Direct and Inverse Proportions - Inverse Proportion

Definition of Inverse Proportion: Two quantities $x$ and $y$ are said to be in inverse proportion if an increase in $x$ causes a proportional decrease in $y$ (and vice versa), such that their product $x \times y$ remains constant. Visually, imagine a balance scale where adding weight to one side requires moving the weight closer to the center to maintain equilibrium.

The Constant Product Rule: For any two quantities in inverse proportion, the product of their corresponding values is always equal to a non-zero constant $k$. This is written as $xy = k$. In a table of values, multiplying the top value by the bottom value in any column will always yield the same result.

Graphical Representation: When values of an inverse proportion are plotted on a graph where the horizontal axis represents $x$ and the vertical axis represents $y$, the points form a smooth, downward-sloping curve called a hyperbola. This curve approaches the axes but never touches them, illustrating that as one variable approaches infinity, the other approaches zero.

The Ratio Relationship: If $(x_1, y_1)$ and $(x_2, y_2)$ are two pairs of values in an inverse proportion, the relationship can be expressed as $x_1 y_1 = x_2 y_2$. This can also be written as a ratio: $\frac{x_1}{x_2} = \frac{y_2}{y_1}$, showing that the ratio of $x$ values is the inverse of the ratio of $y$ values.

Real-world Application - Speed and Time: For a fixed distance, speed and time are inversely proportional. If you double your speed, the time taken for the journey is halved. Visually, a faster-moving object covers the same distance line in a shorter duration.

Real-world Application - Work and Men: The number of workers and the time taken to complete a task are inversely proportional. More workers mean less time is required. If you visualize a wall being built, more hands on the bricks make the height grow faster, reducing the total days on the calendar.