What Is a Percentage?
The word percent comes from the Latin per centum — "per hundred." That's really all a percentage is: a way of expressing a number as a portion of 100. When you see 45%, it means 45 out of every 100.
You already use percentages all the time without thinking about it. A 20% tip at a restaurant. A test score of 85%. A store advertising 30% off. In each case, someone took a number and expressed it as a piece of a hundred — because that makes it easier to understand and compare than fractions or raw counts would.
Percentages are useful precisely because they put everything on the same scale. A class where 18 out of 30 students passed and a class where 24 out of 40 passed — which did better? It's not obvious from the raw numbers. Convert both to percentages and it is: 60% vs. 60%. Same result. That's the whole point.
Part of a Whole
The basic formula for finding a percentage is:
$$\text{percentage} = \frac{\text{part}}{\text{whole}} \times 100\%$$
Say three classrooms have these counts of brown-haired students: 10 out of 20, 12 out of 24, and 9 out of 14. The raw numbers don't tell you much at a glance. Convert each to a percentage and the picture becomes clear:
$$\frac{10}{20} \times 100\% = 50\%$$
$$\frac{12}{24} \times 100\% = 50\%$$
$$\frac{9}{14} \times 100\% \approx 64.3\%$$
The first two classrooms are half brown-haired. The third has fewer brown-haired students in absolute terms, but a higher percentage — because the class is smaller overall. That's what percentages do: they let you compare proportions across different-sized groups.
Percent Change
Percentages also show up when describing how something changes over time. Say a city recorded 44.0 inches of rainfall one year, then 49.0 inches the next. How big was the increase?
Divide the change by the original value:
$$\frac{49.0 - 44.0}{44.0} \times 100\% = \frac{5.0}{44.0} \times 100\% \approx 11.4\%$$
Rainfall increased by about 11.4%. You'll sometimes see this framed as "this year's rainfall was 111.4% of last year's" — that's the same calculation, just without subtracting 100% at the end.
Converting Between Percentages and Decimals
Since percentages are based on hundreds, converting to and from decimals is just a matter of moving the decimal point two places.
Percent to decimal — move the decimal two places left (divide by 100):
$$44\% = 0.44$$ $$13.5\% = 0.135$$ $$100\% = 1.00$$ $$1.5\% = 0.015$$
Decimal to percent — move the decimal two places right (multiply by 100):
$$0.25 = 25\%$$ $$1.00 = 100\%$$ $$2.50 = 250\%$$ $$0.055 = 5.5\%$$
Note that a decimal greater than 1 gives you a percentage greater than 100% — that just means "more than the whole," which comes up with things like percent change or relative comparisons.
Quick check: a store says prices are "125% of last year's prices." Does that mean things got cheaper or more expensive? Show answerMore expensive. 125% of the original price means you're paying the original amount plus an extra 25% — a 25% increase.
Work Through These
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A class of 30 students has 18 who passed the test. What percentage passed? Show answer\(\dfrac{18}{30} \times 100\% = \mathbf{60\%}\)
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A jacket originally costs $80 and is on sale for $60. What is the percent decrease? Show answer\(\dfrac{80 - 60}{80} \times 100\% = \dfrac{20}{80} \times 100\% = \mathbf{25\%}\) decrease
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Convert 0.375 to a percentage. Show answer\(0.375 \times 100 = \mathbf{37.5\%}\) — move the decimal two places right
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Convert 7.5% to a decimal. Show answer\(7.5 \div 100 = \mathbf{0.075}\) — move the decimal two places left