Percentages
Section: Number 2 | Syllabus: Cambridge IGCSE Mathematics (0580)
Percentage of a Quantity and Percentage Change
Percentages appear throughout everyday life - deposits, discounts, profit and earnings all rely on the same core skills: finding a percentage of an amount, comparing one quantity to another, and adjusting a value up or down by a percentage.
- To find a percentage of a quantity, convert the percentage to a decimal and multiply
- To express one quantity as a percentage of another, divide the first by the second and multiply by 100
- To increase or decrease by a percentage, multiply by a multiplier:
- Increase by R%: multiply by 1 + (R/100)
- Decrease by R%: multiply by 1 - (R/100)
- Percentages can go over 100% - a multiplier greater than 1 represents an amount larger than the original
Worked Example: Percentage of a Quantity, and One Quantity as a Percentage of Another
- 35% of 480: 0.35 × 480 = 168
- 84 as a percentage of 240: (84/240) × 100 = 35\%
Worked Example: Percentage Increase and Decrease
- Question: Increase 350 by 18%, and decrease 620 by 15%.
- Increase: multiplier = 1 + (18/100) = 1.18, so 350 × 1.18 = 413
- Decrease: multiplier = 1 - (15/100) = 0.85, so 620 × 0.85 = 527
- Question: A shop's sales this year are 145% of last year's sales. Last year's sales were 2400. Find this year's sales.
- Step 1: 145% as a multiplier is 1.45
- Step 2: 2400 × 1.45 = 3480
- Answer: 3480
Common Mistakes
MistakeWorking out the amount of the increase or decrease, then forgetting to add or subtract it from the original
Fixuse the multiplier method to combine both steps at once: increasing 350 by 18% is 350 × 1.18, not just 350 × 0.18
MistakeDividing by the wrong quantity when expressing one amount as a percentage of another, e.g. working out 240/84 instead of 84/240
Fixalways divide the quantity being compared by the quantity it is being compared to - read the question carefully to identify which is which
Simple and Compound Interest
Simple interest pays the same amount every year, based on the original amount invested. Compound interest pays interest on the growing balance, so the amount earned increases each year.
- Simple interest is calculated on the original amount every year, so the same interest is earned each year
- Interest per year = P × (R/100), where P is the original amount and R is the rate per year
- Compound interest is calculated on the current balance each year, so it includes interest already earned
- After n years: final amount = P × (1 + (R/100))^n
- Neither formula is given in the exam - both must be recalled
Worked Example: Simple Interest
- Question: Calculate the simple interest earned on 2500 invested at 6% per year for 4 years.
- Step 1: Interest per year: 2500 × (6/100) = 150
- Step 2: Total over 4 years: 150 × 4 = 600
- Answer: 600
Worked Example: Compound Interest
- Question: Aisha invests 8000 at a rate of 5% per year compound interest. Find the value of her investment after 3 years.
- Step 1: The multiplier for 5% growth is 1 + (5/100) = 1.05
- Step 2: Apply the multiplier 3 times: 8000 × 1.05^3
- Step 3: 8000 × 1.157625 = 9261
- Answer: 9261
Common Mistakes
MistakeUsing the simple interest method for a compound interest question, e.g. multiplying the original amount by rate × years
Fixcompound interest needs the multiplier raised to the power of the number of years: 8000 × 1.05³, not 8000 × 0.05 × 3
MistakeRecalculating compound interest from the original amount every year, instead of from the growing balance
Fixeach year's compound interest is earned on the previous year's total, which is exactly what raising the multiplier to a power achieves automatically
Reverse Percentages
Extended Only
A reverse percentage problem gives the value after a percentage change and asks for the original value before it. Dividing by the multiplier undoes the percentage change.
- If a value has already been increased or decreased by a known percentage, find the original value by dividing by the multiplier, not by adding or subtracting a percentage of the new value
- original = (new value/multiplier)
Worked Example: Reverse Percentage After an Increase
- Question: After a 15% pay rise, Tomas earns 920 per month. Find his monthly pay before the rise.
- Step 1: 920 represents 115% of the original pay, so the multiplier is 1.15
- Step 2: 920 ÷ 1.15 = 800
- Answer: 800
Worked Example: Reverse Percentage After a Loss
- Question: A shop sells a jacket for 276, making a loss of 20% on the cost price. Find the cost price.
- Step 1: 276 represents 80% of the cost price, so the multiplier is 0.80
- Step 2: 276 ÷ 0.80 = 345
- Answer: 345
Common Mistakes
MistakeFinding the given percentage of the new value and subtracting or adding it, e.g. finding 15% of 920 and subtracting it from 920
Fixthe new value already includes the increase, so 15% of it is not the same as 15% of the original - always divide by the multiplier instead
MistakeUsing the wrong multiplier direction, e.g. multiplying by 1.15 instead of dividing, or using 0.85 instead of 0.80 for a 20% loss
Fixidentify whether the change was an increase or a decrease first, build the correct multiplier, then divide the new value by it
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