Percent of Change

Have you ever wondered how much something increased or decreased? Percent of change tells us exactly that - how much a quantity changed relative to its original amount. This is everywhere in real life: price changes, population growth, sale discounts, and more!

What Is Percent of Change?

Percent of change tells us how much a quantity has increased or decreased as a percentage of the original amount.

Formula:

Percent of Change = (Amount of Change / Original Amount) × 100%

Or: (New - Original) / Original × 100%

Two Types of Percent of Change

Percent Increase

When a value gets larger.

Examples:

• Population grows from 5,000 to 5,500
• Price raises from $20 to $25
• Temperature rises from 60°F to 75°F

Percent Decrease

When a value gets smaller.

Examples:

• Price drops from $50 to $40 (sale!)
• Population shrinks from 10,000 to 8,000
• Weight reduces from 150 lbs to 135 lbs

Calculating Percent Increase

Steps:

1. Find the amount of increase (New - Original)
2. Divide by the original amount
3. Multiply by 100 to convert to percent

Example 1: Price Increase

A shirt costs $20. The price increases to $25. What is the percent increase?

Step 1: Amount of increase

• $25 -$20 = $5

Step 2: Divide by original

• $5 /$20 = 0.25

Step 3: Convert to percent

• 0.25 × 100% = 25%

Answer: The price increased by 25%

Example 2: Population Growth

A town's population was 8,000 last year. This year it's 8,800. Find the percent increase.

Step 1: Increase

• 8,800 - 8,000 = 800

Step 2: Divide by original

• 800 / 8,000 = 0.1

Step 3: Convert to percent

• 0.1 × 100% = 10%

Answer: Population increased by 10%

Example 3: Test Score Improvement

Maria scored 72 on her first test and 90 on her second test. What was her percent increase?

Step 1: Increase

• 90 - 72 = 18 points

Step 2: Divide by original

• 18 / 72 = 0.25

Step 3: Convert to percent

• 0.25 × 100% = 25%

Answer: Her score increased by 25%

Calculating Percent Decrease

Steps:

1. Find the amount of decrease (Original - New)
2. Divide by the original amount
3. Multiply by 100 to convert to percent

Example 1: Sale Price

A jacket originally costs $80 but is on sale for $60. What is the percent decrease?

Step 1: Amount of decrease

• $80 -$60 = $20

Step 2: Divide by original

• $20 /$80 = 0.25

Step 3: Convert to percent

• 0.25 × 100% = 25%

Answer: The price decreased by 25% (25% off!)

Example 2: Weight Loss

Sam weighed 180 pounds and now weighs 162 pounds. Find the percent decrease.

Step 1: Decrease

• 180 - 162 = 18 pounds

Step 2: Divide by original

• 18 / 180 = 0.1

Step 3: Convert to percent

• 0.1 × 100% = 10%

Answer: Weight decreased by 10%

Example 3: Temperature Drop

The temperature was 85°F and dropped to 68°F. What was the percent decrease?

Step 1: Decrease

• 85 - 68 = 17°F

Step 2: Divide by original

• 17 / 85 = 0.2

Step 3: Convert to percent

• 0.2 × 100% = 20%

Answer: Temperature decreased by 20%

Important: Always Use the ORIGINAL Amount!

Common mistake: Using the new amount as the denominator.

Example: Price drops from $50 to $40.

❌ WRONG: $10 /$40 = 0.25 = 25%

✓ RIGHT: $10 /$50 = 0.20 = 20%

Always divide by the original (starting) amount!

Finding the New Amount After a Percent Change

Sometimes you know the percent change and need to find the new amount.

Percent Increase

Method: New Amount = Original × (1 + percent increase as decimal)

Example: A $40 item increases by 15%. What's the new price?

Solution:

• 15% = 0.15
• New price = $40 × (1 + 0.15) =$40 × 1.15 = $46

Answer: $46

Why it works: 100% of original + 15% increase = 115% of original = 1.15 × original

Percent Decrease

Method: New Amount = Original × (1 - percent decrease as decimal)

Example: A $60 item decreases by 20%. What's the new price?

Solution:

• 20% = 0.20
• New price = $60 × (1 - 0.20) =$60 × 0.80 = $48

Answer: $48

Why it works: 100% of original - 20% decrease = 80% of original = 0.80 × original

Real-World Applications

Shopping (Sales)

Problem: A $120 coat is on sale for 35% off. What's the sale price?

