Solve percent increase/decrease problems and use growth factors. | Percentages | SAT Math

Core Idea

A percent increase or decrease is fastest handled with a single multiplication using the growth factor: multiply by (1 + r) for an increase or (1 − r) for a decrease.

Understanding

When a quantity increases by 20%, you don't have to find 20% and then add it. Just multiply by $1.20$ in one step. That multiplier — $1 + 𝑟$ for an increase, $1 - 𝑟$ for a decrease — is called the growth factor.

N e w V a l u e = O r i g i n a l \times (1 \pm 𝑟)

A shirt was $80 and its price rose 15%:

$80 \times 1.15 = 92$

A population of 5,000 fell by 12%:

$5, 000 \times 0.88 = 4, 400$

To find the percent change when you already have both values:

P e r c e n t C h a n g e = N e w - O r i g i n a l O r i g i n a l \times 100 %

The denominator is always the original value — the value you started from. This is where errors cluster. If a price goes from $50 to $60, the increase is based on 50, not 60.

For repeated changes (e.g., 10% increase each year for 3 years), multiply the growth factor repeatedly:

$F i n a l = O r i g i n a l \times (1.10) 3$

Don't add the percents (30% over 3 years) — compounding means each increase builds on the previous result.

Step by Step

Identify the original value and the percent change.
Convert the percent to a decimal r.
Choose the growth factor: (1 + r) for increase, (1 − r) for decrease.
Multiply: New = Original × growth factor. For repeated changes, raise the factor to the appropriate power.

Misconceptions

Adding percents for successive changes — e.g., saying two consecutive 10% increases equal a 20% increase (it's actually 21%).
Using the new value as the base when computing percent change instead of the original value.
Computing the percent of the change correctly but forgetting to add it back (or subtract it) from the original.

Question

Worked Example

The population of a town was 20,000 in 2020. It increased by 10% from 2020 to 2021 and then decreased by 10% from 2021 to 2022. What was the population in 2022?

Select an answer to see the explanation