Quick revision notes · Mathematics 4024 · Chapter 20 of 35
Daily Maths
Everyday maths: percentages, reverse percentages, interest, ratio, scales, rates, units and pie charts.
6 sectionsWorked examples
Common mistakes
At a glance
Everyday maths: percentages, reverse percentages, interest, ratio, scales, rates, units and pie charts.
1Percentages
- Percentage of an amount: change the % to a decimal and multiply. 35% of 240 = 0.35 × 240 = 84.
- One quantity as a percentage of another: (part ÷ whole) × 100.
- Use a multiplier for increases and decreases: +15% → × 1.15; −15% → × 0.85.
- Percentage change = change ÷ original × 100 (profit and loss % work the same way).
Formulas
percentage change = (change ÷ original) × 100
new = original × multiplier
Worked example
A price rises from $40 to $46. Find the percentage increase.
- Change = 6.
- 6 ÷ 40 = 0.15.
15%
2Reverse percentages
- You know the amount after a percentage change and want the original.
- Original = new amount ÷ multiplier. After a 20% discount the multiplier is 0.8.
- Never take the percentage of the new amount.
How to do it
- Write the multiplier for the change (e.g. +15% → 1.15).
- Divide the new amount by it.
Worked example
A phone costs $345 after a 15% price rise. What did it cost before?
- Multiplier = 1.15.
- 345 ÷ 1.15.
$300
3Simple and compound interest
- Simple interest is the same every year: I = PRT ÷ 100 (P = principal, R = rate %, T = years).
- Compound interest is added to the amount each year, so it grows: A = P(1 + r/100)n.
- Total amount with simple interest = P + I. Compound interest earned = A − P.
Formulas
I = PRT/100simple interest
A = P(1 + r/100)ncompound: amount after n years
Worked example
$2000 is invested at 5% per year compound interest for 3 years. Find the interest earned.
- A = 2000 × 1.053 = 2315.25.
- Interest = 2315.25 − 2000.
$315.25
4Ratio and proportion
- Simplify a ratio by dividing all parts by the HCF; use the same units first.
- Sharing in a ratio: add the parts, find one part, multiply.
- Map scale 1 : 50 000 means 1 cm on the map = 50 000 cm = 500 m in real life.
- Best value: compare the price of the same amount (per 100 g, per litre) or the amount per $1.
- Recipes: find the amount for one person (or one unit), then scale up.
Worked example
Share $360 in the ratio 4 : 5.
- 4 + 5 = 9 parts.
- One part = 360 ÷ 9 = 40.
- 4 × 40 = 160 and 5 × 40 = 200.
$160 and $200
5Rates, currency and units
- Exchange rates: multiply to change into the currency the rate is quoted in, divide to change back.
- Rates: speed (km/h), hourly pay ($/h), flow (litres/min), fuel use (km/litre), density (g/cm³), population density (people/km²).
- Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm. Mass: 1 kg = 1000 g; 1 tonne = 1000 kg.
- Area and volume scale with the square and cube: 1 m² = 10 000 cm²; 1 m³ = 1 000 000 cm³; 1 litre = 1000 cm³.
Formulas
density = mass ÷ volume
population density = population ÷ area
Worked example
$1 = 280 rupees. Convert 7000 rupees to dollars.
- Rupees → dollars: divide by 280.
- 7000 ÷ 280.
$25
6Pie charts
- The whole pie is 360°. Angle for a group = frequency ÷ total × 360°.
- From a pie chart: frequency = angle ÷ 360 × total.
- If you know one sector's frequency and angle, find 'degrees per item' and scale.
Worked example
In a survey of 72 students, 18 walk to school. What angle represents walking?
- 18 ÷ 72 = 1/4.
- 1/4 × 360°.
90°
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| Reverse %: 64 + 20% of 64 | Divide by the multiplier: 64 ÷ 0.8 = 80 |
| Percentage change ÷ the new value | Always divide by the original value |
| Compound interest worked as simple interest | Multiply by 1.05 each year: P × 1.05n |
| 1 m² = 100 cm² | 1 m² = 100 × 100 = 10 000 cm² |
| Sharing in 4 : 5 by dividing by 4 and 5 | Divide by the total number of parts, 9 |
★In the exam
- Money answers: give 2 decimal places ($315.25, not $315.3).
- Say clearly what you are comparing in best-value questions (price per 100 g…).
- Read whether the question wants the interest or the total amount.