Math · 13 · Problem-Solving and Data Analysis

Percentages

Notes

Percent = $\tfrac{\text{part}}{\text{whole}}\times100$ .
Increase by $p\%$ → multiply by $1+\tfrac{p}{100}$ ; decrease → $1-\tfrac{p}{100}$ .
Successive changes multiply: +20% then −20% gives 0.96, a 4% loss.
Percent change divides by the original, never the new value.
Reverse: original = new ÷ multiplier.
“ $p\%$ of $q\%$ of $x$ ” = $\tfrac{p}{100}\cdot\tfrac{q}{100}\cdot x$ .

Percent of

p\%\ \text{of}\ x=\dfrac{p}{100}\,x

Percent change

\dfrac{\text{new}-\text{old}}{\text{old}}\times100

Multiplier

\text{new}=\text{old}\left(1\pm\dfrac{p}{100}\right)

Original value

\text{old}=\dfrac{\text{new}}{1\pm\frac{p}{100}}

Repeated change

\text{old}\left(1\pm\dfrac{p}{100}\right)^n

Turn every percent into a multiplier (×1.15, ×0.85) and chain them in one line.

10 official SAT questions on Percentages, easiest first.