The Three Essential Percentage Formulas

Every percentage problem can be solved using one of three fundamental formulas. Understanding these formulas and knowing when to apply each one is the key to mastering percentage calculations in any context, from simple homework problems to complex business scenarios.

Formula 1: Finding the Percentage Amount

Part = (Percentage ÷ 100) × Whole

Use when you know the percentage and the total, and need to find the amount.

Formula 2: Finding What Percentage

Percentage = (Part ÷ Whole) × 100

Use when you know the part and the whole, and need to find the percentage.

Formula 3: Finding the Whole Amount

Whole = Part ÷ (Percentage ÷ 100)

Use when you know the part and the percentage, and need to find the total.

Real-World Examples: Shopping and Discounts

Example 1: Calculating Sale Prices

Problem: A jacket originally costs $120 and is on sale for 30% off. What is the sale price?

Step 1: Calculate the discount amount
Discount = (30 ÷ 100) × $120 = 0.30 × $120 = $36

Step 2: Subtract the discount from the original price
Sale Price = $120 - $36 = $84

Alternative Method: Calculate directly
Sale Price = $120 × (100% - 30%) = $120 × 70% = $120 × 0.70 = $84

Example 2: Determining Original Prices

Problem: After a 25% discount, you paid $75 for a item. What was the original price?

Solution: If 25% was taken off, you paid 75% of the original price
Original Price = $75 ÷ (75 ÷ 100) = $75 ÷ 0.75 = $100

Verification: $100 × 25% = $25 discount, so $100 - $25 = $75 ✓

Financial Examples: Tips, Taxes, and Interest

Example 3: Restaurant Tip Calculations

Problem: Your restaurant bill is $85. You want to leave an 18% tip. How much should you pay in total?

Step 1: Calculate the tip amount
Tip = (18 ÷ 100) × $85 = 0.18 × $85 = $15.30

Step 2: Add tip to the original bill
Total = $85 + $15.30 = $100.30

Quick Method: Total = $85 × (100% + 18%) = $85 × 1.18 = $100.30

Example 4: Sales Tax Calculations

Problem: You're buying a $450 laptop. The sales tax rate is 7.5%. What is the total cost?

Solution:
Tax Amount = (7.5 ÷ 100) × $450 = 0.075 × $450 = $33.75
Total Cost = $450 + $33.75 = $483.75

Direct Method: Total = $450 × 1.075 = $483.75

Business and Academic Examples

Example 5: Grade Calculations

Problem: You scored 42 points out of 50 on a test. What percentage did you achieve?

Solution:
Percentage = (42 ÷ 50) × 100 = 0.84 × 100 = 84%

Interpretation: You achieved 84% on the test.

Example 6: Business Growth Analysis

Problem: A company's revenue increased from $2.4 million to $2.7 million. What was the percentage increase?

Step 1: Find the increase amount
Increase = $2.7M - $2.4M = $0.3M

Step 2: Calculate percentage increase
Percentage Increase = ($0.3M ÷ $2.4M) × 100 = 0.125 × 100 = 12.5%

Advanced Formula Applications

Percentage Change Formula

Percentage Change = ((New Value - Old Value) ÷ Old Value) × 100

Positive result = increase, Negative result = decrease

Example 7: Stock Price Changes

Problem: A stock price dropped from $150 to $135. What is the percentage decrease?

Solution:
Percentage Change = (($135 - $150) ÷ $150) × 100
= (-$15 ÷ $150) × 100 = -0.10 × 100 = -10%

Result: The stock decreased by 10%

Compound Percentage Formula

Final Amount = Initial Amount × (1 + rate)^n

Where rate is the decimal percentage and n is the number of periods

Example 8: Compound Interest

Problem: You invest $1,000 at 6% annual compound interest for 3 years. How much will you have?

Solution:
Final Amount = $1,000 × (1 + 0.06)³
= $1,000 × (1.06)³
= $1,000 × 1.191016
= $1,191.02

Quick Reference: Mental Math Shortcuts

Easy Percentage Calculations

  • 50%: Divide by 2 (50% of 80 = 40)
  • 25%: Divide by 4 (25% of 120 = 30)
  • 20%: Divide by 5 (20% of 75 = 15)
  • 10%: Move decimal point left (10% of 350 = 35)
  • 5%: Half of 10% (5% of 200 = 10)
  • 1%: Move decimal point two places left (1% of 500 = 5)

Building Complex Percentages

  • 15%: 10% + 5% (15% of 60 = 6 + 3 = 9)
  • 30%: 3 × 10% (30% of 40 = 3 × 4 = 12)
  • 12.5%: Half of 25% (12.5% of 80 = 40 ÷ 2 = 10)
  • 75%: 3 × 25% (75% of 60 = 3 × 15 = 45)

Practice Problems with Solutions

Problem 1:

A car depreciates 15% in its first year. If it was worth $25,000 initially, what is it worth after one year?

Click for solution

Depreciated value = $25,000 × 15% = $3,750
New value = $25,000 - $3,750 = $21,250
Answer: $21,250

Problem 2:

In a class of 30 students, 18 passed the exam. What percentage passed?

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Percentage = (18 ÷ 30) × 100 = 0.6 × 100 = 60%
Answer: 60% passed

Problem 3:

If 40% of a number is 120, what is the original number?

Click for solution

Original number = 120 ÷ (40 ÷ 100) = 120 ÷ 0.4 = 300
Answer: 300

Using Technology for Complex Calculations

While understanding formulas is essential, modern tools can help with complex percentage calculations:

  • Online Calculators: Use our percentage calculator for instant results with step-by-step explanations
  • Spreadsheet Functions: Excel and Google Sheets offer percentage functions like =PERCENTILE() and =PERCENT()
  • Mobile Apps: Calculator apps with percentage buttons for quick calculations
  • Programming: Languages like Python can handle complex percentage arrays and datasets

Remember: Technology should supplement, not replace, your understanding of the underlying mathematical principles. Always verify that your results make logical sense in the context of the problem.