K-12 Percentage Formulas

📊 Complete Percentage Formulas Guide 📊

Everything You Need to Know About Percentages (K-12)

🌟 The Three Basic Percentage Formulas 🌟

Percentage = PartWhole × 100
Part = Percentage × Whole100
Whole = Part × 100Percentage

Understanding Percentages

What is a Percentage?
A percentage is a way of expressing a number as a fraction of 100.
The word "percent" means "per hundred" (cent = 100).
Symbol: %

Key Concepts

Part

The portion or piece you're interested in

Example: 25 students passed

Whole

The total or complete amount

Example: 100 total students

Percentage

The part expressed per hundred

Example: 25% passed

Converting Between Forms

1. Percentage to Decimal

Decimal = Percentage100
Or simply: Divide by 100 (move decimal point 2 places left)
• 25% = 25 ÷ 100 = 0.25
• 8% = 8 ÷ 100 = 0.08
• 150% = 150 ÷ 100 = 1.50

2. Decimal to Percentage

Percentage = Decimal × 100
Multiply by 100 (move decimal point 2 places right)
• 0.75 = 0.75 × 100 = 75%
• 0.05 = 0.05 × 100 = 5%
• 1.25 = 1.25 × 100 = 125%

3. Percentage to Fraction

Fraction = Percentage100
Then simplify if possible
• 50% = 50/100 = 1/2
• 25% = 25/100 = 1/4
• 75% = 75/100 = 3/4

4. Fraction to Percentage

Percentage = NumeratorDenominator × 100
Divide numerator by denominator, then multiply by 100
• 3/4 = 0.75 × 100 = 75%
• 1/8 = 0.125 × 100 = 12.5%
• 5/2 = 2.5 × 100 = 250%

Common Conversions Reference

FractionDecimalPercentage
1/20.550%
1/40.2525%
3/40.7575%
1/30.333...33.33%
2/30.666...66.67%
1/50.220%
1/80.12512.5%
1/100.110%
1/1000.011%

Basic Percentage Calculations

1. Finding Percentage of a Number

Result = Percentage100 × Number
Or: Result = Percentage% × Number
Example: What is 30% of 200?
Result = (30/100) × 200 = 0.30 × 200 = 60

2. Finding What Percent One Number is of Another

Percentage = PartWhole × 100
Example: 45 is what percent of 180?
Percentage = (45/180) × 100 = 0.25 × 100 = 25%

3. Finding the Whole When Given Part and Percentage

Whole = PartPercentage × 100
Or: Whole = PartDecimal
Example: 60 is 40% of what number?
Whole = (60/40) × 100 = 1.5 × 100 = 150
Or: Whole = 60/0.40 = 150

Percentage Change & Difference

1. Percentage Increase

% Increase = New Value - Original ValueOriginal Value × 100
New Value = Original × (1 + Percentage100)
Example: A price increases from $50 to $65
% Increase = ((65-50)/50) × 100 = (15/50) × 100 = 30%

2. Percentage Decrease

% Decrease = Original Value - New ValueOriginal Value × 100
New Value = Original × (1 - Percentage100)
Example: A price decreases from $80 to $60
% Decrease = ((80-60)/80) × 100 = (20/80) × 100 = 25%

3. Percentage Change (General)

% Change = |New - Old|Old × 100
Positive result = Increase
Negative result = Decrease
Use absolute value if direction doesn't matter

4. Percentage Difference

% Difference = |Value₁ - Value₂|(Value₁ + Value₂) / 2 × 100
Used when comparing two values without a clear "original"
Example: Compare 80 and 100
% Difference = (|80-100|/((80+100)/2)) × 100 = (20/90) × 100 = 22.22%

5. Percentage Points

Percentage Points = New % - Old %
Important: Percentage points ≠ Percent change!
Example: Interest rate increases from 5% to 8%
• Change in percentage points = 8% - 5% = 3 percentage points
• Percent change = ((8-5)/5) × 100 = 60% increase

Real-World Applications

1. Discount & Sale Price

Discount Amount = Original Price × Discount %100
Sale Price = Original Price - Discount Amount
Sale Price = Original Price × (1 - Discount %100)
Example: 30% off a $50 item
Sale Price = 50 × (1 - 0.30) = 50 × 0.70 = $35

2. Markup & Selling Price

Markup Amount = Cost × Markup %100
Selling Price = Cost + Markup Amount
Selling Price = Cost × (1 + Markup %100)
Example: 40% markup on $30 cost
Selling Price = 30 × (1 + 0.40) = 30 × 1.40 = $42

3. Tax & Total Price

Tax Amount = Price × Tax Rate %100
Total = Price + Tax Amount
Total = Price × (1 + Tax Rate %100)
Example: 8% tax on $100 purchase
Total = 100 × (1 + 0.08) = 100 × 1.08 = $108

