Profit, loss and discount for CDS

Profit and loss is percentage wearing a shopkeeper's apron. Almost every question in this topic comes down to one thing: knowing which price is the base for each percentage. Get the base right and the question is two lines. Get it wrong and you land on a wrong option that UPSC has placed there for exactly that slip.

CDS papers usually carry a few questions from this area, sometimes mixed with percentage or ratio. They reward a candidate who thinks in multipliers, as explained in the percentage guide, rather than one who memorises a separate formula for every story.

The three prices and their bases

The rules follow from the table.

• Profit = SP − CP. Profit % = profit ÷ CP × 100.
• Loss = CP − SP. Loss % = loss ÷ CP × 100.
• Discount = MP − SP. Discount % = discount ÷ MP × 100.

Multipliers do most of the work

A profit of r% means SP = CP × (1 + r/100). A loss of r% means SP = CP × (1 − r/100). A discount of d% means SP = MP × (1 − d/100).

Put the last two together and you get the link between marked price and cost price:

• MP × (1 − d/100) = CP × (1 + r/100)
• So MP ÷ CP = (100 + r) ÷ (100 − d)

This single line answers every "how much above cost should the shopkeeper mark the goods" question.

Shortcuts, and why they work

Successive discounts

Two discounts of a% and b% act like one discount of a + b − ab/100 per cent. Why: the second discount is taken on an already reduced price. The multipliers are (1 − a/100) and (1 − b/100), and their product is 1 − (a + b)/100 + ab/10,000. So 20% and 10% together make 28%, not 30%. The order does not matter, because multiplication does not care about order.

Same selling price, same percentage profit and loss

If two articles are sold at the same price, one at r% profit and the other at r% loss, the overall result is always a loss of r²/100 per cent. Why: the article sold at a loss had a higher cost price, so its loss in rupees is larger than the other's profit in rupees.

Articles bought and sold

If the selling price of x articles equals the cost price of y articles (y > x), the profit % is (y − x) ÷ x × 100. Why: take the CP of one article as ₹1. Then x articles sell for ₹y against a cost of ₹x.

False weights

A dealer who claims to sell at cost price but gives less weight gains (true weight − false weight) ÷ false weight × 100 per cent. Why: the buyer pays for 1,000 g, but the dealer's cost is only for the smaller weight actually handed over. The base is the false weight, because that is what the dealer really spent money on.

Worked questions

Question 1: A shopkeeper marks goods 40% above cost price and allows a discount of 10%. Find the profit per cent.

• Take CP = 100. MP = 140. SP = 140 × 0.9 = 126.
• Profit = 26%. In multipliers: 1.4 × 0.9 = 1.26.

Question 2: A trader wants a profit of 20% after giving a discount of 20%. How far above cost should the goods be marked?

• MP ÷ CP = (100 + 20) ÷ (100 − 20) = 120 ÷ 80 = 1.5.
• The goods should be marked 50% above cost.

Question 3: Two watches are sold for ₹990 each, one at a 10% profit and the other at a 10% loss. Find the overall gain or loss.

• CP of the first = 990 ÷ 1.1 = 900. CP of the second = 990 ÷ 0.9 = 1,100.
• Total CP = 2,000 and total SP = 1,980, so the loss is ₹20, which is 1%.
• The shortcut agrees: r²/100 = 100/100 = 1% loss.

Question 4: The selling price of 12 articles equals the cost price of 15 articles. Find the profit per cent.

• Let the CP of one article be ₹1. Then 12 articles cost ₹12 and sell for ₹15.
• Profit = 3 ÷ 12 × 100 = 25%.

Question 5: A dealer sells at 10% above cost price and also uses a 900 g weight in place of 1 kg. Find the total profit per cent.

• For every 1,000 g the buyer pays 1.1 times the cost of 1,000 g. The dealer spends only the cost of 900 g.
• Profit multiplier = 1.1 × 1,000 ÷ 900 = 1,100 ÷ 900 = 11/9.
• Profit = 2/9 × 100 = 22 2/9% (about 22.2%).

Question 6: By selling an article for ₹1,140, a seller loses 5%. At what price should it be sold to gain 5%?

• CP = 1,140 ÷ 0.95 = 1,200.
• Required SP = 1,200 × 1.05 = ₹1,260.
• A faster route: SP for 5% gain ÷ SP for 5% loss = 105 ÷ 95, so the answer is 1,140 × 105 ÷ 95 = 1,260.

Traps that cost marks

Practice set

1. An article costs ₹1,200 and is sold at a loss of 15%. Find the selling price.
2. The marked price of an item is ₹800 and the discount is 25%. Find the selling price.
3. Find the single discount equal to successive discounts of 20% and 10%.
4. A trader buys oranges at 5 for ₹10 and sells them at 4 for ₹10. Find the profit per cent.
5. An article sold at ₹640 gives a loss of 20%. At what price should it be sold to gain 20%?
6. Goods are marked 25% above cost and sold at a discount of 12%. Find the profit per cent.
7. A dealer claims to sell at cost price but uses an 800 g weight for 1 kg. Find the profit per cent.
8. A shopkeeper earns a profit equal to 20% of the selling price. What is the profit as a percentage of the cost price?

Answers

1. ₹1,020. 1,200 × 0.85 = 1,020.
2. ₹600. 800 × 0.75 = 600.
3. 28%. 20 + 10 − 200/100 = 28. Or 0.8 × 0.9 = 0.72.
4. 25%. CP per orange is ₹2 and SP per orange is ₹2.50, so the profit is 0.5 ÷ 2 × 100.
5. ₹960. CP = 640 ÷ 0.8 = 800, and 800 × 1.2 = 960.
6. 10%. 1.25 × 0.88 = 1.10.
7. 25%. 200 ÷ 800 × 100 = 25. The base is the false weight.
8. 25%. Take SP = 100. Profit = 20, so CP = 80, and 20 ÷ 80 × 100 = 25.

What to do next

• Write the base-price table from memory and check it
• For every question this week, take CP = 100 and work in multipliers
• Solve the profit and loss questions from the last five CDS papers at about a minute each
• Move on to simple and compound interest, which uses the same multiplier habit