How to Calculate Percentage: Complete Guide with Formulas & Examples
Learn how to calculate percentages easily and accurately! Whether you're a student preparing for exams, a professional analyzing data, or someone managing personal finances, understanding percentage calculations is an essential skill. This comprehensive guide, created by math education experts at RevisionTown, covers all percentage formulas, step-by-step methods, practical examples, and real-world applications.
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What is a Percentage?
A percentage is a way of expressing a number as a fraction of 100. The word "percent" comes from the Latin "per centum," meaning "out of one hundred." Percentages are denoted by the symbol % and are used to compare quantities, express proportions, and describe changes.
Key Concept: 1% means 1 out of 100, or \( \frac{1}{100} = 0.01 \)
Understanding this fundamental relationship is crucial for all percentage calculations:
- 50% = 50 out of 100 = \( \frac{50}{100} = 0.5 \) = one half
- 25% = 25 out of 100 = \( \frac{25}{100} = 0.25 \) = one quarter
- 100% = 100 out of 100 = \( \frac{100}{100} = 1 \) = the whole
The Basic Percentage Formula
All percentage calculations stem from one fundamental formula:
The Master Percentage Formula:
\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100 \]
Alternative forms of this formula:
\[ \text{Part} = \frac{\text{Percentage}}{100} \times \text{Whole} \]
\[ \text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100 \]
Memory Aid: Think of these three variables as a triangle. If you know any two, you can find the third.
Method 1: How to Find X% of a Number
This is the most common type of percentage calculation. You need to find what a certain percentage of a given number equals.
The Formula
\[ \text{Result} = \frac{\text{Percentage}}{100} \times \text{Number} \]
Step-by-Step Process:
- Convert the percentage to a decimal by dividing by 100
- Multiply the decimal by the number
- The result is your answer
Example 1: What is 30% of 250?
Solution:
Step 1: Convert percentage to decimal: \( 30\% = \frac{30}{100} = 0.30 \)
Step 2: Multiply: \( 0.30 \times 250 = 75 \)
Answer: 30% of 250 is 75
Verification: \( \frac{75}{250} \times 100 = 30\% \) ✓
Example 2: Calculate 15% of $480
Solution:
\[ \text{Result} = \frac{15}{100} \times 480 = 0.15 \times 480 = 72 \]
Answer: 15% of $480 is $72
Real-world application: This is useful for calculating tips, discounts, or tax amounts.
Method 2: How to Find What Percentage One Number Is of Another
When you need to determine what percentage one number represents of another number, use this method.
The Formula
\[ \text{Percentage} = \frac{\text{Part}}{\text{Total}} \times 100 \]
Step-by-Step Process:
- Divide the part by the total
- Multiply the result by 100
- Add the % symbol
Example 3: 45 is what percentage of 180?
Solution:
Step 1: Divide: \( \frac{45}{180} = 0.25 \)
Step 2: Multiply by 100: \( 0.25 \times 100 = 25 \)
Answer: 45 is 25% of 180
Example 4: You scored 42 out of 50 on a test. What's your percentage?
Solution:
\[ \text{Percentage} = \frac{42}{50} \times 100 = 0.84 \times 100 = 84\% \]
Answer: Your score is 84%
Method 3: How to Find the Total When You Know the Percentage
Sometimes you know the percentage and the part, but need to find the whole (total) amount.
The Formula
\[ \text{Whole} = \frac{\text{Part}}{\text{Percentage}} \times 100 \]
Alternative form:
\[ \text{Whole} = \frac{\text{Part}}{\text{Decimal form of percentage}} \]
Example 5: 60 is 40% of what number?
Solution:
Method 1: \( \text{Whole} = \frac{60}{40} \times 100 = 1.5 \times 100 = 150 \)
Method 2: \( \text{Whole} = \frac{60}{0.40} = 150 \)
Answer: 60 is 40% of 150
Example 6: A $45 shirt is on sale for 25% off. What was the original price?
Solution:
$45 represents 75% of the original price (100% - 25% = 75%)
\[ \text{Original Price} = \frac{45}{0.75} = 60 \]
Answer: The original price was $60
How to Calculate Percentage Increase and Decrease
Percentage change measures how much a value has increased or decreased relative to its original value.
The Formula
\[ \text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{\text{Old Value}} \times 100 \]
For Percentage Increase:
\[ \text{Percentage Increase} = \frac{\text{Increase}}{\text{Original}} \times 100 \]
For Percentage Decrease:
\[ \text{Percentage Decrease} = \frac{\text{Decrease}}{\text{Original}} \times 100 \]
Important: Always use the original (old) value as the denominator, not the new value.
Example 7: A stock price increased from $50 to $65. What's the percentage increase?
