Lecture Notes: Direct Proportionality and Equation Writing

Key Concepts

• Direct Proportionality: When two variables increase together in a consistent ratio. Example: Hours worked (H) and money earned (E).
  • As hours increase, money earned increases proportionally.
  • Expressed as an equation: ( E = 12H )

Example Problem 1: Boiling Water

• Variables

  • ( T ): Time in seconds
  • ( M ): Mass of water in grams

• Given

  • ( T ) is directly proportional to ( M ).
  • ( T = 600 ) when ( M = 200 )

• Objective

  • Find ( T ) when ( M = 450 ).

• Process

  1. Set up proportionality: ( T \propto M )
  2. Convert to equation: ( T = kM )
  3. Find constant ( k ):
     • ( 600 = k \times 200 )
     • ( k = 3 )
  4. Rewrite equation: ( T = 3M )
  5. Find ( T ) for ( M = 450 ):
     • ( T = 3 \times 450 = 1350 )

Example Problem 2: Wire Length

• Variables

  • ( L ): Length of wire in cm
  • ( D ): Diameter of wire in cm

• Given

  • ( L ) is directly proportional to ( D ).
  • Length of 25cm corresponds to diameter of 2cm.

• Objective

  • Calculate the diameter of a wire that is 40cm long.

• Process

  1. Set up proportionality: ( L \propto D )
  2. Convert to equation: ( L = kD )
  3. Find constant ( k ):
     • ( 25 = k \times 2 )
     • ( k = 12.5 )
  4. Rewrite equation: ( L = 12.5D )
  5. Find ( D ) for ( L = 40 ):
     • ( 40 = 12.5D )
     • ( D = 3.2 ) cm

Conclusion

• Equations can describe direct proportionality by incorporating a constant of proportionality.
• Using given values to determine constants allows for solving proportionality problems algebraically.

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This concludes the lecture on writing equations for directly proportional relationships. Review these examples to understand how to solve similar problems.