Percentage Calculator

Five different questions, five different formulas

“Percentage calculator” is really five tools wearing one name, and choosing the wrong one is the most common mistake in this whole topic — including in newspapers, shop signs and other calculator sites.

The question	The formula	Example
What is A% of B?	A ÷ 100 × B	15% of 200 = 30
A is what percent of B?	A ÷ B × 100	30 is 15% of 200
Change from A to B	(B − A) ÷ A × 100	200 → 250 is +25%
A increased by B%	A + (A × B ÷ 100)	200 + 15% = 230
A is B% of what?	A ÷ B × 100	30 is 15% of 200

Notice that rows two and five use identical arithmetic but answer opposite questions. That is not a mistake — it is why writing the question down before reaching for a calculator is worth the five seconds.

The trap that costs people money

A 25% rise is not cancelled by a 25% fall.

Start with 200. Add 25% and you have 250. Now take 25% off 250 — that is 62.50 — and you are left with 187.50. You are below where you started, having gone up then down by the same percentage.

The reason is that each percentage is taken from a different base. The rise was 25% of 200; the fall was 25% of 250, which is a bigger number.

To get from 250 back to exactly 200 you need to lose 50 out of 250, which is a 20% decrease. And the bigger the swing, the worse the asymmetry:

Fall	Rise needed to recover
−10%	+11.1%
−20%	+25%
−33%	+49.3%
−50%	+100%
−80%	+400%

This matters far beyond arithmetic homework. An investment down 50% needs to double just to break even.

Percent versus percentage points

If interest rates go from 5% to 7%, which is right?

• A rise of 2 percentage points — the simple difference.
• A 40% increase — because 2 is 40% of 5.

Both are correct. They are describing the same event in two languages, and reporting usually picks whichever sounds larger. When you see a striking percentage attached to something that is already a percentage — unemployment, interest rates, tax rates, market share — check which of the two is being used.

Removing a percentage: divide, do not subtract

A common and expensive error. A bill is 118, tax is 18%. What was it before tax?

• Wrong: 118 − 18% = 96.76
• Right: 118 ÷ 118 × 100 = 100.00

Subtracting fails because the 18% was charged on the pre-tax amount of 100, not on the 118 you are holding. The general rule: to strip out a percentage that was added, divide by (100 + the percentage) and multiply by 100.

The same applies to discounts in reverse. If a jacket cost 910 after 30% off, the original price was 910 ÷ 70 × 100 = 1,300.

Doing percentages in your head

• 10% — move the decimal point one place left. 10% of 340 is 34.
• 5% — half of 10%. So 17.
• 20% — double 10%. So 68.
• 15% — 10% plus 5%. So 51.
• 1% — move the decimal two places. 3.40.
• Reverse them. 8% of 50 is the same as 50% of 8, which is 4. Percentages are symmetric that way, and one side is often much easier.

A restaurant tip becomes easy: 10% then half again for 15%, or double the 10% for 20%.

Where percentages mislead on purpose

• “Up to 70% off” — one item in the corner is 70% off; the rest are 10%.
• “200% more” means three times as much, not twice. “200% of” means twice.
• Percentages of tiny numbers. A jump from 2 cases to 6 is a 200% increase and still only four cases.
• Stacked discounts. 20% off then a further 10% off is 28% off, not 30% — the second discount applies to the already-reduced price.