Solution:
Step 1: Find the increase: \( 65 - 50 = 15 \)
Step 2: Calculate percentage: \( \frac{15}{50} \times 100 = 0.3 \times 100 = 30\% \)
Answer: The stock price increased by 30%
Example 8: A product's price dropped from $120 to $90. What's the percentage decrease?
Solution:
Step 1: Find the decrease: \( 120 - 90 = 30 \)
Step 2: Calculate percentage: \( \frac{30}{120} \times 100 = 0.25 \times 100 = 25\% \)
Answer: The price decreased by 25%
How to Calculate Percentage Difference
Percentage difference compares two values without considering which is the "original" or "new" value.
\[ \text{Percentage Difference} = \frac{|\text{Value 1} - \text{Value 2}|}{\frac{\text{Value 1} + \text{Value 2}}{2}} \times 100 \]
Or simplified:
\[ \text{Percentage Difference} = \frac{|\text{Value 1} - \text{Value 2}|}{\text{Average of both values}} \times 100 \]
Note: The vertical bars | | indicate absolute value (always positive).
Example 9: Compare values 80 and 100
Solution:
Step 1: Find absolute difference: \( |80 - 100| = 20 \)
Step 2: Find average: \( \frac{80 + 100}{2} = 90 \)
Step 3: Calculate: \( \frac{20}{90} \times 100 = 22.22\% \)
Answer: The percentage difference is 22.22%
Common Percentage Calculations You'll Use
1. Calculating Tips and Gratuities
Formula: \( \text{Tip} = \text{Bill Amount} \times \frac{\text{Tip \%}}{100} \)
Example: 18% tip on a $65 dinner bill
\( \text{Tip} = 65 \times 0.18 = \$11.70 \)
Quick method: For 15%, move the decimal point left one place and add half that amount.
2. Calculating Sales Tax
Formula: \( \text{Tax} = \text{Price} \times \frac{\text{Tax Rate}}{100} \)
Total with tax: \( \text{Total} = \text{Price} \times (1 + \frac{\text{Tax Rate}}{100}) \)
Example: $50 item with 8% tax
\( \text{Total} = 50 \times 1.08 = \$54 \)
3. Calculating Discounts
Formula: \( \text{Discount} = \text{Original Price} \times \frac{\text{Discount \%}}{100} \)
Sale Price: \( \text{Sale Price} = \text{Original} \times (1 - \frac{\text{Discount \%}}{100}) \)
Example: 30% off a $80 jacket
\( \text{Sale Price} = 80 \times 0.70 = \$56 \)
4. Calculating Interest (Simple)
Formula: \( \text{Interest} = \text{Principal} \times \frac{\text{Rate}}{100} \times \text{Time} \)
Example: $1000 at 5% annual interest for 2 years
\( \text{Interest} = 1000 \times 0.05 \times 2 = \$100 \)
Quick Mental Math Tricks for Percentages
These shortcuts can help you calculate percentages quickly without a calculator:
Trick 1: Finding 10%
Move the decimal point one place to the left.
Example: 10% of 450 = 45.0
Trick 2: Finding 1%
Move the decimal point two places to the left.
Example: 1% of 350 = 3.50
Trick 3: Finding 5%
Find 10% and divide by 2.
Example: 5% of 80 = (10% of 80) ÷ 2 = 8 ÷ 2 = 4
Trick 4: Finding 50%
Simply divide by 2.
Example: 50% of 64 = 32
Trick 5: Finding 25%
Divide by 4.
Example: 25% of 200 = 50
Trick 6: The Commutative Property
X% of Y = Y% of X
Example: 4% of 75 = 75% of 4 = 3 (easier to calculate!)