Solution:

• Sale price = $120 × (1 - 0.35) =$120 × 0.65 = $78

Answer: $78

Investing (Gains)

Problem: You invest $1,000 and it grows by 8%. What's your new total?

Solution:

• New total = $1,000 × (1 + 0.08) =$1,000 × 1.08 = $1,080

Answer: $1,080

Sports Statistics

Problem: A player's average went from 15 points per game to 18 points per game. What's the percent increase?

Solution:

• Increase: 18 - 15 = 3 points
• Percent: (3 / 15) × 100% = 0.2 × 100% = 20%

Answer: 20% increase

Population Changes

Problem: A city's population was 250,000 and decreased to 225,000. What's the percent decrease?

Solution:

• Decrease: 250,000 - 225,000 = 25,000
• Percent: (25,000 / 250,000) × 100% = 0.1 × 100% = 10%

Answer: 10% decrease

Markup and Markdown

Markup (Percent Increase)

Markup is when a store adds to the cost to set a selling price.

Example: A store buys a shirt for $20 and marks it up 60%. What's the selling price?

Solution:

• Selling price = $20 × (1 + 0.60) =$20 × 1.60 = $32

Answer: $32

Markdown (Percent Decrease)

Markdown is when a store reduces the price (a discount).

Example: A $50 item is marked down 30%. What's the sale price?

Solution:

• Sale price = $50 × (1 - 0.30) =$50 × 0.70 = $35

Answer: $35

Multiple Percent Changes

When something changes multiple times, you can't just add the percents!

Example: A $100 item increases by 10%, then decreases by 10%. What's the final price?

Wrong way: +10% - 10% = 0%, so still $100? NO!

Right way:

• After first change: $100 × 1.10 =$110
• After second change: $110 × 0.90 =$99

Answer: $99 (not$100!)

Why? The 10% decrease is calculated from $110, not$100!

• 10% of $110 =$11, so $110 -$11 = $99

Common Mistakes to Avoid

❌ Mistake 1: Using the new amount as the original

• Wrong: $60 increases to $75. Change = $15 /$75 = 20%
• Right: $15 /$60 = 25%

❌ Mistake 2: Forgetting to multiply by 100

• Wrong: Percent change = 0.25 (this is a decimal, not a percent!)
• Right: Percent change = 0.25 × 100% = 25%

❌ Mistake 3: Adding percents for multiple changes

• Wrong: +20% then -10% = +10% total
• Right: Calculate each change separately!

❌ Mistake 4: Confusing increase vs decrease

• Increase: New > Original (divide by original)
• Decrease: New < Original (divide by original)
• Always divide by ORIGINAL!

❌ Mistake 5: Sign errors

• Amount of change should always be positive
• The word "increase" or "decrease" tells you the direction

Practice Tips

Tip 1: Use the formula relationship

Think of it like this: Amount of Change = Original × Percent

If you know any two values, you can find the third!

Tip 2: Check reasonableness

• 50% increase means adding half the original (50 → 75)
• 50% decrease means subtracting half the original (50 → 25)
• 100% increase means doubling (50 → 100)
• Does your answer make sense?

Tip 3: Identify what's given

• Original amount?
• New amount?
• Percent change?
• What are you finding?

Tip 4: Set up carefully

• Write the formula
• Identify the original amount
• Calculate the change
• Show all steps

Quick Reference Formulas

Finding Percent of Change:

• Percent Change = (Amount of Change / Original) × 100%
• Percent Change = (|New - Original| / Original) × 100%

Finding New Amount:

• Percent Increase: New = Original × (1 + rate)
• Percent Decrease: New = Original × (1 - rate)

Finding Original Amount:

• From Increase: Original = New / (1 + rate)
• From Decrease: Original = New / (1 - rate)

Summary

Percent of change measures how much something increased or decreased compared to the original amount.

Key Points:

• Always divide by the original amount
• Percent increase: new is larger
• Percent decrease: new is smaller
• Multiple changes are calculated separately
• Used everywhere: sales, sports, population, stocks, etc.

Master percent of change and you'll be able to analyze real-world data, understand sales and discounts, and make informed decisions about money and statistics!