Profit and loss questions, with answers

The questions, with answers

1. 1.What are the core profit and loss formulas, and which price is the percentage taken on?

   Profit = selling price - cost price; loss = cost price - selling price. Profit percent and loss percent are always on the cost price unless the question explicitly says otherwise: profit % = profit / CP × 100. Two rearrangements do most of the work: SP = CP × (100 + p) / 100 for a profit of p%, and SP = CP × (100 - l) / 100 for a loss of l%. A table bought for Rs 1,200 and sold for Rs 1,500 gives a profit of 300, which is 300 / 1200 × 100 = 25% — on the cost, not on the 1,500. Saying "of the selling price" out loud when a question does switch the base is worth marks.

2. 2.How do you find the cost price when you know the selling price and the profit or loss percent?

   Divide, don't subtract. If an item sold for Rs 1,150 at a 15% profit, then SP = CP × 1.15, so CP = 1150 / 1.15 = Rs 1,000. The common error is taking 15% off 1,150 (which gives 977.50), but 15% of the selling price is not the profit — the profit was 15% of the cost. For a loss the divisor is below one: sold at Rs 680 at a 15% loss means CP = 680 / 0.85 = Rs 800. The habit to build: write SP as a multiple of CP first, then solve.

3. 3.How do marked price, discount and profit fit together?

   Marked price is the label; discount is a percentage off the marked price; profit is measured against the cost price. Chain the multipliers. If a shopkeeper marks goods 25% above cost and gives a 12% discount, SP = CP × 1.25 × 0.88 = CP × 1.10, so the profit is 10%. Marking up 25% and discounting 12% is not a 13% profit — the discount is taken on the larger, marked figure. In reverse, to earn 20% after a 20% discount, the markup must satisfy m × 0.8 = 1.2, so m = 1.5: mark up 50%.

4. 4.Why does selling two items at the same price, one at x% profit and one at x% loss, always give a loss?

   Because the two cost prices are different. At the same selling price S, the profitable item cost S / (1 + x/100) and the loss-making item cost S / (1 - x/100), and the second is larger — the loss was on a bigger base than the profit. The net result is always a loss of (x / 10)² percent. Two items sold at Rs 990 each, one at 10% profit and one at 10% loss: costs 900 and 1,100, total cost 2,000 against total sales 1,980, a loss of 20, which is 1% — exactly (10 / 10)². The shortcut is safe to quote; the derivation is what earns the explanation marks.

5. 5.How do successive discounts combine?

   They multiply, they don't add. A 20% discount followed by a 10% discount leaves 0.8 × 0.9 = 0.72 of the price, a total discount of 28%, not 30%. The formula for two discounts a% and b% is a + b - ab / 100, here 20 + 10 - 2 = 28. The order of discounts does not matter for the final price, since multiplication commutes — a 10% then 20% chain also leaves 72%. The same rule covers a markup followed by a discount, and a percentage increase followed by a decrease: a 20% rise then a 20% fall is 1.2 × 0.8 = 0.96, a net 4% drop.

6. 6.How does a dishonest dealer using false weights make a profit?

   By selling less than he charges for. A dealer who claims to sell at cost price but uses a 900 g weight for a kilogram gives 900 g and collects the price of 1,000 g, so on every 900 g of cost he earns the price of 1,000 g. Profit percent = error / (true weight - error) × 100 = 100 / 900 × 100 = 11.11%, or equivalently 1000 / 900 = 10/9, a 1/9 gain. The base is what he actually gave (900), not what he claimed (1,000) — the same cost-price rule as everywhere else. If he also marks up by, say, 10%, chain it: 10/9 × 1.1 = 1.222, a 22.2% profit.

7. 7.How do you solve "had it been sold for Rs y more, the profit would have been z%"?

   The extra rupees equal the difference between the two profit percentages, taken on the cost price. An article sold at 10% profit; had it sold for Rs 60 more, the profit would have been 25%. The gap of 15 percentage points corresponds to Rs 60, so 15% of CP = 60 and CP = Rs 400; the original selling price was 440. The same template covers a loss turning into a profit — sold at a 5% loss, Rs 90 more would give a 10% profit means 15% of CP = 90, CP = 600. Always convert both situations to percentages of the same cost before subtracting.

8. 8.What is the profit percent when the cost price of a items equals the selling price of b items?

   Set both equal to one amount and compare per-item prices. If the cost price of 12 articles equals the selling price of 10, let that amount be Rs 60: cost per article 5, selling price per article 6, profit 1 on a cost of 5, so 20%. In general, profit % = (a - b) / b × 100 when a is the number costed and b the number sold for the same money; if a is smaller than b it is a loss, with the same formula giving a negative value. Here (12 - 10) / 10 × 100 = 20%. The trap is dividing by a instead of b — the base is the selling side's count because the cost per unit comes from it.

How the diagnostic asks it

One question from the Aptitude bank, exactly as a sitting would show it. The bank has 4 on profit & loss and 60 across Aptitude.