A cricket bat is purchased at 20% discount. If the selling price of the bat is ₹ 1,000, what was the original price of the bat?

Correct — D, (d) ₹ 1,250. A discount is always a percentage of the price it is taken off — here the original price of the bat — and never of the amount finally paid. That is the step that decides the item. If the discount is 20% of the original, then what the purchaser pays is the remaining 80% of it, and the ₹ 1,000 in the stem is therefore eighty per cent of the number being asked for, not a hundred per cent of it.

Writing the original price as M, we have 0·80 M = 1,000, so M = 1,000 ÷ 0·80 = ₹ 1,250, which is option (d). The unitary form of the same reasoning is quicker in the examination hall and less prone to slips: if 80 parts are worth ₹ 1,000 then one part is worth ₹ 12·50, and a hundred parts are worth ₹ 1,250.

The forward check confirms it. Twenty per cent of ₹ 1,250 is ₹ 250; taking that off leaves ₹ 1,000, which is exactly what the stem says the bat sold for. Notice how the two figures are related: the discount of ₹ 250 is twenty per cent of the original price but twenty-five per cent of the price actually paid. The same rupee amount answers to two different percentages depending on which figure it is compared with, and that is the entire content of this question. The vocabulary is worth fixing at the same time — the price before the reduction is the marked or list price, the reduction is the discount, and what the customer pays is the selling price. Discount percentages attach to the first of the three.

Why the others are wrong

• (a)₹ 1,100 — Eleven hundred is the option for a candidate who feels that the answer should be somewhat above ₹ 1,000 and takes the smallest of the offered figures that is. Nothing in the arithmetic produces it. The forward test is a single line: twenty per cent of ₹ 1,100 is ₹ 220, so a bat originally priced at ₹ 1,100 would sell after the discount for ₹ 880, and the stem says it sold for ₹ 1,000. The habit this option rewards is worth naming, because it is a general one — on a numerical item where the four options are all of the right order of magnitude, a candidate who has not completed the calculation will drift towards the option nearest to a rough estimate. Since a discount of a fifth is a large one, a rough estimate should already have suggested a figure well above ₹ 1,200, and this option is not in that range at all.
• (b)₹ 1,200 — This is the mistake the item is built to catch: adding twenty per cent to the selling price instead of dividing by 0·80. Twenty per cent of ₹ 1,000 is ₹ 200, and ₹ 1,000 plus ₹ 200 gives the ₹ 1,200 offered here. The step is invalid because the two percentages have different bases — the discount is twenty per cent of the original price, whereas adding twenty per cent to the selling price takes twenty per cent of the smaller figure. The forward check makes the shortfall visible: a bat priced at ₹ 1,200 and discounted by twenty per cent would sell for ₹ 960, which is ₹ 40 short of the stated selling price. The correct relation between the two figures is that the discount is a quarter of the selling price when it is a fifth of the original, which is why the answer is ₹ 1,250 and not ₹ 1,200.
• (c)₹ 1,225 — Twelve hundred and twenty-five is a near miss, placed between the incorrect ₹ 1,200 and the correct ₹ 1,250 so that a candidate who has estimated rather than calculated has somewhere plausible to land. It fails the forward test like the others: twenty per cent of ₹ 1,225 is ₹ 245, leaving a selling price of ₹ 980 rather than ₹ 1,000. The presence of two options only twenty-five rupees apart is itself a signal that the examiner expects the arithmetic to be done exactly, because no estimate can separate them. When an option set is spaced this closely, a candidate should treat it as an instruction to compute rather than to reason approximately — and here the exact computation is nothing worse than dividing a thousand by four fifths.

Commercial arithmetic runs on three prices and the percentages that connect them. The cost price is what the seller paid. The marked price, also called the list or original price, is what the article is labelled at. The selling price is what the buyer actually pays. A discount is a reduction from the marked price, expressed as a percentage of that marked price, so selling price = marked price × (1 − discount rate). Profit and loss, by contrast, are reckoned on the cost price: profit percentage = (selling price − cost price) ÷ cost price × 100. Keeping the two bases apart is the whole discipline of the topic, because the same rupee difference can be quoted as one percentage of the marked price and quite a different percentage of the cost or selling price. When a problem gives the final figure and asks for an earlier one, the operation is division by the multiplier, not subtraction of the percentage — reversing a twenty per cent discount means dividing by 0·8, which multiplies by 1·25, so the original price is a quarter above the discounted price and not a fifth above it. Successive discounts follow the same rule: two discounts of ten per cent one after the other multiply the price by 0·9 twice, giving 0·81, which is a net reduction of nineteen per cent rather than twenty.

Key facts

• A discount is a percentage of the marked or original price, never of the price finally paid.
• Selling price = marked price × (1 − discount rate), so a discount of 20% leaves the buyer paying 80% of the marked price.
• Here 0·80 M = 1,000, so the original price M = 1,000 ÷ 0·80 = ₹ 1,250.
• By the unitary method, 80 parts are ₹ 1,000, one part is ₹ 12·50 and a hundred parts are ₹ 1,250.
• The discount of ₹ 250 is 20% of the original price but 25% of the selling price — the same amount, two different bases.
• Reversing a discount of 20% means dividing by 0·8, which is the same as multiplying by 1·25.
• Two successive discounts of 10% multiply the price by 0·81, a net reduction of 19% and not 20%.
• Profit and loss percentages are reckoned on the cost price, while discount percentages are reckoned on the marked price.

Common traps

• Adding the discount percentage back to the selling price instead of dividing by the complementary multiplier.
• Confusing the base — a discount is on the marked price, while profit and loss are on the cost price.
• Assuming two successive discounts add up; ten per cent twice is nineteen per cent, not twenty.
• Estimating when two options are only twenty-five rupees apart, which is the examiner's signal to calculate exactly.
• Forgetting to run the answer forwards through the story, which rejects every wrong option here in one line each.

• practice — not a real PYQ

  After allowing a discount of 15% on the marked price, a shopkeeper sells an article for ₹ 2,550. What is the marked price of the article?

  • (a)₹ 2,850
  • (b)₹ 2,933
  • (c)₹ 3,000
  • (d)₹ 3,060

  Answer(c) ₹ 3,000

• practice — not a real PYQ

  An article is marked at ₹ 800 and two successive discounts of 10% each are allowed on it. The price paid by the customer is

  • (a)₹ 640
  • (b)₹ 648
  • (c)₹ 660
  • (d)₹ 680

  Answer(b) ₹ 648