Percentage Increase Formulas for K-12 Students

Introduction to Percentage Increase

Percentage increase shows how much a value has grown compared to its original size. It's expressed as a percentage of the original value and is used in many real-world situations like price increases, population growth, and interest calculations.

Elementary School Level (K-5)

Basic Percentage

A percentage is a way to express a number as a fraction of 100:

$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$

For example: If you have 5 red marbles out of 20 total marbles, the percentage is:

$\frac{5}{20} \times 100\% = 25\%$

Simple Percentage Increase

To find how much something has increased:

$\text{Increase} = \text{New Value} - \text{Original Value}$

To find the percentage increase:

$\text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100\%$

Example:

If a toy's price changes from $20 to $25:

Increase = $25 - $20 = $5

Percentage Increase = $5 ÷ $20 × 100% = 25%

Middle School Level (6-8)

Combined Formula for Percentage Increase

A more efficient way to calculate percentage increase:

$\text{Percentage Increase} = \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\%$

Example:

If the population of a town grows from 15,000 to 18,000:

Percentage Increase = ((18,000 - 15,000) ÷ 15,000) × 100% = (3,000 ÷ 15,000) × 100% = 20%

Finding the New Value After an Increase

If you know the original value and the percentage increase, you can find the new value:

$\text{New Value} = \text{Original Value} + (\text{Original Value} \times \frac{\text{Percentage Increase}}{100})$

This can be simplified to:

$\text{New Value} = \text{Original Value} \times (1 + \frac{\text{Percentage Increase}}{100})$

Example:

If a $50 shirt increases by 30% in price, the new price is:

New Value = $50 × (1 + 30/100) = $50 × 1.3 = $65

Finding the Original Value Before an Increase

If you know the new value and the percentage increase, you can find the original value:

$\text{Original Value} = \frac{\text{New Value}}{1 + \frac{\text{Percentage Increase}}{100}}$

Example:

If the current price of a book is $39 after a 30% increase, the original price was:

Original Value = $39 ÷ (1 + 30/100) = $39 ÷ 1.3 = $30

High School Level (9-12)

Percent Change Formula

The general formula for percent change (works for both increases and decreases):

$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$

If the result is positive, it's a percentage increase. If negative, it's a percentage decrease.

Growth Factor and Percent Growth

The growth factor is the ratio of the new value to the original value:

$\text{Growth Factor} = \frac{\text{New Value}}{\text{Original Value}}$

The percent growth can be calculated from the growth factor:

$\text{Percent Growth} = (\text{Growth Factor} - 1) \times 100\%$

Example:

If an investment grows from $1,000 to $1,450:

Growth Factor = $1,450 ÷ $1,000 = 1.45

Percent Growth = (1.45 - 1) × 100% = 0.45 × 100% = 45%

Successive Percentage Changes

When there are multiple percentage changes in succession:

$\text{Final Value} = \text{Original Value} \times (1 + \frac{r_1}{100}) \times (1 + \frac{r_2}{100}) \times ... \times (1 + \frac{r_n}{100})$

where $r_1, r_2, ..., r_n$ are the successive percentage changes.

Example:

If a product's price increases by 10% and then by another 5%:

Final Price = Original Price × (1 + 10/100) × (1 + 5/100) = Original Price × 1.1 × 1.05 = Original Price × 1.155

This is equivalent to a single increase of 15.5%, not simply 15%.

Compound Percentage Increase

When the same percentage increase occurs multiple times (e.g., annual growth):

$\text{Final Value} = \text{Initial Value} \times (1 + \frac{r}{100})^n$

where $r$ is the percentage increase per period, and $n$ is the number of periods.

Example:

If a population grows by 2% annually for 5 years, the final population will be:

Final Population = Initial Population × (1 + 2/100)^5 = Initial Population × (1.02)^5 ≈ Initial Population × 1.104

This represents about a 10.4% total increase over the 5 years.

Average Percentage Increase

To find the average annual percentage increase over multiple years:

$\text{Average Annual Increase} = \left(\sqrt[n]{\frac{\text{Final Value}}{\text{Initial Value}}} - 1\right) \times 100\%$

where $n$ is the number of periods.

Example:

If a house value increases from $200,000 to $250,000 over 4 years, the average annual increase is:

Average Annual Increase = ((250,000 ÷ 200,000)^(1/4) - 1) × 100% ≈ (1.25^0.25 - 1) × 100% ≈ (1.0574 - 1) × 100% ≈ 5.74%

Real-World Applications

Price Increases

Calculate how much more an item costs after a price hike.

Population Growth

Analyze how fast a city or country's population is growing.

Investment Returns

Calculate the growth of investments over time.

Important Note for Students

Remember these key points:

Always use the original value as the denominator when calculating percentage increase
Percentage increases can be greater than 100% if the value more than doubles
Multiple percentage increases are not simply additive—they compound
Percentage increases and decreases are not symmetric (e.g., a 50% increase followed by a 50% decrease does not return to the original value)