Calculator tool
How this calculator works
Use the explanation to understand the formula, assumptions, and practical limits behind the calculator result.
Mode 1: Percentage of a Number
Find X% of Y — useful for calculating discounts, tips, tax amounts, or any proportional portion:
- = the percentage
- = the base number
Example: 18% tip on a 65 bill:
65 × 18 ÷ 100 = $11.70
Mode 2: What Percentage Is A of B?
Find the percentage that number A represents of number B — used for exam scores, market share, completion rates:
Example: 42 correct out of 60 questions:
(42 ÷ 60) × 100 = 70%
Mode 3: Percentage Change
Measure the relative change from an old value to a new value — used for price changes, growth rates, inflation:
- Positive result = increase (growth)
- Negative result = decrease (decline)
Example: price rises from 80 to 95:
(95 − 80) ÷ 80 × 100 = +18.75%
Common Applications
| Calculation | Mode | Example |
|---|---|---|
| Discount amount | Mode 1 | 20% off 150 → 30 off |
| VAT / sales tax | Mode 1 | 16% VAT on 200 → 32 |
| Grade / score | Mode 2 | 54/80 → 67.5% |
| Portfolio return | Mode 3 | 10,000 → 11,500 → +15% |
| Inflation | Mode 3 | 2.50 → 2.80 → +12% |
Percentage Points vs Percent Change
A critical distinction: if interest rates move from 5% to 7%, the change is 2 percentage points — but the percent change in the rate itself is (7−5)/5 × 100 = 40%. Confusing these two measures is one of the most common errors in financial and economic reporting.
Watch the language: Unemployment rising by 2% and rising by 2 percentage points are completely different statements. If unemployment goes from 5% to 7%, it rose by 2 percentage points but by 40% as a percent change. Always ask: percent of what?
Frequently asked questions
What is the difference between percentage change and percentage point change?
A percentage point is the arithmetic difference between two percentage values. A percent change is the relative change expressed as a percentage of the starting value. If a product's market share rises from 20% to 25%, the increase is 5 percentage points. But the percent change in market share is (25 − 20) / 20 × 100 = 25%. Politicians and commentators frequently conflate these — 'increased by 5%' and 'increased by 5 percentage points' mean very different things, and which one is being used dramatically changes the impression conveyed.
How do I calculate the original price before a discount was applied?
If you know the discounted price and the discount percentage, recover the original price by dividing by (1 − discount rate). Example: a 68 item after a 15% discount — original price = 68 ÷ (1 − 0.15) = 68 ÷ 0.85 = 80. A common mistake is to add 15% back to 68, which gives 78.20 — incorrect because 15% of 80 is 12, not 15% of 68.
How do I calculate the percentage increase needed to recover a loss?
Recovery percentage is always larger than the loss percentage. A 20% loss requires a 25% gain to recover: if you start at 100, lose 20% to reach 80, you need to gain 25% of 80 ($20) to get back to 100. The general formula: recovery needed = loss% ÷ (1 − loss%) × 100. For a 50% loss: 50 ÷ 50 × 100 = 100% gain required. For a 33% loss: 33 ÷ 67 × 100 ≈ 49% gain. This asymmetry is why protecting against losses is more valuable than chasing equivalent gains.
What is the base rate error in percentage reasoning?
Base rate errors occur when a percentage is applied to the wrong base. If sales increase from 200k to 250k, the increase is 25% — not 20% (which would be 250k using the new value as the base). Similarly, 'a 50% increase followed by a 50% decrease' does not return to the original: 100 × 1.5 × 0.5 = 75 — a 25% net loss. Sequential percentage changes must be multiplied as factors (1 + r), not added. Compound changes over multiple periods use: final = initial × (1 + r₁) × (1 + r₂) × ... × (1 + rₙ).
