Percentage Change

For Elementary Students

What Is Percentage Change?

Percentage change tells you how much something went UP or DOWN compared to where it started!

Think about it like this: If your allowance was $10 and now it's $12, you want to know "By what percent did it increase?"

Started at: $10
Now at:     $12

How much MORE is it? (as a percentage)

The Big Idea

Percentage change always compares to the ORIGINAL (starting) amount!

Original amount  →  New amount
  (where you      (where you
   started)         ended up)

Two Types of Percentage Change

1. Percentage INCREASE ↑

When the number goes UP!

Price was $20 → now $25
(It increased!)

2. Percentage DECREASE ↓

When the number goes DOWN!

Price was $40 → now $30
(It decreased!)

The Formula (Made Simple!)

Step 1: Find how much it CHANGED

Change = New amount − Original amount

Step 2: Compare the change to the original

Percentage Change = (Change ÷ Original) × 100

Example 1: Percentage Increase

Problem: A shirt was $40, now it costs $50. What's the percentage increase?

Step 1: Find the change

Change = New − Original
Change = $50 − $40 = $10

Step 2: Divide by original

$10 ÷ $40 = 0.25

Step 3: Multiply by 100 to get percentage

0.25 × 100 = 25%

Answer: The price increased by 25%! ↑

Example 2: Percentage Decrease

Problem: A population went from 800 to 680. What's the percentage decrease?

Step 1: Find the change

Change = New − Original
Change = 680 − 800 = −120
(Negative means it went DOWN!)

Step 2: Divide by original (use positive 120)

120 ÷ 800 = 0.15

Step 3: Multiply by 100

0.15 × 100 = 15%

Answer: The population decreased by 15%! ↓

Example 3: Another Increase

Problem: A game was $20, now it's $25. Percentage increase?

Solution:

Step 1: Change = $25 − $20 = $5

Step 2: $5 ÷ $20 = 0.25

Step 3: 0.25 × 100 = 25%

Answer: 25% increase!

Working Backwards: Finding the New Value

Sometimes you KNOW the percentage change and need to find the NEW value!

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

Method 1: Step-by-step

Step 1: Find 20% of $60
20% = 0.20
0.20 × $60 = $12

Step 2: Add the increase to original
$60 + $12 = $72

Answer: $72!

Method 2: The Shortcut!

For a 20% increase, multiply by 1.20!

$60 × 1.20 = $72

Same answer, faster!

Why 1.20?

100% (original) + 20% (increase) = 120% = 1.20

Decrease Shortcut

For a decrease, multiply by LESS than 1!

Example: $200 decreases by 30%. What's the new value?

100% − 30% = 70% = 0.70

$200 × 0.70 = $140

Answer: $140!

Example 4: Sale Price

Problem: A jacket is on sale for 25% off. Original price is $80. What's the sale price?

100% − 25% = 75% (you pay 75% of original)

$80 × 0.75 = $60

Answer: $60!

The VERY Important Rule

ALWAYS divide by the ORIGINAL amount, not the new one!

Why? The percentage depends on your starting point!

Surprising Fact!

Going from 50 to 100 is a 100% increase!

Change: 100 − 50 = 50
50 ÷ 50 = 1.0 = 100%

But going from 100 back to 50 is only a 50% decrease!

Change: 50 − 100 = −50
50 ÷ 100 = 0.5 = 50%

Different starting points = different percentages!

Visual Guide

PERCENTAGE INCREASE:

Original:  ■■■■■ ($100)
New:       ■■■■■■■■ ($120)
Added:     ++++  ($20)

Change: $20
Percent: $20 ÷ $100 = 0.20 = 20% increase

PERCENTAGE DECREASE:

Original:  ■■■■■ ($100)
New:       ■■■ ($60)
Lost:      --  ($40)

Change: $40
Percent: $40 ÷ $100 = 0.40 = 40% decrease

Memory Trick

"Change over Original, times 100!"

