Ratios, Proportions, and Percents (SAT)

Ratios

A ratio compares two quantities: $3:5$ or $\frac{3}{5}$

SAT Trick: Total parts = sum of ratio parts

Example: Ratio of boys to girls is $2:3$

• Total parts = $2 + 3 = 5$
• If 30 students: boys = $\frac{2}{5}(30) = 12$

Proportions

Cross-multiply to solve: $\frac{3}{4} = \frac{x}{12} \quad \Rightarrow \quad 4x = 36 \quad \Rightarrow \quad x = 9$

Percent Problems

Three types:

1. Find percent of a number

What is 25% of 80? $0.25 \times 80 = 20$

2. Percent increase/decrease

Formula: $\text{Percent Change} = \frac{\text{New - Original}}{\text{Original}} \times 100\%$

3. Find the whole

15 is 30% of what number? $15 = 0.30x \quad \Rightarrow \quad x = 50$

SAT Percent Tricks

• Successive percents: Two 10% increases ≠ 20% increase!

  • $100 \times 1.10 \times 1.10 = 121$ (21% increase)

• Discount then tax: Order matters!

  • 20% off then 8% tax ≠ 12% off