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Percent Change
and Growth

Master the formulas, distinctions, and traps behind percent increase, percent decrease, and successive percent changes on the GRE.

GRE Quantitative

What Is Percent Change?

Percent change measures how much a quantity has grown or shrunk relative to its original value. It converts an absolute difference into a relative proportion, making it easy to compare changes across different scales.

The Percent Change Formula

$$\text{Percent Change} = \frac{\text{New} - \text{Original}}{\text{Original}} \times 100\%$$
Positive result
\(\text{New} > \text{Original}\): percent increase
Negative result
\(\text{New} < \text{Original}\): percent decrease

Visualizing Percent Increase

200 ORIGINAL 260 NEW
\(+60\)
\(+30\%\)

Increase vs Decrease: Key Distinction

Percent Increase Percent Decrease
Direction \(\text{New} > \text{Original}\) \(\text{New} < \text{Original}\)
Formula \(\frac{\text{New}-\text{Orig}}{\text{Orig}} \times 100\) \(\frac{\text{New}-\text{Orig}}{\text{Orig}} \times 100\)
Multiplier \(1 + \frac{r}{100}\) \(1 - \frac{r}{100}\)
Example \(+25\%\) → multiply by \(1.25\) \(-20\%\) → multiply by \(0.80\)

The Multiplier Method

$$\text{New Value} = \text{Original} \times \left(1 \pm \frac{r}{100}\right)$$
1
Convert the percent to a decimal: \(r\% \to \frac{r}{100}\).
2
Add for increase: multiplier \(= 1 + \text{decimal}\). Subtract for decrease: multiplier \(= 1 - \text{decimal}\).
3
Multiply the original value by the multiplier to get the new value.

Successive Percent Changes

When a value undergoes two or more percent changes in sequence, you cannot simply add the percentages. Instead, multiply the multipliers together.

$$\text{Final} = \text{Original} \times m_1 \times m_2 \times \cdots \times m_n$$

Why \(+20\%\) then \(-20\% \neq 0\%\)

START 100
\(+20\%\)
 120
\(100 \times 1.20\)
\(-20\%\)
 96
\(120 \times 0.80\)
Net change: \(100 \to 96 = -4\%\)

How to Apply on the GRE

1
Identify the original value and the direction (increase or decrease).
2
Build the multiplier: \(1 + r/100\) for increase, \(1 - r/100\) for decrease.
3
For successive changes, chain the multipliers: multiply them all together before applying to the original.
4
To find the percent change, compute \(\frac{\text{New} - \text{Orig}}{\text{Orig}} \times 100\%\). Remember: the original is always the denominator.
5
Double-check: Does the direction make sense? If you increased then decreased by the same rate, the final value should be less than the original.

Example 1: Simple Percent Increase

A shirt originally costs \(\$80\). After a price increase, it now costs \(\$92\). What is the percent increase?

1
Find the change: \(92 - 80 = 12\).
2
Divide by original: \(\frac{12}{80} = 0.15\).
3
Convert: \(0.15 \times 100\% = \) \(15\%\).

Example 2: Successive Changes

A stock rises \(25\%\) in year 1, then falls \(10\%\) in year 2, then rises \(20\%\) in year 3. If the stock started at \(\$400\), what is its final value?

1
Multipliers: \(1.25\), \(0.90\), \(1.20\).
2
Combined multiplier: \(1.25 \times 0.90 \times 1.20 = 1.35\).
3
Final value: \(400 \times 1.35 = \) \(\$540\).

Example 3: Reverse Percent Change

After a \(30\%\) discount, a jacket costs \(\$56\). What was the original price?

1
A \(30\%\) decrease means the multiplier is \(0.70\).
2
We know: \(\text{Original} \times 0.70 = 56\).
3
Solve: Original = \(\frac{56}{0.70} = \) \(\$80\).

Compound Growth Formula

$$A = P\left(1 + \frac{r}{100}\right)^{n}$$
\(P\)
Principal: starting amount
\(r\)
Rate of change per period (%)
\(n\)
Number of periods
\(A\)
Final amount after \(n\) periods

Visualizing Compound Growth

0 200 400 600 800 0 1 2 3 4 5 6 7 8 9 10 Years
\(20\%\) compound
 Simple (linear)

Common Mistakes

✕ Adding Successive %
Thinking a \(20\%\) increase then a \(10\%\) decrease is a net \(10\%\) increase. You must multiply the multipliers.
✕ Wrong Denominator
Using the new value as the base instead of the original. The original is always the denominator.
✕ Reversing by Adding
To undo a \(30\%\) decrease, you must divide by \(0.70\), not add \(30\%\) of the discounted price.

Summary of Key Formulas

Scenario Formula / Method Example
Percent change \(\frac{\text{New}-\text{Orig}}{\text{Orig}} \times 100\%\) \(\frac{92-80}{80} \to 15\%\)
New from % \(\text{Orig} \times \text{multiplier}\) \(80 \times 1.15 = 92\)
Successive changes \(\text{Orig} \times m_1 \times m_2 \times \cdots\) \(100 \times 1.20 \times 0.80 = 96\)
Compound growth \(P(1+r/100)^n\) \(1000(1.05)^3 \approx 1157.63\)
Reverse (find orig) \(\text{New} \div \text{multiplier}\) \(56 \div 0.70 = 80\)

Question 1 of 5

A laptop's price drops from \(\$1{,}200\) to \(\$960\). What is the percent decrease?

Question 2 of 5

A population increases by \(10\%\) and then increases by another \(10\%\). What is the overall percent increase?

Question 3 of 5

After a \(40\%\) discount, a pair of shoes costs \(\$72\). What was the original price?

Question 4 of 5

A stock rises \(50\%\) in one year and then falls \(50\%\) the next. Compared to the original, the stock is now:

Question 5 of 5

An investment of \(\$2{,}000\) grows at \(5\%\) per year for 3 years (compounded annually). Which expression gives the final value?