EQN-02-07

Percentage Increase & Decrease

Use multipliers to apply, chain, and reverse percentage changes without slips.

~5 min read•Difficulty 2/5

increase by p % \to multiply by (1 + p /100)

decrease by p % \to multiply by (1 - p /100)

A 20% rise is $\times 1.2$ ; a 15% fall is $\times 0.85.$ Multipliers make chaining trivial.

Multiply the multipliers in order — never add the percents.

+ 10% then + 20% \to \times 1.1 \times 1.2 = \times 1.32 (a 32% rise, not 30%)

+ 25% then - 20% \to \times 1.25 \times 0.8 = \times 1.0 (back to the start)

To undo a change, divide by its multiplier (do not apply the opposite percent).

After a 20% rise the price is 120 \to original = 120/1.2 = 100.

percent change = (new - old) / old \times 100

Always divide by the old value — the starting point is the base.

Worked examples

Example 1 (difficulty 2/5)

A 50 item is increased by 20%, then decreased by 10%. What is the final price?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A population grew from 800 to 920. What is the percent increase?

◎ Predict the answer first:

Convert every change to a multiplier $(1 \pm p /100)$ — chaining becomes multiplication.
To reverse a change, divide by its multiplier.
Percent change always divides by the OLD value.

Adding percents for successive changes

Why: They look additive.

Fix: Multiply the multipliers;

+ 10%

then

+ 20%

\times 1.32,

not

\times 1.30.

Dividing by the new value when finding percent change

Why: The new value is the one in front of you.

Fix: The base is the original value; divide the change by it.