1. Basic Percentages
What is a percentage, really?
"Per-cent" literally means "out of 100". So 50% just means "50 out of every 100", which is exactly the same amount as the fraction 50/100 or the decimal 0.5.
Here's the key skill: any fraction can be turned into a percentage by rewriting it with a denominator of 100 (or by dividing top by bottom on your calculator and multiplying by 100). This matters because percentages let you compare things fairly — "3 out of 4" and "75 out of 100" don't look like the same thing at first glance, but once you see both as 75%, the comparison becomes instant.
Fraction → Percentage: multiply the decimal value by 100
Percentage → Decimal: divide by 100 | Decimal → Percentage: multiply by 100
Example: 234/650 = 234 ÷ 650 = 0.36 = 36%
💡 Think of it like this
A percentage is a "universal ruler" — no matter what the original total was (100 people, 650 grams, £8000), converting to percentage puts everything on the same 0–100 scale so you can compare fairly.
Q1. Write 18 out of 24 as a percentage.
Q2. Convert 0.6% into a decimal.
Finding a percentage WITHOUT a calculator
The trick here is to build every percentage out of a few easy "building blocks":
| To find... | Do this |
|---|
| 50% | Halve the amount |
| 25% | Halve it twice (a quarter) |
| 10% | Divide by 10 |
| 1% | Divide by 100 |
Then combine blocks to reach any percentage you need:
- 20% → find 10%, then double it
- 5% → find 10%, then halve it
- 12% → find 10% + 1% + 1% (add them together)
- Over 100% → remember 100% IS the original amount, so 150% = original + 50%
Worked Example
Find 35% of 40. Start with 10% of 40 = 4. So 30% = 4 × 3 = 12, and 5% = 4 ÷ 2 = 2. Add them: 35% of 40 = 12 + 2 = 14.
Q. Find 12% of 650 without a calculator.
Finding a percentage WITH a calculator (multipliers)
This is the method that will save you the most time in exams. A multiplier is just the decimal version of a percentage. To find a percentage of an amount, convert the percentage to a decimal and multiply.
Percentage of an amount = Multiplier × Amount
e.g. to find 12% of 650 → multiplier = 0.12 → 0.12 × 650 = 78
If the percentage is over 100%, the multiplier is simply greater than 1 (e.g. 126% → multiplier 1.26).
Q. Use a multiplier to find 8% of 950.
Expressing one number as a % of another
Write the first number as a fraction of the second, then convert that fraction to a percentage (decimal equivalent × 100).
Worked Example
Amber owes £1200 for a trip and pays a £150 deposit. What % of the trip cost is the deposit?
150/1200 = 0.125 → 12.5%
Q. Express 7 as a percentage of 20.
2. Percentage Increases & Decreases
Increasing by a percentage
A percentage increase makes the amount bigger — you're adding that percentage on top of the original 100%.
Without a calculator: find the percentage using building blocks, then add it to the original amount.
With a calculator: use a multiplier greater than 1, because you are now finding more than the original 100%.
Increase by x% → Multiplier = 1 + (x ÷ 100)
e.g. increase by 15% → multiplier = 1.15 → New amount = Original × 1.15
Worked Example
Increase 200 kg by 21%. Multiplier = 1 + 0.21 = 1.21.
1.21 × 200 = 242 kg
Decreasing by a percentage
A percentage decrease makes the amount smaller — you're taking that percentage away from the original 100%.
Decrease by x% → Multiplier = 1 − (x ÷ 100)
e.g. decrease by 35% → multiplier = 0.65 → New amount = Original × 0.65
Worked Example
An item costing $500 is discounted by 35%. Multiplier = 1 − 0.35 = 0.65.
500 × 0.65 = $325
Q. A jacket costs £64. It is reduced by 30% in a sale. Find the sale price.
Repeated percentage changes
Some questions apply several percentage changes one after another — e.g. a price goes up 10%, then gets discounted 15%, then discounted a further 20%. You can either:
- Apply each multiplier one at a time in sequence, or
- Multiply all the multipliers together first to get one "combined multiplier", then apply it once.
Combined multiplier = m₁ × m₂ × m₃ × ... × mₙ
Worked Example
£80 increased by 10% (×1.10), then decreased by 15% (×0.85), then decreased by a further 20% (×0.80).
