How to Calculate Percentages — % Of & % Change

The Four Core Percentage Calculations

All percentage problems fall into one of four categories. Mastering these eliminates guesswork.

1. Finding “X% of Y” (Part of a Whole)

This is the most common type—calculating a portion of a total.

Formula:
Result = (Percentage ÷ 100) × Base

Steps:

1. Convert the percentage to a decimal (divide by 100).
2. Multiply by the base number.

Example: What is 18% of £250?

• 18 ÷ 100 = 0.18
• 0.18 × 250 = £45

💡 Real-world use: Tips, sales tax, discounts, interest.

2. Finding “X is What Percent of Y?” (Rate or Proportion)

This answers: What share of the whole is this part?

Formula:
Percentage = (Part ÷ Whole) × 100

Steps:

1. Divide the part by the whole.
2. Multiply by 100 to convert to a percentage.

Example: 36 out of 80 students passed. What’s the pass rate?

• 36 ÷ 80 = 0.45
• 0.45 × 100 = 45%

💡 Real-world use: Test scores, market share, success rates.

3. Calculating Percentage Change (Increase or Decrease)

Measures how much a value has grown or shrunk relative to its original amount.

Formula:
% Change = [(New – Original) ÷ |Original|] × 100

Key: The original value is always the base.

Example: A stock rose from £40 to £52.

• (52 – 40) ÷ 40 = 12 ÷ 40 = 0.30
• 0.30 × 100 = +30% (increase)

Example: Energy use dropped from 800 kWh to 680 kWh.

• (680 – 800) ÷ 800 = –120 ÷ 800 = –0.15
• –0.15 × 100 = –15% (decrease)

💡 Real-world use: Inflation, profit/loss, performance metrics.

4. Percentage Difference (Comparing Two Values)

Used when neither value is “original”—e.g., comparing two products or countries.

Formula:
% Difference = [|Value₁ – Value₂| ÷ ((Value₁ + Value₂) ÷ 2)] × 100

Example: Product A costs £60, Product B costs £90.

• Difference = |60 – 90| = 30
• Average = (60 + 90) ÷ 2 = 75
• % Difference = (30 ÷ 75) × 100 = 40%

⚠️ Don’t confuse with percentage change—this is for symmetric comparison.

Reverse Percentages: Finding the Original Value

Often, you know the final amount after a percentage change and need the starting point.

After a Discount:

Original = Final ÷ (1 – Discount Rate)

Example: You paid £84 after a 30% discount.

• Original = 84 ÷ (1 – 0.30) = 84 ÷ 0.70 = £120

After a Markup or Tax:

Original = Final ÷ (1 + Increase Rate)

Example: A bill including 20% VAT is £144.

• Pre-tax = 144 ÷ 1.20 = £120

Pro Tips & Common Mistakes

• Identify the base correctly: In % change, it’s always the original value. Misidentifying this is the #1 error.
• Don’t add percentages: A 10% rise followed by a 10% fall ≠ 0% net change.
  1.10 × 0.90 = 0.99 → 1% net loss.
• Use decimals for multiplication: 25% = 0.25, not 25.
• Round only at the end: Keep intermediate values precise to avoid compounding errors.
• Context matters: “Percent of” vs. “percent more than” are different.
  “20% more than 50” = 60, but “20% of 50” = 10.

Practical Applications

• Personal finance: Calculate interest, loan costs, investment returns
• Shopping: Compare discounts, VAT-inclusive vs. exclusive pricing
• Business: Analyse growth rates, profit margins, cost savings
• Data analysis: Interpret survey results, demographic changes, scientific data
• Education: Grade calculations, statistical reporting

Advanced: Compound Percentages

For successive changes (e.g., annual interest), multiply the factors:

Example: £1,000 grows by 5%, then 3%.

• Final = 1000 × 1.05 × 1.03 = £1,081.50
• Not 1000 × 1.08 = £1,080 (which is approximate but incorrect).