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Unit 3: Ratios, Rates, and Percents

Topic 1

Ratios and Equivalent Ratios

A ratio compares two quantities using division. It can be written in three forms:
a:b,ab,or a to ba:b, \quad \frac{a}{b}, \quad \text{or } a \text{ to } b
where b≠0b \ne 0.
Ratios can compare:
  • Part to part: boys to girls, red to blue.
  • Part to whole: boys to total students.
  • Whole to part: total students to boys.
Two ratios are equivalent if they have the same value when simplified:
ab=cdmeansad=bc\frac{a}{b} = \frac{c}{d} \quad \text{means} \quad ad = bc
This property is the basis for solving proportion problems.

Core Skills

  • Simplify ratios as you would fractions.
  • Find missing terms using cross multiplication.
  • Convert word statements to fraction or “:” form before solving.
  • Express ratios in simplest form and interpret them with correct units.

Example 1: Simplifying Ratios

Simplify the ratio 18:2418:24. Step 1: Divide both terms by their greatest common factor, 6.
1824=34\frac{18}{24} = \frac{3}{4}
Final Answer: 3:4\boxed{3:4}

Example 2: Finding an Equivalent Ratio

Find a ratio equivalent to 4:54:5 where the first term is 12.
45=12x\frac{4}{5} = \frac{12}{x}
Cross multiply: 4x=60⇒x=15.4x = 60 \Rightarrow x = 15. Final Answer: 12:15\boxed{12:15}

Example 3: Solving a Proportion

Solve for xx:
3x=912\frac{3}{x} = \frac{9}{12}
Cross multiply: 3×12=9x⇒36=9x⇒x=4.3 \times 12 = 9x \Rightarrow 36 = 9x \Rightarrow x = 4. Final Answer: x=4\boxed{x = 4}

Example 4: Interpreting Ratios in Context

A recipe uses 2 cups of sugar for every 5 cups of flour. If 20 cups of flour are used, how much sugar is needed?
25=x20\frac{2}{5} = \frac{x}{20}
Cross multiply: 5x=40⇒x=8.5x = 40 \Rightarrow x = 8. Final Answer: 8 cups of sugar\boxed{8\text{ cups of sugar}}

Key Takeaways

  • Simplify ratios by dividing both quantities by the same factor.
  • Equivalent ratios represent the same proportional relationship.
  • Use cross multiplication to find missing terms in a proportion.
  • Always include units and interpret ratios based on context.