A number is 124 more than its one-third. What is that number?

Q: What is the answer to CAPF 2022 Paper I Q31?

D. Correct — D, 186. Let the number be x. 'Is 124 more than its one-third' says x = x/3 + 124. Subtract x/3 from both sides: 2x/3 = 124, so x = 124 × 3 ÷ 2 = 186. Check it against the stem — one-third of 186 is 62, and 186 − 62 = 124, exactly as stated.

(a)194
(b)180
(c)189
(d)186

Answer

Why

Correct — D, 186. Let the number be x. 'Is 124 more than its one-third' says x = x/3 + 124. Subtract x/3 from both sides: 2x/3 = 124, so x = 124 × 3 ÷ 2 = 186. Check it against the stem — one-third of 186 is 62, and 186 − 62 = 124, exactly as stated.

Why the others are wrong

(a)194 — One-third of 194 is not a whole number, and 194 − 194/3 works out to about 129.3, not 124.
(b)180 — 180 − 60 = 120, four short of the required difference.
(c)189 — 189 − 63 = 126, two more than the required difference. Close enough to be tempting if you approximate instead of solving.

Concept

A single-sentence number problem becomes a one-line linear equation once 'more than' is read as addition and 'its one-third' as x/3. The whole difficulty is translation, not algebra.

The fastest route in an examination hall is to work backwards from the options, since each is a small integer: subtract a third and see which leaves 124. That test kills three options in a few seconds. Solving forward is just as quick here, and is safer when the options are not clean.

Key facts

x = x/3 + 124 rearranges to (1 − 1/3)x = 124, that is 2x/3 = 124, giving x = 186.
In general, if a number exceeds its 1/n part by d, then x = dn/(n − 1).
'A is k more than B' means A − B = k; 'A is k times B' means A = kB. Mixing the two is the commonest translation error.
Back-substitution into the original sentence is a complete check and costs one subtraction.

Translate, solve, verify

Sentence→A number is 124 more than its one-third

Equation→x = x/3 + 124

Collect terms→x − x/3 = 124, so 2x/3 = 124

Solve→x = 124 × 3/2 = 186

Verify→186/3 = 62 and 186 − 62 = 124

The two-thirds is the whole trick — the answer is 124 divided by 2/3, not multiplied by 3.

Study next

Forming linear equations from word statementsPercentage equivalents of common fractions and the reverse conversionProblems where a number exceeds a fraction or a percentage of itselfBack-substitution as an option-elimination techniqueRatio and proportion applied to parts of a whole

Common traps

Multiplying 124 by 3 and answering 372, which solves the wrong equation.
Reading 'more than its one-third' as 'one-third more than'.
Choosing 189 because it is close, without checking that the difference is exactly 124.

Asked as a one-sentence relation between a number and a fraction or percentage of itself, with the number as the target.

Related PYQs

CAPF_GAI_2021_Q982021Open & attempt →

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

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

Answer(a) 5200

The same move in percentage clothing — one part of a number is compared with another part of the same number, the difference is given, and the number follows from a single division. There the gap is 8 per cent; here it is two-thirds.

Practice

practice — not a real PYQ
A number is 60 more than its two-fifths. The number is
- (a)80
- (b)100
- (c)120
- (d)150
Answer(b) 100 — x − 2x/5 = 60 gives 3x/5 = 60, so x = 100.
practice — not a real PYQ
If 40% of a number exceeds 25% of the same number by 45, the number is
- (a)240
- (b)280
- (c)300
- (d)320
Answer(c) 300 — the gap is 15% of the number, so 0.15x = 45 and x = 300.