5.4 Ratios and Proportions

Ratios and proportions are powerful tools for comparing quantities and solving real-world problems. They allow us to simplify complex relationships, convert between units, and understand how changes in one quantity affect another.

Mastering these concepts opens doors to applications in finance, science, and everyday life. From scaling recipes to analyzing maps, ratios and proportions help us make sense of the world around us and make informed decisions.

Ratios and Proportions

Ratio notation methods

• Ratios compare two quantities written using "to" ($a$ to $b$), colon ($a:b$), or fraction ($\frac{a}{b}$) notation
• Simplify ratios by dividing both quantities by their greatest common factor (GCF)
• Equivalent ratios have the same simplified form (2:4, 1:2, and 10:20)
• Ratios can be expressed as fractions, decimals, or percents

Solving proportions through cross-multiplication

• Proportions are equations stating two ratios are equivalent ($\frac{a}{b} = \frac{c}{d}$ or $a:b = c:d$)
• Solve proportions by setting up equivalent ratios and cross-multiplying ($ad = bc$)
• Solve the resulting equation for the unknown variable
• Verify the solution by substituting the value back into the original proportion

Applications of constant proportionality

• The constant of proportionality ($k$) is the ratio between two directly proportional quantities ($y = kx$, then $k = \frac{y}{x}$)
• Unit rates compare a quantity to one unit of another quantity (miles per hour, price per item)
  • Determine unit rates by finding the constant of proportionality
  • Use dimensional analysis to convert between different units of measurement
• Scale relationships describe the proportional relationship between corresponding measurements in similar figures
  • Scale factor is the constant of proportionality between corresponding lengths
    • If the scale factor is $k$, all lengths in one figure are $k$ times the corresponding lengths in the other figure
  • Area scale factor is the square of the length scale factor
  • Volume scale factor is the cube of the length scale factor

Types of Proportional Relationships

• Direct proportion: As one quantity increases, the other increases proportionally
• Inverse proportion: As one quantity increases, the other decreases proportionally
• Similar figures maintain the same proportions between corresponding parts, despite differences in size