Direct proportion describes a relationship where two quantities change together in the same way. When one quantity increases, the other increases by the same factor. Mathematically, if y is directly proportional to x, we write y equals k times x, where k is the constant of proportionality. In this example, y equals 2x, so when x is 1, y is 2; when x is 2, y is 4, and so on.
Inverse proportion describes a relationship where as one quantity increases, the other decreases proportionally. Mathematically, if y is inversely proportional to x, we can write y equals k divided by x, or equivalently, x times y equals k, where k is the constant. In this example, y equals 8 divided by x. When x is 1, y is 8; when x is 2, y is 4; when x is 4, y is 2. Notice how the product is always 8.
Now let's compare direct and inverse proportions. In direct proportion, the ratio y over x remains constant, and both quantities change in the same direction. The graph is a straight line passing through the origin. In inverse proportion, the product x times y remains constant, and the quantities change in opposite directions. The graph forms a hyperbola. These two relationships represent fundamentally different ways that quantities can be related.
Let's look at real-world examples. Direct proportion appears when distance increases with time at constant speed, when cost increases with quantity at the same price per item, or when wages increase with hours worked. Inverse proportion occurs when speed and time are related for a fixed distance. In our table, to travel 120 kilometers, if speed doubles from 20 to 40 kilometers per hour, time halves from 6 to 3 hours. The product of speed and time always equals 120.
To summarize what we've learned: Direct proportion means two quantities change together in the same direction, represented by y equals k times x. Inverse proportion means quantities change in opposite directions, represented by y equals k divided by x. Direct proportion creates straight line graphs through the origin, while inverse proportion creates curved hyperbola graphs. Both types of relationships are fundamental in mathematics and appear frequently in real-world situations.