   Change
────────────  × 100 = Percentage Change
  Original

Quick Tips

Tip 1: Find the CHANGE first (New − Original)

Tip 2: ALWAYS divide by the ORIGINAL (starting) amount

Tip 3: Multiply by 100 to get the percentage

Tip 4: For increase: multiply by (1 + rate)

Tip 5: For decrease: multiply by (1 − rate)

For Junior High Students

Understanding Percentage Change

Percentage change quantifies the relative change in a quantity, expressing it as a percentage of the original value.

Definition: For an initial value V₀ and final value V₁, the percentage change is:

Percentage Change = ((V₁ − V₀) / V₀) × 100

Types:

• Percentage increase: V₁ > V₀ (positive change)
• Percentage decrease: V₁ < V₀ (negative change)

Key concept: Percentage change is always relative to the initial (original) value, not the final value.

The General Formula

Formula:

Percentage Change = (Δ / V₀) × 100

where:
Δ = V₁ − V₀ (absolute change)
V₀ = original (initial) value
V₁ = new (final) value

Sign convention:

• Positive result → increase
• Negative result → decrease

Alternative notation:

% Change = ((New − Old) / Old) × 100

Calculating Percentage Increase

Definition: Percentage increase occurs when V₁ > V₀.

Example 1: Price increases from $40 to $50

Change = $50 − $40 = $10
Percentage increase = ($10 / $40) × 100
                    = 0.25 × 100
                    = 25%

Interpretation: The price increased by 25% of its original value.

Example 2: Population grows from 1,200 to 1,500

Change = 1,500 − 1,200 = 300
Percentage increase = (300 / 1,200) × 100
                    = 0.25 × 100
                    = 25%

Example 3: Stock price from $80 to $96

Change = $96 − $80 = $16
Percentage increase = ($16 / $80) × 100
                    = 0.20 × 100
                    = 20%

Calculating Percentage Decrease

Definition: Percentage decrease occurs when V₁ < V₀.

Example 1: Class size from 40 to 34 students

Change = 34 − 40 = −6
Absolute change = 6 (magnitude)
Percentage decrease = (6 / 40) × 100
                    = 0.15 × 100
                    = 15%

Example 2: Temperature from 25°C to 20°C

Change = 20 − 25 = −5
Percentage decrease = (5 / 25) × 100
                    = 0.20 × 100
                    = 20%

Example 3: Revenue from $500,000 to $425,000

Change = $425,000 − $500,000 = −$75,000
Percentage decrease = ($75,000 / $500,000) × 100
                    = 0.15 × 100
                    = 15%

Finding Final Value from Percentage Change

Given: Original value V₀ and percentage change p%

For percentage increase:

V₁ = V₀ + (p/100) × V₀
   = V₀(1 + p/100)

For percentage decrease:

V₁ = V₀ − (p/100) × V₀
   = V₀(1 − p/100)

Example 1: $60 increases by 20%

V₁ = $60 × (1 + 0.20)
   = $60 × 1.20
   = $72

Example 2: $150 decreases by 30%

V₁ = $150 × (1 − 0.30)
   = $150 × 0.70
   = $105

General formula:

V₁ = V₀ × (1 ± p/100)
(+ for increase, − for decrease)

Finding Original Value from Percentage Change

Given: Final value V₁ and percentage change p%

For percentage increase:

V₀ = V₁ / (1 + p/100)

For percentage decrease:

V₀ = V₁ / (1 − p/100)

Example 1: Item sells for $90 after 20% increase. Find original price.

V₀ = $90 / (1 + 0.20)
   = $90 / 1.20
   = $75

Verification: $75 × 1.20 = $90 ✓

Example 2: Sale price $120 after 25% discount. Find original price.

V₀ = $120 / (1 − 0.25)
   = $120 / 0.75
   = $160

Verification: $160 × 0.75 = $120 ✓

Asymmetry of Percentage Changes

Important property: Percentage increase and decrease are NOT symmetric.

Example:

Going from 50 to 100:

Percentage increase = ((100 − 50) / 50) × 100 = 100%

Going from 100 back to 50:

Percentage decrease = ((50 − 100) / 100) × 100 = −50%

Explanation: The base (denominator) changes, so the percentages differ.

