Percentage Math: The Only Guide You Will Ever Need

The one idea behind every percent question

Percent means per hundred. 25% is 25 out of 100, which is the decimal 0.25 and the fraction 25/100. Every percentage problem is asking about three things: the part, the whole, and the percent that connects them.

If you can point at which number is the part and which is the whole, the arithmetic is trivial: part divided by whole, times 100, gives the percent. The hard part is never the math; it is identifying the pieces.

Write the numbers down and label them. "30 is what percent of 150?" Part = 30, whole = 150, so 30/150 x 100 = 20%. That labeling habit prevents nearly every percentage error.

Finding x percent of a number

Multiply the number by the percent as a decimal. 20% of 150 = 150 x 0.20 = 30. To get the decimal, just move the decimal point two places left: 20% becomes 0.20, 7.5% becomes 0.075.

For mental math, build from 10%: 10% of 240 is 24, so 20% is 48 and 5% is 12. Combining these chunks handles most everyday percentages without a calculator.

Common mental shortcuts: 50% is half, 25% is a quarter, 33% is roughly a third, and 1% is just the number divided by 100.

Percent change: increases and decreases

Percent change = (new - old) / old x 100. From 80 to 100: (100-80)/80 x 100 = 25% increase. The old value is always the base, which is where most mistakes happen.

Going from 100 to 80 is a 20% decrease, not 25%: (80-100)/100 x 100 = -20%. Same two numbers, different base, different answer. Direction matters.

This asymmetry surprises people: a 50% rise followed by a 50% fall does not return to the start. 100 becomes 150, then falls to 75. The second percent acts on a bigger base.

Discounts, tips and tax

A 25% discount on $120 means you pay 75%: 120 x 0.75 = $90, saving $30. Compute what you pay directly instead of subtracting; it is one step instead of two.

For tips, 20% is just double the 10%: on an $85 bill, 10% is $8.50, so 20% is $17. Move the decimal for 10%, then scale.

Sales tax works the same in reverse: 8.5% tax on $100 is $100 x 1.085 = $108.50. Multiplying by 1 plus the rate gives the total in one step.

Reversing a percentage

If 30 is 20% of some number, divide: 30 / 0.20 = 150. Reversing means dividing by the decimal instead of multiplying.

Removing tax or discount works the same way. Paid $90 after a 25% discount? The original was 90 / 0.75 = $120.

This is the most useful percentage skill in shopping: working backwards from the price you see to the price that was.

Percentage points vs percent

Going from 10% to 15% is a 5 percentage point increase, but a 50% relative increase. News and politics mix these up constantly, sometimes deliberately.

When someone says "up 200%", they mean triple the original, not double: 100% is the original itself, plus 200% more.

Always ask: percent of what? The base decides everything, and changing the base changes the story.

Frequently asked questions