Percentage Formula Guide
Master percentage calculations, increases, decreases, and practical applications with examples
Percentage Formula Guide - Master Percentage Calculations with Ease
The word "percentage" comes from the Latin "per centum," meaning "by the hundred." A percentage is a way of expressing a number as a fraction of 100. It is denoted using the percent sign (%). For example, 45% is equal to 45/100 or 0.45.
Percentages are used in countless real-world situations: calculating discounts during sales, determining interest rates on loans, analyzing data in statistics, calculating grades in education, and much more. Understanding how to work with percentages is an essential mathematical skill with practical applications in everyday life.
The concept of percentages allows us to compare quantities of different sizes by standardizing them to a common base of 100. This makes it easier to understand proportions and make comparisons across different contexts.
Key Insight: Percentages are essentially fractions with a denominator of 100. Converting between percentages, fractions, and decimals is a fundamental skill in percentage calculations.
Use our advanced percentage calculator to quickly solve percentage problems, calculate increases and decreases, and check your work.
Use Percentage Calculator →There are three fundamental types of percentage problems, each with its own formula:
This calculation answers the question: "What is X% of Y?"
Example: What is 25% of 80?
Solution: (25 ÷ 100) × 80 = 0.25 × 80 = 20
So, 25% of 80 is 20.
This calculation answers the question: "X is what percent of Y?"
Example: 15 is what percent of 60?
Solution: (15 ÷ 60) × 100 = 0.25 × 100 = 25%
So, 15 is 25% of 60.
This calculation answers the question: "X is Y% of what number?"
Example: 30 is 15% of what number?
Solution: (30 ÷ 15) × 100 = 2 × 100 = 200
So, 30 is 15% of 200.
Determine which of the three basic percentage problems you're solving:
Clearly identify and write down:
Label each value to avoid confusion during calculations.
Based on the problem type, choose the correct formula:
Substitute the known values into the formula and calculate step by step:
Check if your answer makes sense in the context of the problem:
Percentage changes are used to describe how much a quantity has grown or reduced relative to its original value. These calculations are essential in finance, economics, statistics, and everyday situations like sales and discounts.
This calculation answers: "By what percentage has a value increased from its original amount?"
Example: A product's price increased from $50 to $65. What is the percentage increase?
Solution: [(65 - 50) ÷ 50] × 100 = (15 ÷ 50) × 100 = 0.3 × 100 = 30%
The price increased by 30%.
This calculation answers: "By what percentage has a value decreased from its original amount?"
Example: A company's revenue decreased from $80,000 to $68,000. What is the percentage decrease?
Solution: [(80,000 - 68,000) ÷ 80,000] × 100 = (12,000 ÷ 80,000) × 100 = 0.15 × 100 = 15%
The revenue decreased by 15%.
If you know the original value and the percentage change, you can calculate the new value:
Use "+" for increase and "-" for decrease.
Example: A $200 item is on sale with a 25% discount. What is the sale price?
Solution: 200 × (1 - 25/100) = 200 × (1 - 0.25) = 200 × 0.75 = $150
The sale price is $150.
Percentages are extensively used in education to calculate grades, scores, and performance metrics. Understanding these calculations is crucial for students, teachers, and parents.
To find the percentage score in an exam:
Example: A student scores 42 out of 50 in a test. What is the percentage score?
Solution: (42 ÷ 50) × 100 = 0.84 × 100 = 84%
The student scored 84%.
Many educational institutions use different grading systems. Here's a common conversion:
| Percentage Range | Letter Grade | GPA (4.0 Scale) |
|---|---|---|
| 97-100% | A+ | 4.0 |
| 93-96% | A | 4.0 |
| 90-92% | A- | 3.7 |
| 87-89% | B+ | 3.3 |
| 83-86% | B | 3.0 |
| 80-82% | B- | 2.7 |
| 77-79% | C+ | 2.3 |
| 73-76% | C | 2.0 |
| 70-72% | C- | 1.7 |
| 67-69% | D+ | 1.3 |
| 65-66% | D | 1.0 |
| Below 65% | F | 0.0 |
When different assignments or exams have different weights in the final grade:
Example: A course has three components: Homework (20% weight, score 85%), Midterm (30% weight, score 78%), Final (50% weight, score 92%). What is the final grade?
Solution: (85×0.20 + 78×0.30 + 92×0.50) ÷ (0.20+0.30+0.50) = (17 + 23.4 + 46) ÷ 1 = 86.4%
The final grade is 86.4%.
When multiple percentage changes occur sequentially, the overall change is not simply the sum of the individual changes.
Example: A price first increases by 20%, then decreases by 15%. What is the overall percentage change?
Solution: Overall change = 100 × [(1 + 20/100) × (1 - 15/100) - 1] = 100 × [(1.20) × (0.85) - 1] = 100 × [1.02 - 1] = 2%
The overall price increased by 2%.
Finding the original value when you know the final value and the percentage change:
Example: After a 25% discount, a item costs $75. What was the original price?
Solution: Original price = 75 ÷ (1 - 25/100) = 75 ÷ 0.75 = $100
The original price was $100.
It's important to distinguish between percentage points and percentages:
Percentage Points: Absolute difference between two percentages. If an interest rate increases from 5% to 7%, it has increased by 2 percentage points.
Percentage Change: Relative difference expressed as a percentage. If an interest rate increases from 5% to 7%, it has increased by 40% (because (7-5)/5 = 0.4 = 40%).
A percentage is a proportion out of 100, while a percentile indicates the value below which a given percentage of observations fall. For example, if you score in the 90th percentile on a test, it means you scored better than 90% of test-takers.
You can use mental math tricks like:
Percentages can exceed 100% when comparing to a original value that is not the maximum possible. For example, if sales increase from $50,000 to $120,000, that's a 140% increase (more than doubling).
You generally don't add or subtract percentages directly unless they're percentages of the same base. Instead, convert to actual values, perform the operation, then convert back to percentages if needed.
For a 15% tip, calculate 10% of the bill and add half of that amount. For 20%, just double the 10% amount. For example, on a $60 bill: 10% = $6, so 15% = $6 + $3 = $9, and 20% = $12.