How Percentage Calculations Work
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A practical guide to calculating percentages, including discounts, increases, decreases, and common mistakes to avoid.
What a Percentage Represents
A percentage is a way of expressing a number as a fraction of 100. The word percent comes from the Latin per centum, meaning per hundred. When you say 25 percent, you mean 25 out of every 100, or one quarter. Percentages are useful because they provide a common scale for comparing quantities of different sizes. A 10 percent increase on 200 and a 10 percent increase on 2,000 are both 10 percent changes, even though the absolute amounts differ.
Calculating a Percentage of a Number
To find a percentage of a number, convert the percentage to a decimal by dividing by 100, then multiply. For example, to find 15 percent of 80: divide 15 by 100 to get 0.15, then multiply by 80 to get 12. This method works for any percentage and any number, including fractional percentages like 0.5 percent or 12.5 percent.
Calculating Percentage Change
Percentage change tells you how much a value has increased or decreased relative to its original amount. The formula is: ((new value minus original value) divided by original value) times 100. If a price goes from 80 to 100, the change is 20, divided by 80 gives 0.25, times 100 gives a 25 percent increase. If the price drops from 100 to 80, the change is negative 20, divided by 100 gives negative 0.20, times 100 gives a 20 percent decrease. Note that the same absolute change (20) produces different percentages depending on the starting value.
Percentage Points vs Percent Change
A common source of confusion is the difference between percentage points and percent change. If an interest rate rises from 5 percent to 7 percent, that is a 2 percentage point increase, not a 2 percent increase. The percent change is actually 40 percent, because 2 divided by 5 is 0.40. Always clarify which you mean when discussing changes in rates or proportions.
Common Mistakes
One frequent error is adding or subtracting percentages directly when the bases differ. For example, if a store offers 20 percent off and then an additional 10 percent off, the total discount is not 30 percent. The second 10 percent applies to the already-discounted price. On a 100 item, the first discount brings it to 80, and the second brings it to 72, for a total discount of 28 percent. Another mistake is forgetting to convert percentages to decimals before multiplying, which leads to results that are off by a factor of 100.