CSAT Solved Papers/ 2024/Q7

2024 CSAT — Q7

Quant Arithmetic & numeracy 2.5 marks Easy

A certain number of men can complete a piece of work in 6k6k days, where kk is a natural number. By what percent should the number of men be increased so that the work can be completed in 5k5k days?

  1. A 10%
  2. B (50/3)%
  3. C 20% Answer
  4. D 25%

Worked rationale

Total work is fixed, so men ×\times days == constant: men ∝1days\propto \dfrac{1}{\text{days}}.

new menold men=old daysnew days=6k5k=65.\frac{\text{new men}}{\text{old men}} = \frac{\text{old days}}{\text{new days}} = \frac{6k}{5k} = \frac{6}{5}.

The kk cancels — it was never relevant. New men are 65\tfrac{6}{5} of the old, i.e. an increase of 15=20%\tfrac{1}{5} = 20\%.

Answer: (c) 20%.

Why the other options miss

  • A
    an arithmetic slip: takes the day-change 6k−5k6k⋅12\tfrac{6k-5k}{6k}\cdot\tfrac{1}{2} or otherwise halves the true fraction; a careless step on the right idea.
  • B
    the wrong base: computes the change the wrong way round — 6−56=16≈16.67%=(50/3)%\tfrac{6-5}{6} = \tfrac16 \approx 16.67\% = (50/3)\%. This is the increase you’d get if men were directly proportional to days (or if you used the new period as the base incorrectly). The seductive near-miss.
  • D
    right ratio, wrong base: uses 65\tfrac{6}{5} but reads it as a 25%25\% jump by taking 54\tfrac{5}{4}-style reasoning, or computes old days−new daysnew days\tfrac{\text{old days}-\text{new days}}{\text{new days}} inconsistently. Right ratio, wrong base.

Specialist insight

The instant you see “men and days,” write men ∝1/\propto 1/days and the problem is half-done — fewer days needs more men, so the ratio of new-to-old men is the reciprocal of the day ratio: 6k:5k6k:5k days ⇒\Rightarrow men ratio 6:56:5. The percentage trap is always which base: a 6→56\to5 change is −16-\tfrac16 on days but the men go 5→65\to6, a +15=20%+\tfrac15 = 20\% rise. Increase is always measured on the original quantity (the old men), so divide the change by 55, not 66. And note the examiner’s flourish — the parameter kk is pure decoration designed to make you hesitate; it cancels every time.

The trap, in one line

Measuring the increase on the wrong base — it is 15=20%\tfrac{1}{5} = 20\% (men go 5→65\to6), not 16=(50/3)%\tfrac{1}{6} = (50/3)\%.

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