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Percent
Increase & Decrease

Learn to calculate how much a value grows or shrinks and interpret what the change really means in everyday situations.

Pre-Algebra

What Is Percent Change?

Percent change describes how much a quantity has grown or shrunk relative to where it started. Instead of just saying "it went up by 10," we express the change as a percentage of the original value so we can compare changes fairly.

Percent Increase
The new value is larger than the original: prices go up, populations grow, scores improve.
Percent Decrease
The new value is smaller than the original: prices drop, temperatures fall, distances shrink.

Why Percent Change Matters

Raw numbers can be misleading. A \(\$5\) price increase on a \(\$10\) item is huge (\(50\%\)), but the same \(\$5\) increase on a \(\$500\) item is tiny (\(1\%\)). Percent change gives us context.

$10 +$5 T-shirt
\(50\%\)
 $500 +$5 Laptop
\(1\%\)

Key Vocabulary

Original Value
The starting amount before the change happens. Also called the "initial" value.
New Value
The amount after the change: the final or ending value.
Amount of Change
\(\text{New} - \text{Original}\). Positive means increase, negative means decrease.

The Core Formula

$$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$$
Result > 0
Positive result means percent increase
Result < 0
Negative result means percent decrease

Breaking Down the Formula

Step 1 Subtract Step 2 Divide Step 3 Multiply
\(\text{New Value} - \text{Original Value}\)
\(=\) Amount of Change
\(\frac{\text{Amount of Change}}{\text{Original Value}}\)
\(=\) Decimal form of the change
\(\text{Decimal} \times 100\)
\(=\) Percent change

Increase vs. Decrease

Percent Increase Percent Decrease
Direction Value goes up Value goes down
New vs. Original New > Original New < Original
Change sign Positive (+) Negative (−)
Example \(\$40 \to \$50\) \(\$50 \to \$40\)
Result \(25\%\) increase \(20\%\) decrease

Step-by-Step Method

1
Identify the original and new values. The original is the value before the change.
2
Find the amount of change: \(\text{New} - \text{Original}\).
3
Divide by the original: \(\dfrac{\text{Change}}{\text{Original}}\).
4
Multiply by \(100\) to convert to a percent.
5
Label your answer as a percent increase or decrease.

Example 1: Price Increase

A pair of shoes was \(\$80\) last month. This month the price is \(\$92\). What is the percent increase?

$80 Original $92 New +$12
\(15\%\)
 increase

Example 2: Population Decrease

A town had \(2{,}000\) residents last year. This year it has \(1{,}700\). What is the percent decrease?

1
Original: \(2{,}000\). New: \(1{,}700\).
2
Change: \(1{,}700 - 2{,}000 = -300\)
3
Divide: \(\dfrac{-300}{2{,}000} = -0.15\)
4
Multiply: \(-0.15 \times 100 = -15\%\)
5
Label: The population decreased by \(15\%\).

Example 3: Test Score Jump

Maya scored \(60\) on her first quiz and \(84\) on the retake. What is the percent increase?

Maya improved her score by \(40\%\). Notice the original value is her first score (\(60\)), not the higher one.

Working Backwards

Sometimes you know the original value and the percent change, and need to find the new value.

$$\text{New Value} = \text{Original} \times \left(1 + \frac{\text{Percent}}{100}\right)$$
Increase
Use \(1 +\) the decimal. A \(20\%\) increase: multiply by \(1.20\).
Decrease
Use \(1 -\) the decimal. A \(20\%\) decrease: multiply by \(0.80\).

Example 4: Finding the New Value

A shirt costs \(\$45\) and goes on sale for \(30\%\) off. What is the sale price?

1
Original: \(\$45\). Percent decrease: \(30\%\).
2
Multiplier: \(1 - 0.30 = 0.70\)
3
New value: \(45 \times 0.70 = \) \(\$31.50\)

Successive Percent Changes

A price increases by \(10\%\) then decreases by \(10\%\). Is it back to the original? No!

1
Start: \(\$100\)
2
After \(+10\%\): \(100 \times 1.10 = \) \(\$110\)
3
After \(-10\%\): \(110 \times 0.90 = \) \(\$99\)

Common Mistakes

✕ Wrong Denominator
Dividing by the new value instead of the original. Always divide by the starting value.
✕ Forgetting Direction
Writing "\(25\%\)" without saying increase or decrease. Always label the direction.
✕ Canceling Changes
Assuming \(+10\%\) then \(-10\%\) returns to the start. It does not: the bases differ.

Summary Cheat Sheet

Task Formula Example
Percent change \(\dfrac{\text{New} - \text{Orig}}{\text{Orig}} \times 100\) \(\dfrac{92-80}{80}\times 100 = 15\%\)
New after increase \(\text{Orig} \times (1 + r)\) \(80 \times 1.15 = 92\)
New after decrease \(\text{Orig} \times (1 - r)\) \(45 \times 0.70 = 31.50\)
Original from new \(\dfrac{\text{New}}{1 \pm r}\) \(\dfrac{92}{1.15} = 80\)

Question 1 of 5

A jacket's price went from \(\$60\) to \(\$75\). What is the percent increase?

Question 2 of 5

A town's population fell from \(5{,}000\) to \(4{,}250\). What is the percent decrease?

Question 3 of 5

A laptop costs \(\$800\). After a \(15\%\) discount, what is the sale price?

Question 4 of 5

A stock price rises by \(20\%\) and then falls by \(20\%\). What is the overall percent change?

Question 5 of 5

After a \(25\%\) increase, an item costs \(\$150\). What was the original price?