Implication: A 50% decrease does NOT reverse a 100% increase.

Verification:

Start: 50
Increase 100%: 50 × 2 = 100
Decrease 50%: 100 × 0.5 = 50 (returns to original)

To reverse a 100% increase, you need a 50% decrease, not 100%.

Successive Percentage Changes

When multiple percentage changes occur sequentially:

V_final = V_original × (1 + p₁/100) × (1 + p₂/100) × ...

Example: Price increases by 10%, then by 20%

Starting price: $100

After first increase: $100 × 1.10 = $110
After second increase: $110 × 1.20 = $132

Overall change:

Change = $132 − $100 = $32
Percentage change = ($32 / $100) × 100 = 32%

Note: 10% + 20% ≠ 32% (not simply additive!)

Calculation: Combined factor = 1.10 × 1.20 = 1.32 (32% increase)

Applications

Finance: Investment returns

Initial investment: $5,000 Final value: $5,750

Return = (($5,750 − $5,000) / $5,000) × 100
       = ($750 / $5,000) × 100
       = 15%

Business: Sales growth

Last year's sales: $200,000 This year's sales: $230,000

Growth rate = (($230,000 − $200,000) / $200,000) × 100
            = 15%

Science: Measurement error

Actual value: 50.0 cm Measured value: 51.5 cm

Percentage error = ((51.5 − 50.0) / 50.0) × 100
                 = 3%

Retail: Markups and markdowns

Cost to retailer: $60 Selling price: $90

Markup = (($90 − $60) / $60) × 100
       = 50%

Percentage Point vs. Percentage Change

Critical distinction:

Percentage point: Absolute difference in percentages

Percentage change: Relative change in percentages

Example: Interest rate changes from 5% to 7%

Percentage point increase: 7% − 5% = 2 percentage points

Percentage change:

((7 − 5) / 5) × 100 = 40% increase

Context matters: "Increase from 5% to 7%" is:

• 2 percentage point increase
• 40% relative increase

Common Errors

Error 1: Using final value as base

❌ From $100 to $120: (20 / 120) × 100 = 16.67% ✓ From $100 to $120: (20 / 100) × 100 = 20%

Always use original (initial) value as denominator.

Error 2: Assuming symmetry

❌ After 50% increase, a 50% decrease returns to original ✓ After 50% increase (× 1.5), need 33.33% decrease (÷ 1.5) to return

Error 3: Adding successive percentages

❌ 10% increase then 15% increase = 25% total increase ✓ 10% then 15%: 1.10 × 1.15 = 1.265 (26.5% total increase)

Error 4: Confusing percentage points with percentage change

When interest rates rise from 4% to 5%, the increase is 1 percentage point, not 1%.

The percentage change is: ((5 − 4) / 4) × 100 = 25%

Tips for Success

Tip 1: Always identify the original (baseline) value clearly

Tip 2: Use the formula: ((New − Old) / Old) × 100

Tip 3: For finding new value: multiply by (1 + rate) for increase, (1 − rate) for decrease

Tip 4: Verify calculations by working backwards

Tip 5: Remember percentage changes are not symmetric

Tip 6: For successive changes, multiply the factors: (1 + r₁)(1 + r₂)

Tip 7: Distinguish between percentage points (absolute) and percentage change (relative)

Extensions: Compound Growth

Percentage change over multiple periods follows compound growth formula:

V_final = V_initial × (1 + r)ⁿ

where r = growth rate per period, n = number of periods

Example: $1,000 invested at 5% annual growth for 3 years

V_final = $1,000 × (1.05)³
        = $1,000 × 1.157625
        = $1,157.63

Total percentage increase:

((1157.63 − 1000) / 1000) × 100 = 15.76%

Summary

Key formula:

Percentage Change = ((New − Original) / Original) × 100

Finding new value:

New = Original × (1 ± rate)

Finding original value:

Original = New / (1 ± rate)

Key principles:

• Always use original value as base
• Percentage increase and decrease are asymmetric
• Successive changes multiply, don't add
• Distinguish percentage points from percentage change

Practice