10. Percent Proportions and Percent Models

Percent means "per hundred," so many percent problems are proportions:
\[ \frac{\text{part}}{\text{whole}}=\frac{\text{percent}}{100}. \]

Example: What is 30% of 80?
\[ \frac{x}{80}=\frac{30}{100}. \]
Cross-multiply:
\[ 100x=2400,\qquad x=24. \]

Percent increase and decrease also use part-whole thinking. If a price goes from 50 dollars to 65 dollars, the increase is 15 dollars. The percent increase is:
\[ \frac{15}{50}=\frac{x}{100}. \]
So \(x=30\%\).

Percent proportions connect to ratio language. A class with 18 out of 24 students passing has:
\[ \frac{18}{24}=\frac{75}{100}=75\%. \]

Common misconception: using the new amount as the denominator for percent change. Percent change uses the original amount as the denominator.

Academic habit: label part, whole, and percent before writing the proportion.