Combined multiplier: 1.10 × 0.85 × 0.80 = 0.748
80 × 0.748 = £59.84 — an overall 25.2% decrease from the original.
⚠️ Watch out
Repeated percentage changes do NOT simply add or subtract! A 10% increase followed by a 20% decrease is NOT the same as a 10% overall decrease. You must multiply the multipliers, never add/subtract the raw percentages.
Finding the percentage change itself
Sometimes you're given the "before" and "after" values and asked to describe the change as a percentage. There are two equivalent methods:
Method 1: m = After ÷ Before
m > 1 → percentage increase | m < 1 → percentage decrease
Method 2: Percentage Change = [(After − Before) ÷ Before] × 100
Positive answer → increase | Negative answer → decrease
Worked Example
Students in a school go from 250 to 310.
310 ÷ 250 = 1.24 → a percentage increase of 24%
Check with Method 2: (310−250)/250 × 100 = 24 ✓
Q. A plant's height changes from 40 cm to 34 cm. Find the percentage change.
Percentage profit or loss
Exactly the same idea as percentage change, but using shop language: items are bought at a cost price and sold at a selling price.
m = Selling Price ÷ Cost Price
m > 1 → profit | m < 1 → loss
Percentage Profit = [(Selling Price − Cost Price) ÷ Cost Price] × 100
Positive → profit | Negative → loss
Worked Example
Sophie buys a car for $8000 and sells it for $5600.
5600 ÷ 8000 = 0.7 → selling price is only 70% of cost price → a loss of 30%
✅ Sanity check
Always use common sense to check your answer — if something sold for MORE than it cost, you should be expecting a profit, not a loss. If your working gives a loss in that situation, you've made a sign error somewhere.
Q. A trader buys a bike for £120 and sells it for £150. Find the percentage profit.
3. Reverse Percentages
What is a reverse percentage?
This is the question type that trips up the most students, so let's be really clear about it. A reverse percentage question gives you the value AFTER a percentage increase or decrease happened, and asks you to work backwards to find the value BEFORE it happened.
🔎 How to spot one
Look for phrases like "find the old amount", "original price", or "price before the increase/discount". Any time the question asks you to travel backwards in time, it's a reverse percentage question.
How to solve them
The golden rule: the percentage change was originally applied to the before amount, not the after amount. So to undo it, you must divide the after amount by the multiplier — never apply the "opposite" percentage to the after amount.
Before × p = After → Before = After ÷ p
where p is the multiplier for the original change (e.g. +4% → p = 1.04, −5% → p = 0.95)
⚠️ The #1 mistake in this whole chapter
Increasing an amount by a percentage is NOT undone by decreasing the result by that same percentage!
Example: Increase 100 by 10% → 110. Now decrease 110 by 10% → you get 99, NOT the original 100!
This is because the 10% decrease is 10% of the NEW (bigger) number, not the original. This is exactly why we always divide by the multiplier instead.
Worked Example
Jennie gets a 5% pay rise. Her new salary is £31 500. What was her salary before?
Multiplier: p = 1 + 0.05 = 1.05
Before = 31 500 ÷ 1.05 = £30 000
Q1. A shop discounts an item by 20%. The sale price is £64. Find the original price.
Q2. After a 12% increase, a population is 2016. What was the population before the increase?
Exam Tips
❌ Common Mistake
Treating a percentage increase and its "reverse" decrease as opposites (e.g. thinking a 10% increase is undone by a 10% decrease). It isn't — the percentages are taken from different base amounts.
❌ Common Mistake
Adding/subtracting percentages directly across repeated changes (10% up then 20% down ≠ 10% down overall). Always multiply the multipliers.
✅ Examiner Tip
Spot reverse percentage questions by looking for "original", "before", or "old" amount in the wording — this signals you need to divide, not multiply.
✅ Examiner Tip
Use common sense to sanity-check profit/loss answers: if the selling price is higher than the cost price, your answer MUST be a profit.
📝 Mark Scheme Habit
Always write down your multiplier explicitly (e.g. "multiplier = 1.15") — method marks are often awarded just for showing you identified the correct multiplier, even if the final arithmetic slips.
📝 Mark Scheme Habit
When a question says "hence" or gives you a fraction/decimal equivalent already, use it directly rather than re-deriving from scratch — it saves time and reduces error risk.