Direct and Inverse Proportions

• Introduction to Proportion

  • Meaning of Variation
  • Comparison of two quantities
  • Real-life applications

• Direct Proportion (Direct Variation)

  • Definition
    • Two quantities, $x$ and $y$, change together
    • If one increases, the other increases proportionally
    • If one decreases, the other decreases proportionally
  • Mathematical Representation
    • $\frac{x}{y} = k$ (where $k$ is positive constant of proportionality)
    • $x_1 y_2 = x_2 y_1$ or $\frac{x_1}{y_1} = \frac{x_2}{y_2}$
  • Graphical Representation
    • Straight line passing through the origin
  • Common Examples
    • More items bought $\rightarrow$ More cost
    • More speed (at constant time) $\rightarrow$ More distance covered
    • More working hours $\rightarrow$ More work done

• Inverse Proportion (Indirect Variation)

  • Definition
    • Two quantities, $x$ and $y$, move in opposite directions
    • If one increases, the other decreases proportionally
    • If one decreases, the other increases proportionally
  • Mathematical Representation
    • $x \times y = k$ (where $k$ is constant product)
    • $x_1 y_1 = x_2 y_2$
  • Common Examples
    • More workers $\rightarrow$ Less time taken to finish work
    • More speed $\rightarrow$ Less time taken to cover a fixed distance
    • Population of grazers $\rightarrow$ Faster consumption of food stock

• Problem-Solving Techniques

  • Step 1: Identify the given quantities
  • Step 2: Determine if it is direct or inverse proportion
  • Step 3: Set up the appropriate formula/equation
  • Step 4: Solve for the unknown variable ($x$ or $y$)
  • Alternative Method: Unitary Method integration