Key Takeaways
- To find X% of Y: Multiply Y by (X/100), or move decimal left 2 places and multiply
- To find what % X is of Y: Divide X by Y and multiply by 100
- Percent change: ((New - Old) / Old) x 100, can be positive (increase) or negative (decrease)
- Quick trick: 10% is just moving the decimal one place left; 5% is half of that
- Percentage increases and decreases are not symmetric--a 50% increase followed by 50% decrease doesn't return to the original
Understanding Percentages
A percentage is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin "per centum," meaning "by the hundred." Percentages are everywhere in daily life--from sales discounts and tax rates to test scores and financial returns.
The percentage symbol (%) essentially means "divided by 100." So 25% = 25/100 = 0.25 as a decimal. This simple relationship is the foundation of all percentage calculations.
Percentage = (Part / Whole) x 100
Types of Percentage Calculations
Finding X% of a Number
This is the most common percentage calculation. To find X% of Y, multiply Y by X/100 (or equivalently, by the decimal form of X%).
Example: What is 20% of 150?
Finding What Percentage One Number is of Another
To find what percentage X is of Y, divide X by Y and multiply by 100. This tells you what portion X represents of the total Y.
Example: 45 is what % of 180?
Calculating Percentage Change
Percentage change measures the relative change between an old value and a new value. A positive result indicates an increase; a negative result indicates a decrease.
% Change = ((New - Old) / Old) x 100
Example: Price increased from $50 to $65
Percentage Difference
Percentage difference compares two values without designating either as the "original." It's useful when comparing two independent measurements where neither is clearly the baseline.
% Difference = (|V1 - V2| / ((V1 + V2) / 2)) x 100
Pro Tip: Mental Math Shortcuts
To find 10% of any number, just move the decimal point one place left (10% of 450 = 45). For 5%, take half of 10%. For 15%, add 10% and 5% together. For 1%, move the decimal two places left.
Common Percentage Applications
| Percentage | Decimal | Fraction | Common Use |
|---|---|---|---|
| 10% | 0.10 | 1/10 | Quick estimates, easy tip |
| 15% | 0.15 | 3/20 | Standard tip |
| 20% | 0.20 | 1/5 | Good tip, sales discounts |
| 25% | 0.25 | 1/4 | Quarter portion, discounts |
| 33.33% | 0.333... | 1/3 | Splitting in thirds |
| 50% | 0.50 | 1/2 | Half off sales, 50/50 splits |
| 75% | 0.75 | 3/4 | Three-quarter marks |
| 100% | 1.00 | 1/1 | The whole, full amount |
Frequently Asked Questions
Percent change measures the change from a specific starting value (old to new), so it has a clear direction (increase or decrease). Percent difference compares two values without designating either as the original--it uses their average as the reference and is always positive. Use percent change when you have a clear "before" and "after"; use percent difference when comparing two independent values.
Percentage changes are calculated based on the current value, not the original. If $100 increases by 50%, you have $150. Then a 50% decrease is calculated from $150, which is $75 reduction, leaving you with $75--not $100. This asymmetry is why losses hurt more than gains of the same percentage.
To find the original price before a percentage discount: divide the sale price by (1 - discount%). For example, if an item costs $60 after a 20% discount: $60 / (1 - 0.20) = $60 / 0.80 = $75 original price.
A percentage point is an absolute difference between two percentages, while a percent change is a relative change. If interest rates rise from 5% to 8%, that's a 3 percentage point increase but a 60% increase in the rate ((8-5)/5 x 100). This distinction is important in finance and statistics.
For compound growth over multiple periods, use: Final = Initial x (1 + rate)^periods. For example, $100 growing at 5% per year for 3 years: $100 x (1.05)^3 = $115.76. You cannot simply add 5% three times (which would give $115).
Master Percentage Calculations
Whether you're calculating discounts, analyzing data, or solving homework problems, understanding percentages is an essential skill. Bookmark this calculator for quick access whenever you need it.