Basic Numeracy: Interest, profit and loss, and mixtures

Interest, profit and loss, and mixtures

Think of a pocket-money jar. If your parents add 10% to whatever is in the jar every year, the jar grows faster each year because last year’s bonus also earns a bonus. That is compound interest.

Principal (P) means the money you start with. Rate (R) is the yearly per cent. Time (T) is the number of years.

Simple interest

SI = P × R × T ÷ 100. Amount (A) = P + SI.

Example 1. ₹5,000 at 8% a year for 3 years. SI = 5,000 × 8 × 3 ÷ 100 = ₹1,200. Amount = ₹6,200.

Simple interest grows by the same amount every year, like adding the same number of marbles each day.

Compound interest

A = P(1 + R/100)^n, where n is the number of periods. CI = A − P.

Example 2. ₹10,000 at 10% a year for 2 years. A = 10,000 × 1.1 × 1.1 = ₹12,100. CI = ₹2,100. SI for the same would be ₹2,000.

Shortcut: for 2 years, CI − SI = P(R/100)² = 10,000 × 0.01 = ₹100. That matches 2,100 − 2,000.

Half-yearly compounding. Halve the rate and double the periods. Example 3. ₹8,000 at 10% a year, compounded half-yearly, for 1 year: 8,000 × 1.05 × 1.05 = ₹8,820.

Rule of 72. Money roughly doubles in 72 ÷ R years. At 8%, about 9 years (the exact answer is 9.006 years). Use it only to check options quickly.

Profit and loss

• Cost price (CP): what the seller paid. Selling price (SP): what the buyer pays.
• Profit % = (SP − CP) ÷ CP × 100. Loss % = (CP − SP) ÷ CP × 100.
• Marked price (MP): the price on the tag. A discount is always taken on the marked price.

Example 4. CP ₹400, SP ₹500. Profit = ₹100. Profit % = 100 ÷ 400 × 100 = 25%.

Example 5. MP ₹1,000, discount 20%, so SP = ₹800. If CP was ₹640, profit = 160 ÷ 640 × 100 = 25%.

Example 6, successive discounts. 20% then 10% is not 30%. Single discount = 20 + 10 − (20 × 10 ÷ 100) = 28%. Check: 100 → 80 → 72, a 28% cut.

Example 7, the false weight. A shopkeeper sells at cost price but gives 900 g for every 1 kg. He gains 100 g on every 900 g he gives, so profit = 100 ÷ 900 × 100 = 11 1/9%, about 11.11%.

Example 8, same SP trap. Two phones sell for ₹1,200 each, one at 20% profit and one at 20% loss. CP of the first = 1,200 ÷ 1.2 = ₹1,000. CP of the second = 1,200 ÷ 0.8 = ₹1,500. Total CP ₹2,500, total SP ₹2,400, so a loss of ₹100, which is 4%. Shortcut: x²/100 = 20 × 20 ÷ 100 = 4% loss.

Mixtures and alligation

Alligation is a quick cross method to find the ratio in which two things at different prices must be mixed to get a middle price.

Write the cheaper price and the dearer price on top, the mean (target) price in the middle, then subtract across:
Cheaper : Dearer = (Dearer − Mean) : (Mean − Cheaper)

Example 9. Rice at ₹30/kg and ₹45/kg is mixed to sell at ₹36/kg with no profit or loss. Ratio = (45 − 36) : (36 − 30) = 9 : 6 = 3 : 2. So 3 kg of the ₹30 rice for every 2 kg of the ₹45 rice. Check: (3 × 30 + 2 × 45) ÷ 5 = 180 ÷ 5 = ₹36.

Example 10, replacement. A can has 40 litres of milk. 4 litres are taken out and replaced with water, and this is done twice. Milk left = 40 × (36/40)² = 40 × 0.81 = 32.4 litres. Water = 7.6 litres.

Example 11, mixing two mixtures. Jar A has milk and water in the ratio 3 : 1, jar B has 1 : 3. Equal amounts are mixed. Milk share = (3/4 + 1/4) ÷ 2 = 1/2, so the new ratio is 1 : 1.

Common traps

• Taking profit % on SP instead of CP.
• Adding successive discounts directly.
• Forgetting to halve the rate and double the time for half-yearly interest.
• In alligation, putting the ratio the wrong way round; always check with a quick weighted average.

What UPSC asks: short word problems on interest, discounts and mixtures, often with one trap. Check every answer by putting it back into the question.