If 35% of a number is 416 more than 27% of the same number, then the number is

Q: What is the answer to CAPF 2021 Paper I Q98?

A. Correct — A, 5200. Both percentages are taken of the same number, so they can be subtracted directly. 35 per cent less 27 per cent is 8 per cent, and that 8 per cent is the stated excess of 416. One per cent is therefore 416 ÷ 8 = 52, and the whole number is 52 × 100 = 5,200. Verifying: 35 per cent of 5,200 is 1,820, 27 per cent is 1,404, and 1,820 − 1,404 = 416. (Provisional — pending the official UPSC key.)

(a)5200
(b)2600
(c)3900
(d)3328

Answer

Why

Correct — A, 5200. Both percentages are taken of the same number, so they can be subtracted directly. 35 per cent less 27 per cent is 8 per cent, and that 8 per cent is the stated excess of 416. One per cent is therefore 416 ÷ 8 = 52, and the whole number is 52 × 100 = 5,200. Verifying: 35 per cent of 5,200 is 1,820, 27 per cent is 1,404, and 1,820 − 1,404 = 416.

Why the others are wrong

(b)2600 — Exactly half the answer, and it halves the gap too — 35 per cent of 2,600 is 910, 27 per cent is 702, and the difference is 208.
(c)3900 — Three-quarters of the answer. It gives 1,365 and 1,053, a gap of 312 rather than 416.
(d)3328 — This is 416 multiplied by 8 instead of divided by it — the classic reversal when converting 8 per cent back to the whole.

Concept

Percentages of one and the same base behave like ordinary numbers, so a difference of percentages is itself a percentage of that base. Turning the sentence into the equation 0.35x − 0.27x = 416 removes every step that could go wrong, and 0.08x = 416 is solved by a single division.

The one move that matters is recognising that both percentages sit on the same number. Once that is seen, the arithmetic is trivial and the only remaining risk is the direction of the final division. Multiplying 416 by 8 gives 3,328, which is sitting there as option (d) waiting for exactly that slip. A quick sanity test settles it: 8 per cent of a number is small, so the number must be much larger than 416, not four-fifths of eight times it.

Key facts

35 per cent − 27 per cent = 8 per cent of the same number.
8 per cent of x = 416, so 1 per cent of x = 52 and x = 5,200.
The check runs 1,820 − 1,404 = 416.
Multiplying rather than dividing by 8 produces 3,328, the planted distractor.
One per cent of the answer is a whole number here, which is a signal the intended path is the short one.

Same base, so subtract the percentages

Sentence→35% of x exceeds 27% of x by 416

Equation→0.35x − 0.27x = 416

Collect→0.08x = 416

Solve→x = 416 ÷ 0.08 = 5,200

Verify→1,820 − 1,404 = 416

Divide by 8 per cent, not multiply by 8 — the second route gives 3,328, which is option (d).

Study next

Translating percentage sentences into linear equationsPercentage increase and decrease on a common baseSuccessive percentage changes and their combined effectProblems where one quantity is expressed as a percentage of anotherRatio problems solved by finding the value of one unit

Common traps

Multiplying by 8 instead of dividing by 8 per cent.
Treating the two percentages as though they sat on different bases.

A one-equation percentage item whose only difficulty is the direction of the last division.

Related PYQs

No directly related past PYQ was found.

Practice

practice — not a real PYQ
If 40% of a number exceeds 25% of the same number by 90, the number is
- (a)450
- (b)600
- (c)540
- (d)360
Answer(b) 600 — the gap is 15 per cent, so 0.15x = 90 and x = 600.
practice — not a real PYQ
12% of a number is 96. What is 30% of the same number?
- (a)240
- (b)260
- (c)280
- (d)320
Answer(a) 240 — one per cent is 8, so the number is 800 and 30 per cent of it is 240.