The Percentage Formula, Explained from First Principles

A percentage is a fraction with a fixed denominator: 100. That’s it. Once you internalise that, the rest follows.

This article walks through the percentage formula, the five most common variations, and the mental shortcuts that turn percent calculations into something you can do at the dinner table.

The base formula

X% of Y = (X / 100) × Y

20% of 850 is (20/100) × 850 = 0.20 × 850 = 170. The ”%” symbol is just a marker that says: divide this number by 100 first, then multiply.

Why does it work? Because “20%” is shorthand for “20 per hundred” — twenty parts out of every hundred. If 850 is your “every hundred”, then 20 parts of every hundred is 20/100 × 850.

Five variations

1. Percentage of a value

The base case. What is 15% of 480?

0.15 × 480 = 72

2. Percentage increase

Increase 200 by 25%. You add the percentage of the original to the original.

new = old × (1 + X/100)
    = 200 × 1.25 = 250

The shortcut: if the increase is 100%, the new value is double. If 50%, it’s 1.5×. If 10%, it’s 1.1×.

3. Percentage decrease

Discount $80 by 30%. Subtract instead of add.

new = old × (1 − X/100)
    = 80 × 0.70 = 56

4. Reverse percentage

X is what percent of Y? Divide and multiply by 100.

percent = (X / Y) × 100

For instance, 12 is (12/40) × 100 = 30% of 40.

5. Percentage difference

When you want to compare two quantities symmetrically — neither is “old” or “new” — use percentage difference.

difference = |A − B| / ((|A| + |B|) / 2) × 100

This treats both values equally. The percentage change from 80 to 100 is 25%; the percentage difference between them is 22.2%.

Mental shortcuts

Most percentages can be computed in your head with two anchors:

• 10% is just shifting the decimal. 10% of 240 = 24. 10% of 1,750 = 175.
• 1% is two shifts. 1% of 240 = 2.4.

From these you can build anything:

• 5%: half of 10%. (5% of 240 = 12)
• 15%: 10% + 5%. (15% of 240 = 36)
• 20%: double 10%. (20% of 240 = 48)
• 25%: a quarter — divide by 4. (25% of 240 = 60)
• 50%: half. (50% of 240 = 120)
• 75%: half plus a quarter. (75% of 240 = 180)

Restaurant tip? 18% on a $54 bill = (10% × 54) + (8% × 54) = 5.40 + 4.32 = 9.72. Done in your head, no phone needed.

When percentages mislead

A 50% increase followed by a 50% decrease does not return you to the start.

100 → 100 × 1.5 = 150 → 150 × 0.5 = 75

You lose 25% of your original. This is why investment returns are usually compared by their geometric mean, not their arithmetic mean. A stock that drops 50% needs to double (100% gain) to recover.

Another classic trap: percentage points vs percentages. If interest rates rise from 4% to 6%, that’s a 2 percentage point increase but a 50% relative increase. News headlines routinely confuse the two.

Doing it instantly

If you want to skip the mental math, the OurDailyCalc percentage calculator handles all five variations with a single click. Or open the command palette anywhere on the site (⌘K) and type “20% of 500” — it’ll jump straight to the answer.

TL;DR

• A percentage is a fraction with denominator 100. X% of Y = (X/100) × Y.
• Five common variations: percent of, increase, decrease, reverse, and difference.
• Anchor on 10% (decimal shift) and 1% (two shifts) — the rest follows.
• Watch out for percentage points vs percentages, and asymmetric inverse operations.