Welcome to inverse proportion! When two quantities are inversely proportional, as one increases, the other decreases proportionally. Their product always remains constant. Let's see this relationship in action with the equation x times y equals 12.
The mathematical relationship for inverse proportion can be expressed in several ways. First, y is proportional to one over x. This gives us the equation y equals k over x, where k is a constant. Rearranging this, we get x times y equals k. Let's look at an example where k equals 24. As x increases from 2 to 8, y decreases from 12 to 3, but their product always equals 24.
Inverse proportion appears frequently in real life. Consider speed and time for a fixed distance. If you need to travel 120 kilometers, driving at 30 kilometers per hour takes 4 hours, but at 60 kilometers per hour takes only 2 hours. Similarly, with workers completing a job, if 2 workers need 24 hours to finish 48 person-hours of work, then 4 workers would complete it in just 12 hours. The product always remains constant.
It's important to distinguish between direct and inverse proportion. In direct proportion, both quantities change in the same direction - when one increases, the other also increases. The relationship is y equals k times x. In inverse proportion, the quantities change in opposite directions - when one increases, the other decreases. Here we see both relationships graphed together, showing how they behave very differently.
Let's summarize the key points about inverse proportion. Remember that y equals k over x, which means the product x times y always equals the constant k. When one quantity increases, the other decreases proportionally. The graph forms a hyperbola curve. We see inverse proportion in many real-world scenarios like speed and time, workers and completion time, gas pressure and volume, and light brightness with distance. Understanding inverse proportion is essential for solving practical problems in science and daily life.