Percentage aptitude questions, with answers

The questions, with answers

1. 1.Which fraction-to-percentage conversions should you know by heart?

   Enough that you never reach for long division: 1/2 = 50%, 1/3 = 33⅓%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16⅔%, 1/7 ≈ 14.28%, 1/8 = 12.5%, 1/9 = 11⅑%, 1/10 = 10%, 1/11 ≈ 9.09%, 1/12 = 8⅓%, 1/16 = 6.25%, 1/20 = 5%. Their multiples follow: 3/8 = 37.5%, 5/6 = 83⅓%, 7/12 = 58⅓%. Most percentage questions are written so the numbers land on one of these; recognising 37.5% as three-eighths turns a calculation into a glance.

2. 2.How do you handle successive percentage changes?

   Two successive changes of a% and b% give a net change of a + b + ab/100, with signs carried. It is not a + b, because the second change applies to the already-changed amount.

   Example: a price rises 25% and then falls 20%. Net = 25 − 20 + (25 × −20)/100 = 5 − 5 = 0%. The price is back where it started. The same formula works for area when both sides of a rectangle change: 10% longer and 10% shorter gives 10 − 10 − 1 = −1%, a smaller area.

3. 3.Why doesn't a 25% increase followed by a 25% decrease bring you back to the start?

   Because the decrease is taken from a larger base. 100 becomes 125 after a 25% rise; 25% of 125 is 31.25, so the fall lands at 93.75. To undo a rise of x%, the fall must be x / (100 + x) × 100 percent: undoing +25% needs −20%, undoing +50% needs −33⅓%, undoing +100% needs −50%. The general result: a rise and fall of the same percentage always leaves you below the start, by (x/10)² percent — 25% each way loses 6.25%.

4. 4.If A is x% more than B, by what percentage is B less than A?

   By x / (100 + x) × 100, not x. If A is 25% more than B, take B = 100 and A = 125; B is 25 less on a base of 125, which is 20%. The mirror statement: if A is x% less than B, then B is x / (100 − x) × 100 more than A — 20% less one way is 25% more the other. Whenever a question changes which quantity is the base, recompute; the percentage travels with the base.

5. 5.If the price of a commodity rises by x%, by how much must consumption fall to keep spending the same?

   By x / (100 + x) × 100 percent — the same expression as the previous question, because it is the same situation in disguise: spending = price × quantity, and to hold the product fixed the quantity must shrink by the factor the price grew. A 25% price rise needs a 20% cut in consumption; a 50% rise needs a 33⅓% cut. If instead the price falls by x%, consumption can rise by x / (100 − x) × 100 for the same spend.

6. 6.How do population growth and depreciation questions work?

   Both are compounding: after n periods at r% per period, the amount is P × (1 + r/100)ⁿ for growth and P × (1 − r/100)ⁿ for depreciation. For two periods, (1 + r/100)² = 1 + 2r/100 + (r/100)², which is why 10% growth for two years is 21%, not 20%.

   Example: a town of 8,000 grows 10% a year. After two years: 8,000 × 1.1 × 1.1 = 8,000 × 1.21 = 9,680. Working backwards — "the population is now 9,680 after two years of 10% growth" — divide by 1.21.

7. 7.What is the difference between a percentage change and a change in percentage points?

   If a pass rate moves from 10% to 15%, it has risen by 5 percentage points but by 50 percent, because 5 is half of the original 10. Questions and news headlines both exploit the ambiguity. When a question says "increased by 5%" about a quantity that is itself a percentage, check the options: if one of them is 10.5% the setter meant a relative change; if one is 15% they meant points. Interviewers who ask this in person want to hear you name the two readings.

8. 8.How do you solve pass-mark questions?

   Turn each statement into an equation with the maximum marks M as the unknown, then subtract. "A scores 30% and fails by 20 marks" means pass mark = 0.30M + 20. "B scores 40% and passes by 10 marks" means pass mark = 0.40M − 10. Equate: 0.30M + 20 = 0.40M − 10, so 0.10M = 30 and M = 300. Pass mark = 0.30 × 300 + 20 = 110. Check with B: 0.40 × 300 − 10 = 110. The pattern generalises to any "same target, two percentages, two gaps" question: the difference in percentages equals the sum of the gaps.

How the diagnostic asks it

One question from the Aptitude bank, exactly as a sitting would show it. The bank has 2 on percentages and 30 across Aptitude.