Inversely Proportional Quantities
Two quantities are inversely proportional if their product remains constant, represented by \( k \). Mathematically, this relationship is expressed as: $$ x \cdot y = k $$ or, equivalently, by isolating \( y \): $$ y = \frac{k}{x} $$
Here, \( k \) is the inverse proportionality constant, which defines the relationship between the two variables.
This means that when \( x \) increases, \( y \) decreases in such a way that their product remains unchanged, and vice versa.
But what does this mean in real-world terms? Let’s consider a practical example.
A Practical Example
Imagine a town square that needs to be cleaned after a community festival.
If one person takes on the task alone, it will take 100 hours to clean the entire square.
However, if more people join in, the time required decreases proportionally to the number of workers, following this formula:
$$ t = \frac{100}{n} $$
Where \( t \) represents the time required to finish the job (in hours), \( n \) is the number of workers, and \( 100 \) is the total number of work hours needed (a constant).
The table below illustrates how the time changes based on the number of workers:
| Number of Workers (n) | Time to Complete the Task (t) [hours] | Product n * t (total hours) |
|---|---|---|
| 1 | 100.0 | 100.0 |
| 2 | 50.0 | 100.0 |
| 4 | 25.0 | 100.0 |
| 5 | 20.0 | 100.0 |
| 10 | 10.0 | 100.0 |
| 20 | 5.0 | 100.0 |
The table clearly shows how the time required to complete a 100-hour job decreases as the number of workers increases, while the product \( n \cdot t = 100 \) stays constant.
For example, with two workers, the cleaning time drops to 50 hours. With four workers, it decreases further to 25 hours. If twenty people are involved, the task is completed in just 5 hours, and so on.
If we plot this relationship with \( n \) on the horizontal axis and \( t \) on the vertical axis, the result is a hyperbolic curve.
This type of curve has a characteristic “open arms” shape, reflecting the principle that as one quantity increases, the other decreases, without ever reaching zero.
Inversely proportional quantities describe scenarios where two variables adjust to balance each other.
Many natural phenomena follow this type of relationship.
A classic example of inverse proportionality is Boyle’s Law, which states that the product of the pressure (\( P \)) and volume (\( V \)) of a gas at a constant temperature is always equal to a constant: $$ P \cdot V = k $$ When the volume of a gas is reduced, its pressure increases proportionally, and vice versa. Put simply, compressing a gas raises its pressure, while allowing it to expand reduces the pressure.
