There was a hike in petrol price by 12%. By how much percentage should a person decrease his petrol consumption such that there is no change in his expenditure on petrol?

Correct — C, 10·7%. Expenditure is price multiplied by quantity. If the price rises from 100 to 112 and the bill is to stay the same, the quantity must fall to 100/112 of its old value, that is to 89.29 per cent. The cut is therefore 100 − 89.29 = 10.71 per cent, which rounds to the printed 10·7. The formula worth carrying is: for a price rise of R per cent, the consumption cut that keeps the bill unchanged is 100R/(100 + R), here 1200/112 = 10.71.

• (a)8·6% — 8.6 per cent is close to 12/140 and matches no step of this calculation. Cutting consumption by 8.6 per cent would leave the bill about 2.4 per cent higher than before.
• (b)9·8% — A cut of 9.8 per cent leaves 90.2 per cent of the old quantity, and 1.12 × 0.902 is 1.010 — the bill still rises by about one per cent.
• (d)12% — This is the trap the item is built on: a 12 per cent rise is not undone by a 12 per cent cut, because the two percentages are taken on different bases. 1.12 × 0.88 = 0.9856, so the bill would fall by 1.44 per cent.

A percentage increase and the percentage decrease that reverses it are never equal, because each is measured on a different base. If a quantity rises by R per cent, restoring the original value needs a fall of 100R/(100 + R) per cent. The mirror rule applies to a fall of R per cent, which needs a rise of 100R/(100 − R) per cent to be undone.

The clean way is to fix the expenditure at 100 units and let the price be the multiplier. New price 112, same bill 100 means the quantity index falls from 1 to 100/112 = 0.8929. Quoting the change as a fraction of the old quantity gives 10.71 per cent, not 12. Reading the printed options helps: three of them cluster around ten per cent and one repeats the 12 from the stem, which is the tell that the examiner is testing exactly this asymmetry.

• For a price rise of R per cent, the consumption cut that leaves expenditure unchanged is 100R/(100 + R) per cent.
• A 12 per cent rise needs a cut of 1200/112 = 10.71 per cent.
• A rise of R per cent followed by a fall of R per cent leaves the value lower by R²/100 per cent.
• For a 25 per cent rise the cut is 20 per cent; for a 100 per cent rise it is 50 per cent.
• Expenditure equals price multiplied by quantity, so holding it fixed makes the two inversely proportional.

• Answering 12 per cent, on the assumption that the same percentage reverses the change.
• Computing 12/112 and forgetting to express it as a percentage.
• Taking the cut on the new consumption rather than the old.

A percentage item that offers the number from the stem as a distractor, testing whether a candidate knows that increases and decreases are not symmetric.

• practice — not a real PYQ

  The price of a commodity rises by 25 per cent. By how much must consumption fall to keep expenditure unchanged?

  • (a)16.67 per cent
  • (b)20 per cent
  • (c)25 per cent
  • (d)33.33 per cent

  Answer(b) 20 per cent — 100 × 25 ÷ 125 = 20.

• practice — not a real PYQ

  A number is increased by 20 per cent and then decreased by 20 per cent. The net change is

  • (a)no change
  • (b)a 4 per cent fall
  • (c)a 4 per cent rise
  • (d)a 2 per cent fall

  Answer(b) a 4 per cent fall — 1.20 × 0.80 = 0.96, and the loss is R²/100 = 4 per cent.