Ratio
A ratio compares two quantities of the same kind, in the same units, by division. If a class has 18 girls and 12 boys, the ratio of girls to boys is 18 : 12, which simplifies to 3 : 2 (divide both by 6).
Order matters: boys to girls is 2 : 3.
Convert to the same units first. The ratio of 50 paise to ₹2 is 50 : 200 = 1 : 4, not 50 : 2.
A ratio has no units.
Proportion
Four quantities are in proportion if the two ratios are equal. We write a : b :: c : d, and it is true exactly when a × d = b × c.
Example. Are 3, 5, 12, 20 in proportion? 3 × 20 = 60 and 5 × 12 = 60. Yes.
The unitary method
Find the value of one unit first, then multiply.
Example. 6 notebooks cost ₹150. What do 10 cost? One notebook costs 150 ÷ 6 = ₹25, so 10 cost 25 × 10 = ₹250.
Dividing in a given ratio
Example. Share ₹840 between Aman and Bela in the ratio 3 : 4.
There are 3 + 4 = 7 parts, each worth 840 ÷ 7 = ₹120. Aman gets 3 × 120 = ₹360 and Bela gets 4 × 120 = ₹480.
Percentage
Per cent means "per hundred", and the symbol is %. 35% means 35/100.
Convert | Method | Example |
|---|---|---|
Fraction → % | multiply by 100 | 3/8 = 3/8 × 100% = 37.5% |
% → fraction | put over 100 and simplify | 45% = 45/100 = 9/20 |
Decimal → % | multiply by 100 | 0.07 = 7% |
% → decimal | divide by 100 | 125% = 1.25 |
Percentage of a quantity: 15% of 240 = 15/100 × 240 = 36.
What percentage? 18 out of 24 is 18/24 × 100% = 75%.
Percentage increase and decrease
Percentage change = (change ÷ original value) × 100%.
Example. A price rises from ₹400 to ₹460. The increase is ₹60, and the percentage increase is 60/400 × 100% = 15%.
Profit and loss
Cost price (CP): what the seller paid. Selling price (SP): what the seller received.
If SP > CP, profit = SP − CP. If SP < CP, loss = CP − SP.
Profit % and loss % are always worked out on the cost price.
Example. A bicycle bought for ₹2500 is sold for ₹2800. Profit = ₹300, and profit % = 300/2500 × 100% = 12%.
Example. An item costing ₹600 is sold at a 10% loss. Loss = 10% of 600 = ₹60, so SP = ₹540.
Simple interest
When you borrow money, the amount borrowed is the principal (P). The extra money paid for using it is interest, charged at a rate (R) per cent per year for a time (T) in years.
Simple interest (SI) = P × R × T ÷ 100, and the amount repaid = P + SI.
Example. ₹5000 at 8% per year for 3 years: SI = 5000 × 8 × 3 ÷ 100 = ₹1200, and the amount is ₹6200.
Summary
A ratio compares like quantities in the same units; a : b :: c : d when ad = bc.
Percentage means per hundred; percentage change uses the original value.
Profit and loss percentages are on the cost price; SI = PRT/100.