Direct and Inverse Proportions - Concept and Problems of Inverse Proportion

Inverse Proportion refers to the relationship between two quantities $x$ and $y$ such that if one quantity increases, the other decreases in the same ratio, keeping their product constant. This is represented visually as a downward sloping curve on a graph (a rectangular hyperbola) that never touches the $x$ or $y$ axes.

The constant of variation, denoted by $k$, is the fixed product of the two inversely related variables. For any pair of values $(x, y)$, the relationship is defined by the equation $x \times y = k$. Visually, if you imagine a rectangle with sides $x$ and $y$, the area of this rectangle remains constant regardless of how the individual sides change.

The inverse relationship can be written as $x \propto \frac{1}{y}$, which is read as '$x$ is inversely proportional to $y$'. This means that $x$ is directly proportional to the reciprocal of $y$.

In a tabular representation of inverse proportion, as you move from left to right, the values in the first row ($x$) might increase while the values in the second row ($y$) decrease, but multiplying the top value by the bottom value in any column will always yield the same result.

A common real-world application is the relationship between Speed and Time for a fixed distance. If you double your speed, the time taken is halved. On a speed-time graph, this is shown as a curve where the product of the coordinates at any point equals the total distance traveled.

Another classic example is Men and Days in work-related problems. If more workers are employed for a task, the number of days required to complete it decreases proportionally, assuming the rate of work remains constant.

When comparing two states of an inverse relationship, we use the ratio equation $x_1 y_1 = x_2 y_2$. This can also be visualized as a balance: as the value of $x$ 'weight' increases, the 'distance' $y$ must decrease to keep the system in equilibrium.