Quick Reference: Percentage, Decimal, and Fraction Conversions
| Percentage | Decimal | Fraction | Common Use |
|---|---|---|---|
| 1% | 0.01 | \( \frac{1}{100} \) | Small increments |
| 5% | 0.05 | \( \frac{1}{20} \) | Sales tax (some states) |
| 10% | 0.10 | \( \frac{1}{10} \) | Basic tip |
| 15% | 0.15 | \( \frac{3}{20} \) | Standard tip |
| 20% | 0.20 | \( \frac{1}{5} \) | Generous tip |
| 25% | 0.25 | \( \frac{1}{4} \) | One quarter |
| 33.33% | 0.3333 | \( \frac{1}{3} \) | One third |
| 50% | 0.50 | \( \frac{1}{2} \) | One half |
| 66.67% | 0.6667 | \( \frac{2}{3} \) | Two thirds |
| 75% | 0.75 | \( \frac{3}{4} \) | Three quarters |
| 100% | 1.00 | \( \frac{1}{1} \) | The whole |
Real-World Applications of Percentage Calculations
In Finance and Business
- Profit Margins: \( \text{Profit Margin} = \frac{\text{Profit}}{\text{Revenue}} \times 100 \)
- Return on Investment (ROI): \( \text{ROI} = \frac{\text{Gain - Cost}}{\text{Cost}} \times 100 \)
- Interest Rates: Calculating loan payments and investment returns
- Market Changes: Stock price fluctuations
- Budget Allocation: Distributing funds across categories
In Education
- Test Scores: Converting raw scores to percentages
- Grade Point Averages: Weighted calculations
- Attendance Rates: Tracking student participation
- Improvement Tracking: Measuring progress over time
In Daily Life
- Shopping: Calculating sale prices and discounts
- Restaurants: Determining tips and splitting bills
- Nutrition: Daily value percentages on food labels
- Fitness: Body fat percentage, workout improvements
- Surveys: Understanding poll results and statistics
Common Mistakes to Avoid When Calculating Percentages
Mistake 1: Using the Wrong Base
Wrong: Calculating percentage increase using the new value as the denominator
Correct: Always use the original (old) value as the base for percentage change
Example: Increase from 50 to 75:
❌ Wrong: \( \frac{25}{75} \times 100 = 33.33\% \)
✓ Correct: \( \frac{25}{50} \times 100 = 50\% \)
Mistake 2: Confusing Percentage Points with Percentages
Example: If interest rates go from 5% to 8%:
✓ The rate increased by 3 percentage points
✓ The rate increased by 60% (relative increase)
These are different measures!
Mistake 3: Multiple Percentage Changes
Wrong: Adding consecutive percentage changes
Example: A 10% increase followed by a 10% decrease does NOT return to the original value
Starting with 100:
After 10% increase: \( 100 \times 1.10 = 110 \)
After 10% decrease: \( 110 \times 0.90 = 99 \)
You end up with 99, not 100!
Mistake 4: Rounding Too Early
Tip: Keep as many decimal places as possible during calculations and round only at the end
This prevents accumulation of rounding errors, especially in multi-step problems
Practice Problems with Solutions
Practice Problem 1
Question: A store offers a 40% discount on a $125 item. What is the sale price?
Solution:
Discount amount: \( 125 \times 0.40 = \$50 \)
Sale price: \( 125 - 50 = \$75 \)
Or directly: \( 125 \times (1 - 0.40) = 125 \times 0.60 = \$75 \)
Practice Problem 2
Question: Your salary increased from $45,000 to $52,000. What's the percentage increase?
Solution:
Increase: \( 52,000 - 45,000 = \$7,000 \)
Percentage: \( \frac{7,000}{45,000} \times 100 = 15.56\% \)
Practice Problem 3
Question: If 35% of students passed an exam and 91 students passed, how many students took the exam?
Solution:
\( \text{Total Students} = \frac{91}{0.35} = 260 \)
Therefore, 260 students took the exam.
Practice Problem 4
Question: A population of 12,000 grew to 15,600. Calculate both the absolute increase and percentage increase.
Solution:
Absolute increase: \( 15,600 - 12,000 = 3,600 \)
Percentage increase: \( \frac{3,600}{12,000} \times 100 = 30\% \)
Expert Tips for Mastering Percentage Calculations
Tip 1: Memorize Key Percentages
Know common percentages like 10%, 25%, 50%, and 75% by heart. This makes mental calculations much faster.
Tip 2: Check Your Answer
Always verify if your answer makes sense. If you calculate 120% of 50 and get 25, something's wrong!
Tip 3: Practice with Real Scenarios
Use percentage calculations in your daily life. Calculate tips, discounts, and price comparisons while shopping.
Tip 4: Understand the Context
Read problems carefully to understand what percentage you're calculating and what the base value should be.
Tip 5: Use the Triangle Method
Draw a triangle with "Part" at the top, "Whole" at bottom-left, and "Percentage" at bottom-right. Cover what you're looking for to see the formula.
Quick Summary: Percentage Calculation Formulas
| What You're Finding | Formula | Example |
|---|---|---|
| X% of a number | \( \frac{X}{100} \times \text{Number} \) | 25% of 80 = 20 |
| What % is X of Y | \( \frac{X}{Y} \times 100 \) | 15 is 30% of 50 |
| X is Y% of what | \( \frac{X}{Y} \times 100 \) | 20 is 25% of 80 |
| Percentage increase | \( \frac{\text{New} - \text{Old}}{\text{Old}} \times 100 \) | 50→75 is 50% increase |
| Percentage decrease | \( \frac{\text{Old} - \text{New}}{\text{Old}} \times 100 \) | 100→80 is 20% decrease |
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