Percentages

Percent means per hundred. So $25\% = \frac{25}{100} = 0.25$.

Three core formulas — every SAT percent question reduces to one of these:

1. Find the part: $\text{Part} = \text{Percent} \cdot \text{Whole}$. What is 30% of 80? → $0.30 \times 80 = 24$.

2. Find the percent: $\text{Percent} = \frac{\text{Part}}{\text{Whole}}$. 24 is what percent of 80? → $\frac{24}{80} = 0.30 = 30\%$.

3. Find the whole: $\text{Whole} = \frac{\text{Part}}{\text{Percent}}$. 24 is 30% of what number? → $\frac{24}{0.30} = 80$.

Translating the words. Of almost always means multiply. Is means equals. What is the unknown.

"What is 30% of 80?" → $x = 0.30 \times 80$. Read it left to right.

"24 is what percent of 80?" → $24 = p \times 80$, where $p$ is the unknown percent.

Percent change. Use this formula:

$\text{Percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%$

If a stock goes from $50 to $65: $\frac{65 - 50}{50} = 0.30 = 30\%$ increase.

If it then drops back to $50: $\frac{50 - 65}{65} \approx -0.231 = -23.1\%$. A 30% increase followed by a 30% decrease does NOT bring you back to the start — the percentages are computed against different baselines.

Compound increases / discounts. A 20% discount followed by a 10% discount is NOT 30% off. Multiply the factors:

$0.80 \times 0.90 = 0.72 \Rightarrow 28\% \text{ total off}$

The successive discounts apply to a smaller and smaller base each time.