Percentage Formulas Reference — All the Percentage Calculations You Need

Complete Percentage Formula Reference

How to Use the Percentage Calculator

1. Open the percentage calculator.
2. Select the type of percentage calculation you need (percentage of a number, percentage change, reverse percentage, etc.).
3. Enter the known values into the appropriate fields.
4. Click Calculate. Results are shown with the formula used and step-by-step working.

Percentage Change in Detail

Percentage change expresses how much a value has changed relative to its original value:

% change = ((New − Old) / Old) × 100

Important points:

• A positive result = percentage increase (value went up)
• A negative result = percentage decrease (value went down)
• Always divide by the original (old) value, not the new value
• The formula measures change relative to where you started

Example: A stock price fell from £120 to £90. % change = ((90 − 120) / 120) × 100 = (−30/120) × 100 = −25%. The stock fell 25%.

Reverse Percentage

A reverse percentage problem gives you the value after a percentage change and asks for the original value before the change:

Original = Current value / (1 + rate/100) [for an increase]

Original = Current value / (1 − rate/100) [for a decrease]

Example: After a 30% increase, a product costs £91. What was the original price?

Original = 91 / 1.30 = £70.

Common mistake: Subtracting 30% from the final value: 91 − 30% of 91 = 91 − 27.3 = £63.70. This is wrong because it takes 30% of the final price, not the original price. The correct answer is £70.

VAT and Tax Calculations

VAT (Value Added Tax) is one of the most common reverse percentage applications. In the UK, standard VAT is 20%.

Price including VAT: Price exc. VAT × 1.20

Price excluding VAT (from a VAT-inclusive price): Price inc. VAT / 1.20

VAT amount: Price exc. VAT × 0.20 (or: Price inc. VAT − Price exc. VAT)

Example: A product costs £240 including 20% VAT. VAT amount = 240 − (240/1.20) = 240 − 200 = £40. Price without VAT = £200.

Percentage Points vs. Percentages

This is one of the most misunderstood distinctions in everyday numerical communication:

• A percentage point is the arithmetic difference between two percentages: from 3% to 5% is a 2 percentage point increase.
• A percentage change between two percentages is relative: from 3% to 5% is a (5−3)/3 × 100 = 66.7% increase (the rate increased by two-thirds of its original value).

Both can be true simultaneously and mean different things. "Unemployment fell by 2 percentage points (from 8% to 6%)" and "unemployment fell by 25% (relative decrease from 8%)" describe the same fact. The choice of framing can significantly affect perceived magnitude.

Compound Percentage Changes

Compound percentage changes multiply together rather than adding. For n periods with rate r% each:

Final = Start × (1 + r/100)ⁿ

This is compound interest, compound inflation, compound population growth — any situation where a percentage change builds on the previous result.

Example: £1,000 invested at 7% per year for 10 years: Final = 1000 × 1.07¹⁰ = 1000 × 1.9672 = £1,967.15.

The Rule of 72 gives a quick estimate of doubling time: divide 72 by the annual rate. At 7%, money doubles in approximately 72/7 ≈ 10.3 years (the exact value above confirms this: nearly doubles in 10 years).

Common Percentage Mistakes

Percentage in Statistics

Relative frequency

Relative frequency = (frequency of event / total observations) × 100. If 45 out of 200 students passed a test, the pass rate = (45/200) × 100 = 22.5%.

Percentage error

Percentage error measures how far an estimated or measured value is from the true value: % error = |measured − true| / true × 100. If you measured 48 cm when the true value was 50 cm: % error = |48−50|/50 × 100 = 4%.

Confidence intervals as percentages

Polls report percentages with a margin of error, e.g., "45% support, ±3%." This means the true value is likely between 42% and 48%. The margin covers the uncertainty from sampling — a random sample of 1,000 people has a margin of error of approximately ±3%.

Discounts and Savings

Percentage discount problems are a daily use case for percentage calculations:

• Discount amount: Discount = Original price × (discount%/100)
• Sale price: Sale price = Original × (1 − discount%/100)
• What percentage was the discount? % = (discount amount / original price) × 100

Example: A jacket originally at £150 is now £99. Discount = 150 − 99 = £51. % discount = (51/150) × 100 = 34%.

Common Questions

What is 0% of any number?

0% of any number = 0. 0% means "no portion of" — taking none of the value. This may sound obvious but is sometimes confused in contexts like "0% APR" (no interest charged) or "0% fat" (no fat content claimed).

What is 100% of a number?

100% of a number is the number itself. 100% means "the whole thing." 100% of 75 = 75. A 100% increase doubles the original: 75 + 100% of 75 = 75 + 75 = 150. A "100% increase" and "double" are the same thing.

How do you add two percentages of the same value?

Simply add them: 15% of 200 + 20% of 200 = 35% of 200 = 70. This works because both percentages reference the same base. But when percentages reference different bases (different totals), you cannot simply add them — weight each percentage by its base first.

Can a percentage be greater than 100%?

Yes. A 200% increase means the value tripled (it increased by 200% of itself: 100 + 200 = 300). A percentage of a number can exceed 100% when the part is larger than the whole (which happens when measuring against a smaller reference). For example, if sales this year are £150 and last year were £100, this year's sales are 150% of last year's.