Percentage Calculator

The three questions people actually ask

Almost every percentage problem is one of these, and they are easy to confuse:

1. What is X% of Y? → multiply. 15% of 80 is 0.15 × 80 = 12.
2. X is what percent of Y? → divide. 12 out of 80 is 12 ÷ 80 = 15%.
3. What is the percent change from X to Y? → difference over the *original*.

From 80 to 92 is (92 − 80) ÷ 80 = 15%.

The third is where most mistakes happen, because it matters which number is the starting point.

Percentage change is not symmetrical

This trips up almost everyone. A price that falls 50% and then rises 50% does not return to where it started:

• 100 falls 50% → 50
• 50 rises 50% → 75

The reason is that each percentage is taken of a different base. To undo a 50% fall you need a 100% rise. Generally, to reverse a drop of *p*, you need a rise of p ÷ (1 − p).

Percent versus percentage point

A rate moving from 4% to 6% has risen 2 percentage points, which is a 50% increase. Both statements are true and they mean different things. Financial and political reporting routinely conflates them, and the difference is often the entire story.

Reverse percentages

If a price is £96 *after* a 20% discount, the original was not £96 × 1.20. The £96 represents 80% of the original, so:

original = 96 ÷ 0.80 = £120

The same logic recovers a pre-tax figure from a tax-inclusive one — divide by 1 + rate, never multiply. This is the single most common percentage error in everyday finance.

Compounding

Successive percentages multiply rather than add. Three consecutive 10% rises give 1.10³ = 1.331, a 33.1% increase, not 30%. Over long periods this gap becomes the dominant effect, which is the whole basis of compound interest.

Quick mental methods

• 10% — move the decimal one place.
• 5% — half of 10%.
• 15% — 10% plus half of it again.
• 20% — double 10%.
• Reversal trick — 8% of 25 is awkward, but 25% of 8 is obviously 2. a% of b

always equals b% of a.