puts z in the denominator, as in option (b), and reduces the answer as z grows. With the same numbers it gives 10 ÷ 800 = 0·0125. It is also the only option in which the numerator carries a single letter, and dimensional sense alone rules it out: the answer has to grow with both x and z.

Concept

A percentage is a fraction with denominator one hundred, so 'p per cent of q' is simply pq/100. Once every phrase in a problem is rewritten that way, percentage questions become ordinary algebra, and the hundreds usually cancel. Two habits make this family reliable. First, translate each clause into an equation as you read it rather than after reading the whole question. Second, when the options are algebraic expressions, substitute small numbers and test — a numerical check is faster and safer than re-deriving, and it catches transposition errors that are almost invisible in symbols. The same approach handles 'A is what per cent of B', successive percentage changes, and percentage-based comparisons of two quantities.

Percentage items appear several times in every paper in this family, and the algebraic form used here is the version that separates candidates who can manipulate the relation from those who have learnt only numerical drills. The four options are all built from the same three letters, so nothing can be eliminated by inspection alone — but a single substitution disposes of three of them in seconds.

Algebraic option sets built from the same three letters cannot be narrowed by inspection, so the technique that pays is substitution with small distinct numbers. It is faster than deriving the expression symbolically, and it is far more reliable, because a transposition error is invisible in symbols and glaring in numbers. Choose values that are unequal and not too small — 10, 20 and 40 rather than 1, 2 and 3 — so that no two options coincide by accident, and compute the required quantity directly before testing any expression against it.

Key facts

'p per cent of q' means pq/100.
From (x/100)a = (y/100)b it follows that xa = yb and b = xa/y.
z per cent of b = (z/100)(xa/y) = (zx/y) × (a/100), which is (zx/y) per cent of a.
The answer must be directly proportional to both x and z, and inversely proportional to y.
A numerical substitution — x = 10, y = 20, a = 100, z = 40 — gives 20, matching zx/y.
Choose distinct small numbers when testing options, so that no two expressions coincide by accident.
x/yz denotes x divided by the product yz, not (x/y) multiplied by z.
In percentage problems the factors of 100 almost always cancel, so they can be carried lightly.

Study next

Percentage increase and decrease, and successive changesExpressing one quantity as a percentage of another, in both directionsPercentage change and its effect on ratiosProfit, loss and discount as applications of percentageSubstitution as a technique for algebraic multiple-choice questions

Common traps

Placing the applied percentage z in the denominator.
Interchanging x and y when transposing the first equation.
Misreading a two-letter denominator as a product in the numerator.
Testing options with numbers that are equal or too small to distinguish them.

Percentage questions in EPFO papers come in both numerical and algebraic dress. The algebraic ones are best attacked by substitution; the numerical ones by translating each clause into an equation as it is read. Either way, write the relation down before looking at the options.

Related PYQs

EPFO_EOAO_2017_Q112

In a city, 80% population eat rice and 90% of the rice eaters are non-vegetarians. Then what percent of the population are vegetarian rice eaters?

(a) 7·2
(b) 8
(c) 9
(d) 10

Answer(b) 8

A percentage problem from the earlier paper worked by the same translation of each clause into an equation — what proportion of a city's population are vegetarian rice eaters, given two overlapping percentages.

Practice

practice — not a real PYQ
If 20 percent of p equals 30 percent of q, then q is what percent of p ?
- (a)50
- (b)66 2/3
- (c)120
- (d)150
Answer(b) 66 2/3
practice — not a real PYQ
If a percent of m equals b percent of n, then n is equal to which one among the following ?
- (a)abm
- (b)am/b
- (c)bm/a
- (d)ab/m
Answer(b) am/b

Why