Lecture Notes: Directly Proportional and Inversely Proportional

Directly Proportional

• Definition: As one amount increases, another amount increases at the same rate.
• Symbol: Directly Proportional (not to be confused with infinity symbol).

Example: Earnings

• Pay Rate: $20 per hour.
• Earnings are directly proportional to hours worked:
  • If hours worked = 2, then earnings = $40.
  • If hours worked = 3, then earnings = $60.

Constant of Proportionality

• Definition: Value that relates the two amounts.
• Example Calculation:
  • Earnings = 20 * Hours Worked
  • Formula: y = kx (where k is the constant of proportionality).*

Example Calculation for Constant of Proportionality:

• Given: y is directly proportional to x, when x = 3, y = 15.
• Calculation:
  • 15 = k * 3
  • k = 15 / 3 = 5.
• Therefore, y = 5x.*

Finding Values Using the Constant:

• Example:
  • To find y when x = 9:
    • y = 5 * 9 = 45.
  • To find x when y = 2:
    • 2 = 5x → x = 2/5 = 0.4.*

Inversely Proportional

• Definition: When one value decreases at the same rate that the other increases.

Example: Speed and Travel Time

• As speed increases, travel time decreases:
  • This relationship can be expressed as y is inversely proportional to x, or y = k/x.

Example Calculation:

• Scenario: 4 people can paint a fence in 3 hours.
• Calculation to find time for 6 people:
  • Use: t = k/n (where t = hours, k = constant, n = people).
  • Given: 3 = k/4 → k = 12, so t = 12/n.
  • For 6 people: t = 12/6 = 2 hours.

Finding Number of People Needed:

• To complete the job in 0.5 hours:
  • 12/n = 0.5 → n = 12/0.5 = 24 (24 people needed).

Other Types of Proportionality

Proportional to a Square

• Example: Distance fallen by a stone dropped from a height is proportional to the square of the time.
• Formula: d = kt².
• Specific Calculation:
  • Given: d = 19.6 m at t = 2 s → 19.6 = k * 4 → k = 4.9.
  • Therefore, d = 4.9t²; for t = 3, d = 4.9 * 3² = 44.1 m.

Inverse Square

• Definition: One value decreases as the square of another value.
• Example: Light and distance:
  • Brightness decreases as the square of the distance from the light source increases.
  • Example: Brightness of 1 at 1 meter is 0.25 at 2 meters (doubled distance = quarter of brightness).

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