What is a Percentage?
Percentages are everywhere , sale prices, tax rates, interest, test scores, poll results. The concept is simple, but there are a few calculation patterns that come up constantly, and it's worth having them down cold.
Per cent means per hundred. 45% means 45 out of every 100. To find a percentage of a number, convert to a decimal and multiply: 30% of 80 = 0.30 x 80 = 24. To convert a decimal to a percentage, multiply by 100. To go the other way, divide by 100.
What percent actually means
"Per cent" comes from Latin , per centum , meaning per hundred. A percentage is just a fraction with 100 in the denominator, expressed in a standardized way.
60% means 60/100. 8% means 8/100. 150% means 150/100, which is 1.5 , more than the whole thing. 0.5% means 0.5/100, or 5 out of every thousand.
The reason percentages are useful is that they put everything on the same scale. A class where 18 out of 30 students passed and a class where 120 out of 200 passed are both 60% , same success rate, different raw numbers. Percentages strip away the different totals so you can compare the underlying proportions.
Converting between percentages, decimals, and fractions
You need to be fluid with all three forms. Percentages are for communicating with people. Decimals are for calculating. Fractions are sometimes the clearest way to see what's happening.
Percentage to decimal: divide by 100. Or just move the decimal point two places left. 75% becomes 0.75. 8% becomes 0.08. 130% becomes 1.30.
Decimal to percentage: multiply by 100. Move the decimal point two places right. 0.4 becomes 40%. 0.035 becomes 3.5%. 1.25 becomes 125%.
Fraction to percentage: divide numerator by denominator to get a decimal, then multiply by 100. 3/8 = 0.375 = 37.5%.
Common ones worth knowing by heart: 1/4 = 25%, 1/2 = 50%, 3/4 = 75%, 1/3 ≈ 33.3%, 2/3 ≈ 66.7%, 1/5 = 20%, 1/8 = 12.5%.
The three percentage calculations
Most percentage problems are one of three types. Once you recognize which type you're dealing with, the calculation is straightforward.
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Percentage increase and decrease
Percentage change = (New - Old) / Old x 100. Positive means an increase. Negative means a decrease.
To apply a percentage increase directly: multiply by (1 + rate as a decimal). A $80 item increasing by 25% costs 80 x 1.25 = $100. The 1 keeps the original amount; the 0.25 adds the increase.
To apply a percentage decrease: multiply by (1 - rate). An item at $80 with a 25% discount costs 80 x 0.75 = $60.
One thing that surprises people: a 20% increase followed by a 20% decrease does not return you to where you started. $100 + 20% = $120. $120 - 20% = $96. The second percentage is applied to a different base number. This is why percentage changes don't simply cancel, and why credit card companies love minimum payment structures.
Everyday shortcuts
For tips at a restaurant, the 10% method is faster than any calculator. Move the decimal one place left to find 10%, then adjust. A $65 bill: 10% = $6.50, so 20% = $13, 15% = $9.75.
For sales tax, the multiplier method is cleaner than adding separately. 8.5% tax on a $45 item: 45 x 1.085 = $48.83. One step instead of two.
For discounts, same idea. 30% off $120: 120 x 0.70 = $84. The 0.70 represents what's left after the discount.