Percent Increase
& Decrease

Master the formulas and strategies for solving percent change problems: from price markups to population drops and everything in between.

ISEE Upper Level

What Is Percent Change?

Percent change measures how much a quantity grows or shrinks relative to its starting value. When a store raises a price from $\$50$ to $\$65$, we say the price had a percent increase. When a population drops from $10{,}000$ to $8{,}500$, we call that a percent decrease.

Two Directions of Change

Percent Increase

The new value is greater than the original. The change amount is positive.

Percent Decrease

The new value is less than the original. The change amount is positive but represents a loss.

Visualizing Percent Change

The Core Formula

Increase

$\text{New} > \text{Original}$, so the difference is positive.

Decrease

$\text{New} < \text{Original}$, so we take the absolute value of the difference.

Denominator

Always the original value: the starting point before any change.

The Multiplier Approach

Instead of computing change and dividing, you can use a single multiplier:

Increase by $r\%$

$$\text{New} = \text{Original} \times \left(1 + \frac{r}{100}\right)$$

Decrease by $r\%$

$$\text{New} = \text{Original} \times \left(1 - \frac{r}{100}\right)$$

Increase vs. Decrease: Key Differences

	Percent Increase	Percent Decrease
Direction	Value goes up	Value goes down
Multiplier	$1 + \frac{r}{100} > 1$	$1 - \frac{r}{100} < 1$
Example ($25\%$)	Multiply by $1.25$	Multiply by $0.75$
Denominator	Always the original value

Step-by-Step Strategy

1

Identify the original value. This is the starting amount before any change occurs.

2

Find the change. Compute $|\text{New} - \text{Original}|$.

3

Divide by the original. This gives the decimal form of the percent change.

4

Multiply by $100$. Convert the decimal to a percentage.

5

State direction. Label the answer as an increase or decrease.

Example 1: Price Increase

A jacket's price rises from $\$80$ to $\$100$. What is the percent increase?

1

Original: $\$80$

2

Change: $\$100 - \$80 = \$20$

3

Divide: $\dfrac{20}{80} = 0.25$

4

Convert: $0.25 \times 100 = 25\%$

Example 2: Population Decrease

A town's population falls from $12{,}000$ to $9{,}600$. Find the percent decrease.

1

Original: $12{,}000$

2

Change: $12{,}000 - 9{,}600 = 2{,}400$

3

Divide: $\dfrac{2{,}400}{12{,}000} = 0.20$

4

Convert: $0.20 \times 100 = 20\%$

Example 3: Finding the Original

After a $30\%$ increase, a value becomes $260$. What was the original value?

1

Set up the equation: $\text{Original} \times 1.30 = 260$

2

Solve: $\text{Original} = \dfrac{260}{1.30}$

3

Compute: $\text{Original} = 200$

Example 4: Successive Changes

A stock increases by $20\%$, then decreases by $20\%$. Does it return to its original price?

1

Start: Let the original price be $\$100$.

2

After $+20\%$: $100 \times 1.20 = \$120$

3

After $-20\%$: $120 \times 0.80 = \$96$

Why They Don't Cancel

Successive Changes Formula

Where $a$ and $b$ are the successive percent changes (use negative values for decreases).

+10% then +20%

$1.10 \times 1.20 = 1.32$
Net: +32%

+10% then -10%

$1.10 \times 0.90 = 0.99$
Net: -1%

-25% then +25%

$0.75 \times 1.25 = 0.9375$
Net: -6.25%

Common Mistakes to Avoid

✕ Wrong Denominator

Dividing the change by the new value instead of the original. The original is always the denominator.

✕ Cancelling Percents

Assuming $+20\%$ then $-20\%$ returns to the start. They apply to different bases.

✕ Subtracting from New

To find the original after a $30\%$ increase, divide by $1.30$: don't subtract $30\%$ from the new value.

Summary of Key Formulas

Task	Formula	Example
Find % change	$\frac{\|\text{New} - \text{Orig}\|}{\text{Orig}} \times 100$	$\frac{20}{80} \times 100 = 25\%$
Find new value	$\text{Orig} \times (1 \pm \frac{r}{100})$	$80 \times 1.25 = 100$
Find original	$\frac{\text{New}}{1 \pm \frac{r}{100}}$	$\frac{260}{1.30} = 200$
Successive changes	$\text{Orig} \times m_1 \times m_2$	$100 \times 1.20 \times 0.80 = 96$

Question 1 of 5

A shirt's price drops from $\$60$ to $\$45$. What is the percent decrease?

The change is $\$60 - \$45 = \$15$. Dividing by the original: $\frac{15}{60} = 0.25 = 25\%$. The close distractor $33.\overline{3}\%$ results from incorrectly dividing by the new value $45$.

Question 2 of 5

After a $40\%$ increase, a value becomes $350$. What was the original value?

A $40\%$ increase means the multiplier is $1.40$. So the original $= \frac{350}{1.40} = 250$. Choice (a) comes from subtracting $40\%$ of $350$ directly, which is incorrect.

Question 3 of 5

A price increases by $50\%$ and then decreases by $50\%$. What is the overall percent change from the original price?

The net multiplier is $1.50 \times 0.50 = 0.75$. That means the final value is $75\%$ of the original: a $25\%$ decrease. The changes do NOT cancel because the $50\%$ decrease applies to the larger, already-increased value.

Question 4 of 5

A store marks up an item by $60\%$, then offers a $25\%$ discount. If the final price is $\$96$, what was the original cost?

The net multiplier is $1.60 \times 0.75 = 1.20$. So the original $= \frac{96}{1.20} = 80$. The item's cost was $\$80$.

Question 5 of 5

The number of students in a club increases from $40$ to $52$. Which of the following is the percent increase?

Change: $52 - 40 = 12$. Percent increase: $\frac{12}{40} \times 100 = 30\%$. The distractor $23.1\%$ comes from dividing by the new value $52$ instead of the original $40$.

Task	Formula	Example
Find % change	\(\frac{\|\text{New} - \text{Orig}\|}{\text{Orig}} \times 100\)	\(\frac{20}{80} \times 100 = 25\%\)
Find new value	\(\text{Orig} \times (1 \pm \frac{r}{100})\)	\(80 \times 1.25 = 100\)
Find original	\(\frac{\text{New}}{1 \pm \frac{r}{100}}\)	\(\frac{260}{1.30} = 200\)
Successive changes	\(\text{Orig} \times m_1 \times m_2\)	\(100 \times 1.20 \times 0.80 = 96\)

Percent Increase& Decrease

What Is Percent Change?

Two Directions of Change

Visualizing Percent Change

The Core Formula

The Multiplier Approach

Increase vs. Decrease: Key Differences

Step-by-Step Strategy

Example 1: Price Increase

Example 2: Population Decrease

Example 3: Finding the Original

Example 4: Successive Changes

Why They Don't Cancel

Successive Changes Formula

Common Mistakes to Avoid

Summary of Key Formulas

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Percent Increase
& Decrease