1. Interpreting Ratios

• Definition: A ratio is a comparison of two quantities, expressed as a fraction, with a colon (:), or with words.

  • Example: Ratio of cats to dogs is 3:2, or $$\frac{3}{2}$$.

• Simplification: Ratios should always be simplified to their lowest terms.

  • Example: 10:15 → 2:3.

• Scaling ratios: Multiplying/dividing both parts of a ratio by the same number gives an equivalent ratio.

  • Example: 2:3 → 4:6.

• Part-to-part vs. part-to-whole:

  • Part-to-part: Ratio of boys to girls is 3:2.

  • Part-to-whole: Boys are 3 out of 5 total students → 3/5.

ACT Tip: Pay attention to whether the question asks for part-to-part or part-to-whole; this is a common trap.

2. Direct Proportion

• Definition: Two quantities are directly proportional if one increases/decreases at the same rate as the other.

  • Formula: $$\frac{x_1}{y_1}=\frac{x_2}{y_2}$$ or $$y=kx$$, where k is the constant of proportionality.

• Example: If 5 pencils cost $2, how much do 15 pencils cost?

  • Ratio: $$\frac{5}{2}=\frac{15}{x}$$ ​⟹ $$x=6$$

• Graph: A direct proportion is a straight line through the origin.

3. Inverse Proportion

• Definition: Two quantities are inversely proportional if their product is constant.

  • Formula: $$x_1y_1=x_2y_2$$​ or $$y=\frac{k}{x}.$$

• Example: If 6 workers finish a job in 10 hours, how many hours would 12 workers take?

  • $\$\$6\backslash cdot10=12\backslash cdot\; x\$\$\; ⟹\; \$\$x=5\$\$$

• Graph: An inverse proportion is a curve (hyperbola).

4. Applications

• Mixtures: Ratios are used in combining mixtures (e.g., paint or food problems).

• Speed, distance, and time: These problems often rely on direct or inverse proportionality.

• Probability: Often expressed in ratio form (favorable outcomes : total outcomes).

5. Common ACT Question Types

1. Simplifying a ratio or converting part-to-part into part-to-whole.

2. Solving word problems involving direct or inverse proportions.

3. Working with multiple ratios (e.g., if A:B = 2:3 and B:C = 4:5, find A:C).

4. Proportional reasoning in geometry (similar triangles, scale drawings, etc.).

Key Strategies for the ACT

• Always reduce ratios to simplest form.

• Be careful whether the ratio is part-to-part or part-to-whole.

• Write proportions clearly and cross-multiply to solve quickly.

• When stuck, plug in simple numbers to test the ratio.

• Make to complete all the School of Mathematics ACT quizzes on this topic, and score at least 80% or more on each quiz.