Ratios, rates, proportions

Ratios, rates, and proportions describe proportional relationships between quantities, allowing you to compare parts to parts, parts to wholes, or scale amounts across different contexts.

Understanding Ratios and Rates

A ratio compares two or more quantities with the same units, written as $a:b$ or $\frac{a}{b}$. A rate compares two quantities with different units.

• Part-to-Part Ratio: Compares one subgroup to another subgroup.

  Example: In a class of $12$ boys and $18$ girls, the ratio of boys to girls is $12:18 = 2:3$.

• Part-to-Whole Ratio: Compares a subgroup to the total amount.

  Example: In the same class of $30$ students, the ratio of boys to total students is $12:30 = 2:5$.

• Unit Rates: A rate expressed for a single unit of measure (where the denominator is $1$), calculated as $\frac{\text{Quantity } A}{\text{Quantity } B}$.

  Example: Driving $150\text{ miles}$ on $5\text{ gallons}$ of gas yields a unit rate of $\frac{150}{5} = 30\text{ miles per gallon}$.

Proportions and Cross-Multiplication

A proportion is an equation stating that two ratios or rates are equal: $\frac{a}{b} = \frac{c}{d}$.

• Solving Proportions: Use cross-multiplication to set product pairs equal: $a \cdot d = b \cdot c$.

  Example: Solve $\frac{x}{12} = \frac{5}{4} \implies 4x = 60 \implies x = 15$.

• Scale Factors: When scaling geometric figures or maps, every dimension is multiplied by a constant ratio $k$.

  Example: If $1\text{ cm}$ on a map represents $5\text{ miles}$, then $4.5\text{ cm}$ represents $4.5 \times 5 = 22.5\text{ miles}$.

Direct and Inverse Variation

Proportional relationships describe how one variable changes when another variable is scaled.

• Direct Variation ($y = kx$): As $x$ increases, $y$ increases proportionally. The constant of proportionality is $k = \frac{y}{x}$.

  Example: If $y$ varies directly with $x$, and $y = 12$ when $x = 3$, then $k = \frac{12}{3} = 4$, so the equation is $y = 4x$.

• Inverse Variation ($y = \frac{k}{x}$ or $x \cdot y = k$): As $x$ increases, $y$ decreases proportionally so their product remains constant.

  Example: If $4$ workers build a wall in $6\text{ hours}$, $k = 4 \times 6 = 24$. Then $8$ workers will build it in $\frac{24}{8} = 3\text{ hours}$.

1. Always set up proportions with matching units in both numerators and denominators.

   Example: A recipe requires $3$ cups of flour for $12$ cookies. How many cups of flour are needed for $20$ cookies?

   • $\frac{\text{flour}}{\text{cookies}}: \frac{3}{12} = \frac{x}{20}$

   • Cross-multiply: $12x = 60 \implies x = 5\text{ cups}$

2. Define ratio multiplier variables ($x$) for multi-part ratio word problems.

   Example: A recipe requires $3$ cups of flour for $12$ cookies. How many cups of flour are needed for $20$ cookies?

   • $\frac{\text{flour}}{\text{cookies}}: \frac{3}{12} = \frac{x}{20}$

   • Cross-multiply: $12x = 60 \implies x = 5\text{ cups}$

3. Distinguish between part-to-part and part-to-whole ratios before calculating.

   Example: The ratio of dogs to cats at a shelter is $3:4$. What fraction of the total animals are dogs?

   • Part-to-part: $\text{dogs} = 3$, $\text{cats} = 4$

   • Total parts: $3 + 4 = 7$

   • Fraction of total: $\frac{\text{dogs}}{\text{total}} = \frac{3}{7}$

4. Convert compound rates using dimensional analysis step-by-step.

   Example: Convert $60\text{ miles per hour}$ to $\text{feet per second}$ ($1\text{ mile} = 5,280\text{ feet}$).

   • Rate expression: $\frac{60\text{ miles}}{1\text{ hour}} \times \frac{5280\text{ feet}}{1\text{ mile}} \times \frac{1\text{ hour}}{3600\text{ seconds}}$

   • Simplify: $\frac{60 \times 5280}{3600} = \frac{316,800}{3,600} = 88\text{ ft/s}$

5. Use $xy = k$ for inverse variation tasks involving work rates or time.

   Example: A journey takes $3\text{ hours}$ at $40\text{ mph}$. How long does it take at $60\text{ mph}$?

   • Constant distance ($k$): $3 \times 40 = 120\text{ miles}$

   • New time: $60 \cdot t = 120 \implies t = 2\text{ hours}$