Percentage Calculator

A percentage is a fraction with a denominator of 100. Every correct calculation comes from finding the right part and the right whole - and knowing which is which.

• "20% of 150" - the base is 150, so (20/100) x 150 = 30.
• "What percent of 150 is 30?" - the base is still 150, so (30 / 150) x 100 = 20%.
• "150 is 20% of what?" - here 150 is the part, so 150 / (20/100) = 750.
• Three questions, three arrangements - the base is whichever quantity the percentage is expressed "of".

Note. The most common error is dividing the wrong way round. Ask which number follows the word "of" - that is your base, and it stays in the denominator when finding a percentage of it.

The old value is always the base. Reversing it gives a different answer, and the two directions are not symmetric.

A 50% gain followed by a 33.3% loss returns you exactly to where you started - not to a break-even of 50%. This asymmetry is why losses are more damaging than equivalent gains are helpful: after a 50% fall, a 100% gain is required merely to recover.

Applying 10% growth and then a 10% decline does not return you to the start.

• They multiply, they do not add. +10% then -10% is 1.10 x 0.90 = 0.99, a net loss of 1%.
• Order does not matter for the result - 1.10 x 0.90 equals 0.90 x 1.10 - but it matters enormously for how the path feels and when you might act.
• Three moves of +10% compound to 1.331, not 30%... it is 33.1%.
• For averaging returns across periods, multiply the factors together and take the root - do not average the percentages.

When a discounted price or a tax-inclusive total is shown, the original requires division rather than multiplication.

The divisor is always 1 plus the rate for an increase, or 1 minus the rate for a decrease. Multiplying instead of dividing is the standard error, and it gives a number that is wrong by exactly the rate.

Two expressions that look similar and mean different things.

• Percentage points - an arithmetic difference between two percentages. If a rate rises from 4% to 6%, it rose by 2 percentage points.
• Percentage change - relative change. From 4% to 6% is a 50% increase in the rate itself.
• Why it matters - "rates rose by 50%" and "rates rose by 2 points" describe the same move, and confusing them exaggerates or understates the change dramatically.
• Applied to interest rates - a mortgage rate moving from 5% to 6% is one percentage point but a 20% increase in the cost of borrowing.

Note. In financial reporting, always check whether a quoted percentage change refers to the absolute difference or the relative one. The figure is frequently the larger-sounding option.