Understanding Percentages: The Complete Guide
A percentage is a way of expressing a number as a fraction of 100. The word itself comes from the Latin per centum, meaning "by the hundred." Percentages are one of the most universally used mathematical concepts in daily life, appearing in everything from supermarket discounts and VAT calculations to interest rates, exam scores, and statistical data.
Despite their ubiquity, percentages are frequently misunderstood — particularly when it comes to the difference between a percentage and a percentage point, or when calculating percentage change versus absolute change. This guide explains the four most common percentage calculations and when to use each one.
The Four Core Percentage Calculations
1. What Percentage Is X of Y?
This is the most fundamental percentage calculation. The formula is: (X ÷ Y) × 100. Use this when you want to express one number as a proportion of another — for example, a student who scores 72 out of 90 on a test has achieved (72 ÷ 90) × 100 = 80%. Similarly, if a business spends £15,000 on marketing out of a total budget of £60,000, marketing represents 25% of the budget.
2. Finding a Percentage of a Number
To find X% of Y, the formula is: (X ÷ 100) × Y. This is used constantly in retail (calculating discounts), finance (computing interest), and taxation (working out VAT). For example, 20% VAT on a £450 item: (20 ÷ 100) × 450 = £90 in tax, making the total price £540.
3. Percentage Change (Increase or Decrease)
Percentage change measures how much a value has grown or shrunk relative to its original size. The formula is: ((New Value − Original Value) ÷ Original Value) × 100. A positive result indicates an increase; a negative result indicates a decrease. For example, if a property was worth £200,000 and is now worth £230,000, the percentage increase is ((230,000 − 200,000) ÷ 200,000) × 100 = 15%.
A common mistake is confusing percentage change with percentage point change. If a savings account interest rate rises from 1.5% to 2.5%, that is a 1 percentage point increase — but a 66.7% relative increase. The distinction is critical when interpreting financial news and economic data.
4. Adding or Removing a Percentage
To add a percentage to a number, multiply by (1 + percentage/100). To remove a percentage, multiply by (1 − percentage/100). Adding 15% to £80: 80 × 1.15 = £92. Removing 15% from £92 does not return £80 — it gives 92 × 0.85 = £78.20. This asymmetry is important to understand when working with discounts and markups.
Practical Applications
| Scenario | Calculation Type | Example |
|---|---|---|
| Exam score | What % is X of Y? | 72 out of 90 = 80% |
| VAT calculation | X% of Y | 20% of £450 = £90 |
| Salary increase | Add % | £30,000 + 5% = £31,500 |
| Sale discount | Remove % | £120 − 25% = £90 |
| Investment return | % Change | £1,000 → £1,150 = +15% |
| Budget allocation | What % is X of Y? | £15k of £60k = 25% |