What a percentage is
“Percent” means “per hundred”. A percentage is a way of expressing a part of a whole as a number out of 100. So 25% means 25 out of 100, which is 25/100, or 0.25, or 1/4. Percentages make it easier to compare amounts that come from different totals: 18 out of 72 and 30 out of 120 are hard to compare at a glance, but as percentages they are both 25%.
The three forms move into each other easily:
- Percentage to decimal: divide by 100. So 15% = 0.15.
- Decimal to percentage: multiply by 100. So 0.6 = 60%.
- Percentage to fraction: put it over 100 and simplify. So 40% = 40/100 = 2/5.
The conversions between decimals and fractions are covered in more detail in the article on fraction conversion.
Finding a percentage of a number
To find a percentage of an amount, convert the percentage to a decimal and multiply.
15% of 80: 0.15 × 80 = 12.
A mental method is to build up from 10%. Ten percent of 80 is 8, and 5% is half of that, which is 4. Adding gives 8 + 4 = 12, the same answer.
35% of 200: 10% is 20, so 30% is 60, and 5% is 10. Total: 60 + 10 = 70. By multiplication: 0.35 × 200 = 70. ✓
Useful anchors are 50% (half), 25% (a quarter), 10% (divide by 10) and 1% (divide by 100).
Finding what percentage one number is of another
Divide the part by the whole and multiply by 100.
18 out of 72: 18 ÷ 72 = 0.25, which is 25%.
What percentage of 40 is 14? 14 ÷ 40 = 0.35, so 35%.
The most common mistake is dividing the wrong way round. The “whole” is the number after the word “of” and goes on the bottom.
Percentage change
Percentage change tells you how much a quantity grew or shrank compared with where it started.
percentage change = (new value − old value) ÷ old value × 100
Increase. A price goes from 50 to 65. The change is 65 − 50 = 15. Divided by the old value, 15 ÷ 50 = 0.30, so it is a 30% increase.
Decrease. A quantity goes from 80 to 60. The change is −20. Divided by the old value, −20 ÷ 80 = −0.25, so it is a 25% decrease.
It is the original value that goes on the bottom. That is why going from 80 to 60 is a 25% decrease, but going back from 60 to 80 is an increase of 20 ÷ 60, which is about 33.3%. The same two numbers give different percentages depending on which one you start from.
Increasing and decreasing by a percentage
To increase a number by a percentage, multiply by 1 plus the percentage as a decimal. To increase 200 by 10%, multiply by 1.10: 200 × 1.10 = 220.
To decrease, multiply by 1 minus the decimal. To reduce 200 by 20%, multiply by 0.80: 200 × 0.80 = 160.
This single-multiplication method is quicker than working out the percentage and then adding or subtracting, and it becomes important with the next topic.
Reverse percentages
Reverse percentage problems give you the final amount after a percentage change and ask for the original.
A discount example. After a 20% discount, an item costs 48. What was the original price?
A 20% discount means the new price is 80% of the original, so the multiplier is 0.8. Therefore original × 0.8 = 48, and original = 48 ÷ 0.8 = 60.
Check: 60 × 0.8 = 48. ✓
A common mistake is to take 20% of 48 and add it back (48 + 9.60 = 57.60). That is wrong, because the 20% was taken from the original price, not from 48.
An increase example. After adding 10%, a total is 55. The multiplier is 1.10, so original = 55 ÷ 1.10 = 50. Check: 50 × 1.10 = 55. ✓
This is solving a simple equation, a skill from the article on linear equations: the unknown is multiplied by the multiplier, so you divide both sides by it.
Successive percentage changes
Percentage changes do not simply add up, because each one applies to a new base.
Up 50% then down 50%. Start at 100. Increase by 50%: 100 × 1.5 = 150. Decrease by 50%: 150 × 0.5 = 75. The result is 75, not back to 100.
Two discounts, 20% then 10%. The multipliers are 0.8 and 0.9, and 0.8 × 0.9 = 0.72. The final price is 72% of the original, which is a 28% total reduction, not 30%.
Percentages and percentage points
If a rate rises from 10% to 12%, it has risen by 2 percentage points. In relative terms, the rise is 2 ÷ 10 = 20%. These are different statements, and news reports sometimes blur them. Check which one is meant before drawing conclusions.
Reading percentages in context
A percentage hides the size of the underlying numbers. “Sales rose 100%” could mean from 2 to 4 or from 2,000 to 4,000. A percentage out of a small group may be unreliable, and a vivid percentage can look impressive while the actual amount is tiny. Charts that use percentages can also exaggerate differences, a point developed in misleading charts. Percentages relate closely to ratios and proportions, since both compare quantities by scaling.
For money questions such as loans, taxes or investments, the formulas above are general principles only. The details depend on rules and terms that differ by place and product, so check official sources or the provider’s documents for your own situation. The numbers in this article are hypothetical.