4. Tip Calculation

Tip Amount = Bill × Tip %100
Total = Bill + Tip Amount
Example: 20% tip on $60 bill
Tip = 60 × 0.20 = $12
Total = $60 + $12 = $72

5. Profit & Loss

Profit/Loss = Selling Price - Cost Price
Profit % = ProfitCost Price × 100
Loss % = LossCost Price × 100
Example: Bought for $80, sold for $100
Profit = $100 - $80 = $20
Profit % = (20/80) × 100 = 25%

6. Commission

Commission = Sales Amount × Commission %100
Example: 5% commission on $5,000 sales
Commission = 5000 × 0.05 = $250

Advanced Percentage Concepts

1. Successive Percentage Changes

Final = Original × (1 ± p₁100) × (1 ± p₂100)
Important: You CANNOT just add the percentages!
Example: Price increases 20%, then decreases 20%
Final = Original × 1.20 × 0.80 = Original × 0.96
Result: 4% decrease (not 0%!)

2. Net Percentage Change

Net % = a + b + ab100
For two successive changes of a% and b%
Use + for increase, - for decrease
Example: +10% then +20%
Net = 10 + 20 + (10×20)/100 = 30 + 2 = 32%

3. Reverse Percentage (Finding Original)

Original = Final1 ± (Percentage/100)
Example: After 25% increase, price is $150. Find original.
Original = 150 / 1.25 = $120

4. Percentage of Percentage

Result = p₁ × p₂100 of Total
Example: 30% of 40% of 200
= (30 × 40)/100 of 200
= 12% of 200 = 24

5. Percentage Allocation

Part = Total × Individual %Sum of all %
Example: Divide $1000 in ratio 2:3:5
Total parts = 2+3+5 = 10
First: 1000 × 2/10 = $200
Second: 1000 × 3/10 = $300
Third: 1000 × 5/10 = $500

Comprehensive Worked Examples

Example 1: Finding Percentage

Question: In a class of 40 students, 32 passed. What percentage passed?
1 Identify Part and Whole: Part = 32, Whole = 40
2 Apply formula: Percentage = (Part/Whole) × 100
3 Calculate: (32/40) × 100 = 0.8 × 100 = 80%
Answer: 80% of students passed

Example 2: Percentage of a Number

Question: What is 35% of 240?
1 Convert percentage to decimal: 35% = 0.35
2 Multiply: 0.35 × 240
3 Calculate: 0.35 × 240 = 84
Answer: 84

Example 3: Finding the Whole

Question: 45 is 15% of what number?
1 Use formula: Whole = (Part/Percentage) × 100
2 Substitute: Whole = (45/15) × 100
3 Calculate: 3 × 100 = 300
Answer: 300

Example 4: Percentage Increase

Question: A salary increased from $40,000 to $46,000. Find the percentage increase.
1 Find the increase: $46,000 - $40,000 = $6,000
2 Use formula: % Increase = (Increase/Original) × 100
3 Calculate: (6,000/40,000) × 100 = 0.15 × 100 = 15%
Answer: 15% increase

Example 5: Discount Problem

Question: A $180 jacket is on sale for 40% off. What is the sale price?
1 Method 1: Calculate discount: 180 × 0.40 = $72
2 Subtract from original: $180 - $72 = $108
OR
1 Method 2: Direct calculation: 180 × (1 - 0.40) = 180 × 0.60 = $108
Answer: Sale price is $108

Example 6: Successive Changes

Question: A price increases by 25%, then decreases by 20%. What is the net change?
1 Let original = 100 (for easy calculation)
2 After 25% increase: 100 × 1.25 = 125
3 After 20% decrease: 125 × 0.80 = 100
4 Or use formula: Net = 25 + (-20) + (25×(-20))/100 = 5 - 5 = 0%
Answer: No net change (back to original price)

Example 7: Profit and Loss

Question: An item bought for $150 is sold for $195. Find profit percentage.
1 Calculate profit: $195 - $150 = $45
2 Use formula: Profit % = (Profit/Cost Price) × 100
3 Calculate: (45/150) × 100 = 0.3 × 100 = 30%
Answer: 30% profit

🧮 Interactive Percentage Calculator

🎯 Quick Tips & Shortcuts

💡 Mental Math Shortcuts

• 10% → Divide by 10

• 5% → Half of 10%

• 20% → Double 10%

• 25% → Divide by 4

• 50% → Divide by 2

• 1% → Divide by 100

💡 Common Mistakes to Avoid

✗ Adding/subtracting percentages directly

✗ Confusing percentage points with percent

✗ Using wrong base for calculations

✓ Always identify part, whole, percentage

✓ Check if answer makes sense

💡 Percentage > 100%

Yes, percentages can exceed 100%!

Example: 150% means 1.5 times

200% means double

300% means triple

💡 Percentage of Increase/Decrease

ALWAYS use the ORIGINAL value as the denominator!

% Change = (Difference/Original) × 100

Not (Difference/New) × 100