POP

G3.8 Percentage Change

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01 — Explanation

How do I calculate the percentage change between two values?

Percentage change always measures the difference relative to the original (starting) value — never the new value:

\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100

Example: Height increases from 95 cm to 114 cm. Difference $=19$ . $\dfrac{19}{95}\times100 = 20\%$ increase.

Example (decrease): A running time falls from 72 s to 63 s. Difference $=9$ . $\dfrac{9}{72}\times100=12.5\%$ decrease.

Common Misconception: Dividing the change by the new value instead of the original. The original value is always the 100% baseline you're measuring against — in profit/loss questions, that's always the cost price, never the selling price.

02 — Explanation

How do I do this without a calculator?

Form the fraction $\dfrac{\text{change}}{\text{original}}$ , then simplify until the denominator is a factor of 100 so it converts easily.

Example: Enrollment changes from 250 to 310. $\dfrac{60}{250}=\dfrac{6}{25}$ . Since $25\times4=100$ , multiply top and bottom by 4: $\dfrac{24}{100}=24\%$ .

03 — Explanation

How do I find the new value after a percentage change, quickly?

Use a single multiplier rather than calculating the change separately and adding/subtracting it:

Increase by $X\%$ : multiplier $=1+\frac{X}{100}$ .
Decrease by $X\%$ : multiplier $=1-\frac{X}{100}$ .

Example: Increase 78 by 15%: $78\times1.15=89.7$ . Decrease 120 by 19%: $120\times0.81=97.2$ .

This also works in reverse — if you're told the multiplier, you can read off the percentage change directly: a multiplier of $1.05$ means a $5\%$ increase; a multiplier of $0.75$ means a $25\%$ decrease.

Exam Tip: The three-step "difference ÷ original × 100" method is slower than spotting the multiplier directly, but it's safer — it forces you to explicitly calculate the change and identify the original value, which is exactly where most marks are lost.

04 — Explanation

Why can't I just add or subtract percentage changes that happen one after another?

Because each percentage change is calculated on whatever the value currently is, not on the original amount — the 100% baseline shifts every time. A 10% increase followed by a 30% decrease is not an overall 20% decrease.

Using multipliers makes this clear: $1.10\times0.70=0.77$ , which is an overall decrease of $1-0.77=0.23$ , i.e. 23%, not 20%.

Common Misconception: Treating consecutive percentage changes as additive. Percentage change is multiplicative — always multiply the multipliers together, never add the percentages.

05 — Explanation

How do I find the original value before a percentage change (a "reverse percentage")?

Don't take a percentage of the new value — that uses the wrong baseline. Instead, divide the new value by the multiplier that produced it.

Example: A sale price of £68 is after a 15% reduction. Find the original price.

The multiplier for a 15% decrease is $0.85$ .
Since $\text{original}\times0.85=68$ , the original price is $68\div0.85=£80$ .

Common Misconception: Calculating 15% of £68 and adding it back on. This finds 15% of the wrong amount (the already-reduced price), not the true original — reverse percentage questions always require dividing by the multiplier, never multiplying the new value by a percentage.

06 — Worked example

Worked Example

A shop increases a £45 item's price by 20%, then has a sale offering 20% off the new price. Is the final price equal to the original £45? If not, find it.

After the increase: $45\times1.20=£54$ .
After the 20% decrease on £54 (not on £45!): $54\times0.80=£43.20$ .
The final price is £43.20, which is less than the original £45 — the two 20% changes don't cancel out, because the second 20% was taken from a larger amount than the first.