4. Ratios, Proportions & Percentages

This is the heart of SAT® Problem-Solving and Data Analysis. A ratio compares quantities; a proportion sets two ratios equal. Solve proportions by cross-multiplying: \(\dfrac{3}{4} = \dfrac{x}{20}\) gives \(4x = 60\), so \(x = 15\).

Percent means 'out of 100'. The core relationship is:
\[ \text{part} = \text{percent} \times \text{whole}. \]

• 'What percent of 120 is 30?' → \(30 \div 120 = 0.25 = 25\%\).
• 'Find 15% of 80' → \(0.15 \times 80 = 12\).
• Percent increase/decrease: a 25%-off \$40 shirt costs \(40 - 0.25(40) = \$30\); raising 80 by 15% gives \(80 \times 1.15 = 92\).

A unit rate is the amount per one unit: 150 miles in 3 hours is \(150 \div 3 = 50\) miles per hour.

⚠️ Common trap: for a percent change, divide by the original amount, not the new one. And a 20% raise followed by a 20% cut does not return to the start.

💡 Tip: turn a percent into a decimal (move the point two places left) and multiply — fast and calculator-friendly.