Half the Everyday Mathematics section of an IMO paper is here. Every one of these problems is the same idea: a part compared with a whole.
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A ratio compares two quantities of the same kind; a proportion says two ratios are equal.
⚖️ How to share in a ratio
Add the parts. To share ₹600 in the ratio 2 : 3, the parts total 5.
One part is 600 ÷ 5 = ₹120.
Multiply back: the shares are ₹240 and ₹360.
Joining two ratios. If a : b = 2 : 3 and b : c = 4 : 5, make the b values match. Multiply the first by 4 and the second by 3: a : b : c = 8 : 12 : 15.
In a proportion a : b :: c : d, the product of the extremes equals the product of the means — a × d = b × c. That single line solves most "find x" ratio questions.
Everything here is measured on the cost price unless the question says otherwise.
💰 The formulas
| Quantity | Formula |
|---|---|
| Profit | SP − CP |
| Profit % | (profit ÷ CP) × 100 |
| Loss % | (loss ÷ CP) × 100 |
| SP from CP | CP × (100 + profit%) ÷ 100 |
| Discount | marked price − selling price |
| Discount % | (discount ÷ marked price) × 100 |
Profit is on cost price, discount is on marked price. Mixing the two bases is the single commonest error in this topic, and Olympiad papers set questions that punish it.
A 20% rise followed by a 20% fall is not back where you started. 100 → 120 → 96, a net 4% loss. The second percentage is taken on a bigger number.
Simple interest is the same every year; compound interest is charged on the growing amount.
🏦 The two formulas
Simple interest = P × R × T ÷ 100. The amount is P + SI.
Compound interest: first find the amount, A = P × (1 + R/100)ᵀ, then CI = A − P.
For interest compounded half-yearly, halve the rate and double the time.
Worked example. ₹5000 for 2 years at 10% compounded yearly: A = 5000 × 1.1 × 1.1 = ₹6050, so CI = ₹1050. The simple interest would have been only ₹1000 — the extra ₹50 is interest on the first year’s interest.
Shortcut for 2 years: the difference between compound and simple interest is P × (R ÷ 100)². For the example above, 5000 × 0.01 = ₹50, which matches.
12 Olympiad questions on ratio, percentage, profit and interest. 🍀
