Percentages Made Easy: Four Common Problems With Worked Examples
Percentages appear in discounts, tax, tips, grades and pay rises, yet many people still guess or reach for a pencil. The trick is knowing which of four question types you are facing. Once you can name the type, the formula is simple, and a calculator can do the arithmetic instantly.
What a percentage really is
A percentage is a fraction out of 100. So 25 percent means 25 out of 100, or 0.25 as a decimal. Converting between the two forms is the key step in every problem: divide by 100 to go from percent to decimal, and multiply by 100 to go back.
Always decide what the whole is. Most percentage mistakes come from using the wrong base number.
Type one: what is X percent of Y
Multiply the number by the percentage as a decimal. To find 15 percent of 80, calculate 80 multiplied by 0.15, which equals 12. This is the type you need for tips, tax amounts and the size of a discount.
A quick mental shortcut helps you sense-check any result. Ten percent of a number is simply that number with the decimal point moved one place to the left. Five percent is half of that, and twenty percent is double. For example, ten percent of 340 is 34, so fifteen percent is 34 plus 17, giving 51. If the calculator shows something far from your rough estimate, you probably chose the wrong mode or typed a digit incorrectly.
Type two: X is what percent of Y
Divide the part by the whole and multiply by 100. If you scored 42 out of 60, then 42 divided by 60 is 0.7, which is 70 percent. Use it for grades, completion rates and budget shares.
Type three: percentage change
Subtract the old value from the new one, divide by the old value, then multiply by 100. If rent goes from 500 to 560, the change is 60, and 60 divided by 500 is 0.12, so the rent rose by 12 percent. A negative result means a decrease.
Note that going up 20 percent and then down 20 percent does not return you to the start, because the second percentage is taken from a larger base.
Type four: add or subtract a percentage
To add a percentage, multiply by one plus the decimal. A 200 price with 18 percent added is 200 multiplied by 1.18, which is 236. To subtract, multiply by one minus the decimal, so a 25 percent discount on 80 is 80 multiplied by 0.75, which is 60.
Use the PDFImgTools percentage calculator
The percentage calculator has a mode for each of the four types, so you do not need to remember formulas. Everything runs in your browser.
- Open the Percentage Calculator and pick the mode that matches your question.
- Type the two numbers into the labelled fields.
- Read the result, which updates as you type.
- Swap the numbers if you picked the wrong mode and compare.
- Round the answer sensibly, especially when it involves money.
A practical scenario
A freelancer wants to invoice 1,200 plus 14 percent tax, and offer a 10 percent early payment discount on the total. Using the add mode she gets 1,368. The subtract mode then gives 1,231.20. She checks it with the other modes and sends a clear invoice with no guesswork.
Frequently asked questions
Why is a 50 percent increase followed by a 50 percent decrease not neutral?
The decrease is calculated on the larger new value, so you end up below the start. Starting at 100, up 50 percent gives 150, and down 50 percent gives 75.
How do I find the original price before a discount?
Divide the sale price by one minus the discount decimal. If 60 is the price after 25 percent off, then 60 divided by 0.75 gives 80.
What is the difference between percent and percentage points?
Percentage points are the plain gap between two percentages. Going from 10 percent to 12 percent is 2 points but a 20 percent relative increase.
How do I calculate a tip quickly?
Find ten percent by moving the decimal one place left, then add half of it for fifteen percent or double it for twenty percent.
