Notes on Chapter 4: Ratio, Proportion, and Percent

Chapter 4: Ratio, Proportion, and Percent

Section 4.2: Special Applications of Ratio and Proportion

Objectives

  • Solve problems involving proportions

Scale Drawings

  • Application of Proportions: Proportion equations are used in practical scenarios, especially in scaling drawings of objects accurately while fitting them on paper.

    • Example: Drafter reducing dimensions of machine parts, building layouts, etc., using a fixed ratio known as the scale factor.

Example of Scale Drawings

  • Life Size vs Scale Drawings: For example, drawing an automobile at a reduced size where the ratio of actual dimensions (length and width) equals the ratio of scaled dimensions.

Setting Up Ratios

  • To solve for unknown dimensions:

    • Ensure all quantities are in the same units (e.g., converting feet to inches: 1 ft = 12 in.).

  • Define the ratio of actual dimensions to scale dimensions.

Solving Proportions

  • Example Problem: Rectangular room dimensions (Length: 18 ft, Width: 12 ft) are scaled for a drawing (Drawing length: 4.5 in.).

    • Convert 18 ft to inches (216 in); convert 12 ft to inches (144 in).

    • Set up the proportion for width (x for the drawing width):
      [
      \frac{Width{drawing}}{Width{actual}} = \frac{Length{drawing}}{Length{actual}}
      ]

    • Solve the proportion by cross-multiplying.

Finding Scale Factor

  • Scale Factor Calculation: Determine how each inch on the drawing corresponds to the actual object’s size: 1 in. = 4 ft defines scale factor.

Similar Figures

  • Definition: Two figures are similar if they share the same shape but differ in size (e.g., blueprints vs actual objects, photographs).

  • Properties: Corresponding dimensions maintain the same scale ratio.

Example Problem Using Similar Figures

  • Finding heights or lengths in similar triangles or figures.

    • Example: If the height of the landscaper is 6 ft and their shadow is 8 ft while the house will have a height of 15 ft, set up a proportion to find the shadow's extent.

    • Solution: Use similar triangles to draw relationships and find shadow extensions.

Direct Proportion

  • Definition: Two quantities are directly proportional when an increase in one leads to a rise in the other.

    • Example: Electrical resistance in a wire, where resistance increases with the length of the wire.

Solving Direct Proportion Problems

  • Example Problem: If 1 ft of wire has a resistance of 1.65 ohms, set a proportion to determine the length needed for a resistance of 19.8 ohms.

Inverse Proportion

  • Definition: Two quantities are inversely proportional if an increase in one results in a decrease in the other (e.g., travel speed vs time taken).

  • Example Problem: If a trip takes 2 hours at 50 mph, use proportions to calculate the time at different speeds, like 60 mph.

Gears and Pulleys

  • Inverse Proportions: The relationship between gear sizes and rotation speeds. E.g., a gear with more teeth turns slower than a smaller gear.

  • Application in RPM: Gears with fewer teeth rotate faster than those with more teeth; use this to calculate operational speeds.

Example Using Gears and Pulleys

  • Determine the speed of gear A given gear B’s turning speed and their tooth configurations.

Pulleys

  • Operation: Pulleys function similarly to gears; their speed is inversely proportional to their diameter: a larger diameter results in slower rotation compared to a smaller one.

    • Example: If pulley B (diameter 16 in.) is at 240 rpm, calculate pulley A's speed (diameter 20 in.) by setting up the inverse proportion.