Ratio, proportion and percentages
Sharing in a ratio, percentage change, reverse percentages, interest, direct and inverse proportion, rates and money.
📌 Key points
- Simplify a ratio by dividing every part by the HCF. To share an amount in the ratio a : b, divide it by a + b to find one part, then multiply.
- Percentage change = changeoriginal × 100%. Always divide by the original amount.
- Multipliers: an increase of 15% multiplies by 1.15; a decrease of 15% multiplies by 0.85.
- Reverse percentages: divide the new amount by the multiplier. If a price after a 20% increase is $72, the original was 72 ÷ 1.2 = $60.
- Simple interest I = PRT100 (the same amount each year). Compound interest: value = P × (1 + r100)n.
- Depreciation (losing value) each year by r% uses the multiplier (1 − r100)n; growth and decay questions work the same way.
- Direct proportion: y = kx. Inverse proportion: y = kx. Other forms include y = kx2 and y = kx2. Find k from the given values first.
- Speed = distance ÷ time. To convert m/s to km/h multiply by 3.6; from km/h to m/s divide by 3.6.
- Best buy: compare the cost of the same amount (e.g. per 100 g). Currency: multiply or divide by the exchange rate.
- Time: 1 hour = 60 minutes, so 2.5 hours is 2 hours 30 minutes, not 2 hours 50 minutes.
📖 Key terms
- Ratio
- A comparison of two or more quantities, written a : b.
- Percentage change
- The change as a percentage of the original value.
- Multiplier
- The number you multiply by to apply a percentage change, e.g. 1.08 for +8%.
- Simple interest
- Interest paid only on the original amount, the same each year.
- Compound interest
- Interest paid on the original amount plus interest already earned.
- Depreciation
- A fall in value over time.
- Direct proportion
- y = kx: when one quantity doubles, so does the other.
- Inverse proportion
- y = kx: when one quantity doubles, the other halves.
✏️ Worked example
A car bought for $18 000 loses 12% of its value each year. Find its value after 3 years.
- A 12% decrease each year gives a multiplier of 1 − 0.12 = 0.88.
- Value after 3 years = 18 000 × 0.883.
- 0.883 = 0.681472, so the value = 18 000 × 0.681472 = 12 266.496.
- Answer: $12 266.50 (to the nearest cent).
In reverse percentage questions never take the percentage of the new amount: divide by the multiplier, then check by working forwards.
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