22. Percent Increase and Percent Decrease

Percent change compares the change to the original amount.

\[ \text{percent change}=\frac{\text{new}-\text{old}}{\text{old}}\times100\%. \]

Increase example: A price rises from 50 dollars to 65 dollars.
\[ \text{change}=65-50=15. \]
\[ \frac{15}{50}=0.30=30\%. \]
This is a 30% increase.

Decrease example: A price drops from 80 dollars to 60 dollars.
\[ \text{change}=60-80=-20. \]
The amount of decrease is 20:
\[ \frac{20}{80}=0.25=25\%. \]
This is a 25% decrease.

Case study - population: A town grows from 2,000 people to 2,500 people. Change \(=500\). Percent increase:
\[ \frac{500}{2000}=0.25=25\%. \]

Common mistake: dividing by the new value. Percent change is measured from the original value unless the problem clearly says otherwise.

---

Deep reasoning: percent change measures change relative to the start

Percent change is

\[ \frac{\text{new}-\text{original}}{\text{original}}\times100\%. \]

The original value is the comparison base because the question asks how large the change is relative to where the quantity started. You may report direction with a positive/negative sign or with the words increase/decrease.

GED® worked example

A monthly bill falls from ($80) to ($68). The decrease is ($12):

\[ \frac{12}{80}=0.15=15\%. \]

The bill decreased 15%. Check with a multiplier: (80(0.85)=68).

Now compare a rate rising from 12% to 15%. The increase is 3 percentage points. Relative percent increase is

\[ \frac{15-12}{12}=0.25=25\%. \]

Both statements can be correct, but they answer different questions.

Repeated percent changes compound. A 10% increase followed by a 10% decrease gives multiplier \(1.10\times0.90=0.99\), so the final amount is 1% below the original.

Error clinic

Dividing by the new value is the most common error. A rise from 50 to 60 is \(10/50=20\%\), not \(10/60\\). Also do not confuse the absolute change of 10 units with a 10% change.

Mastery summary

Find signed or absolute change, divide by the original, convert to percent, state direction, and use a multiplier to verify.