A reduction of 21% in the price of rice enables a person to buy 10.5 kg more for ₹ 1,000. What is the reduced price of rice per kg ?

Correct — option (1), ₹ 20.

Step 1 — find the money the price cut frees up. The rice that ₹ 1,000 bought before now costs 21% less.
Saving = 21% of ₹ 1,000 = ₹ 210

Step 2 — that ₹ 210 is spent at the reduced price and buys the extra 10.5 kg.
Reduced price = ₹ 210 ÷ 10.5 kg = ₹ 20 per kg

Why Step 1 works: the old quantity now costs 79% of ₹ 1,000 = ₹ 790. The rest of the budget, ₹ 1,000 − ₹ 790 = ₹ 210, pays for the additional rice.

Check with full working:
At ₹ 20 per kg, ₹ 1,000 buys 1,000 ÷ 20 = 50 kg
Before the cut, ₹ 1,000 bought 790 ÷ 20 = 39.5 kg
Extra = 50 − 39.5 = 10.5 kg, as the stem says

The original price was ₹ 20 ÷ 0.79, about ₹ 25.32 per kg; it is not among the options.

The idea to remember: with a fixed budget, the amount saved by the price cut, divided by the extra quantity, gives the new price.

Why the others are wrong

• (2)
  ₹ 30 — At ₹ 30 per kg, the ₹ 210 saved would buy 210 ÷ 30 = 7 kg extra, not the 10.5 kg the stem gives.

  ₹ 30 is also not the original price. Working back from ₹ 20 at a 21% cut gives about ₹ 25.32 per kg before the reduction.

• (3)
  ₹ 40 — At ₹ 40 per kg, the ₹ 210 saved would buy 210 ÷ 40 = 5.25 kg extra, half of the 10.5 kg in the stem.

  Check another way: ₹ 1,000 buys 25 kg at ₹ 40, while at the old price it bought 19.75 kg. The gain is 5.25 kg, not 10.5 kg.

• (4)
  ₹ 15 — At ₹ 15 per kg, the ₹ 210 saved would buy 210 ÷ 15 = 14 kg extra, more than the 10.5 kg the stem gives.

  A lower price means more extra rice for the same saving, so ₹ 15 overshoots. The price that gives exactly 10.5 kg is ₹ 20.

Expenditure = price × quantity. When the amount spent is fixed, quantity moves in inverse proportion to price: a lower price buys more, a higher price buys less.

If the price falls by r%, a fixed budget buys r/(100 − r) × 100% more. For a 21% cut this is 21/79, about 26.6% more rice: 39.5 kg becomes 50 kg.

The reverse case uses r/(100 + r) × 100%. If a price rises by 25%, consumption must fall by 25/125 = 20% to keep spending unchanged.

The percentage saving on a fixed budget is an amount of money, and that money is spent at the new price.

Key facts

• Expenditure = price × quantity; with expenditure fixed, quantity changes in inverse proportion to price.
• A price cut of r% lets a fixed budget buy r/(100 − r) × 100% more; a 21% cut gives about 26.6% more.
• A price rise of r% needs a consumption cut of r/(100 + r) × 100% to keep spending unchanged; a 25% rise needs a 20% cut.
• In this item ₹ 1,000 buys 39.5 kg before the cut and 50 kg after it, at ₹ 20 per kg.
• Dividing the saving (₹ 210) by the extra quantity (10.5 kg) gives the reduced price, because the extra rice is bought at that price.

Common traps

• Calling ₹ 210 ÷ 10.5 = ₹ 20 the original price. The extra 10.5 kg is bought at the new price, so ₹ 20 is the reduced price; the original was about ₹ 25.32.
• Treating a 21% price cut as 21% more rice. Quantity is inversely proportional to price, so it becomes 1 ÷ 0.79 of the old amount, a rise of 21/79, about 26.6%.
• Taking 21% of a price per kg instead of 21% of the ₹ 1,000 spent. The money that buys the extra rice is 21% of the whole budget.

UnlockIAS will link similar questions from RAS Pre 2016 here once that paper is published on this site.

• practice — not a real PYQ

  A 20% fall in the price of sugar enables a buyer to get 5 kg more for ₹ 800. What was the original price of sugar per kg?

  • (a)₹ 32
  • (b)₹ 40
  • (c)₹ 38.40
  • (d)₹ 36
  Answer(2) — Saving = 20% of ₹ 800 = ₹ 160; reduced price = 160 ÷ 5 = ₹ 32; original = 32 ÷ 0.8 = ₹ 40.

  Option (1) is the reduced price; option (3) adds 20% to ₹ 32 instead of dividing by 0.8; option (4) fits no step.

• practice — not a real PYQ

  If the price of petrol rises by 25%, by what percentage must a car owner reduce consumption so that the spending on petrol does not change?

  • (a)25%
  • (b)20%
  • (c)15%
  • (d)33⅓%
  Answer(2) — New consumption = 1 ÷ 1.25 = 0.8 of the old, a cut of 25/125 = 20%. Option (1) repeats the price rise; option (4) is 25/75, the formula for a price fall; option (3) fits no step.