# Direct Variation

### example quantity speed varies

If one quantity increases (or decreases) each **time** another quantity increases (or decreases), the two quantities are said to vary together. The most common form of this is direct variation in which the **ratio** of the two amounts is always the same. For example, speed and **distance** traveled vary directly for a given time. If you travel at 4 mph (6.5 kph) for three hours, you go 12 mi (19.5 km), but at 6 mph (9.5 kph) you go 18 mi (28.5 km) in three hours. The ratio of distance to speed is always 3 in this case.

The common ratio is often written as a constant in an equation. For example, if s is speed and d is distance, the **relation** between them is direct variation for d = ks, where k is the constant. In the example above, k = 3, so the equation becomes d = 3s. For a different time interval, a different k would be used.

Often, one quantity varies with respect to a power of the other. For example, of y = kx^{2}, then y varies directly with the square of x. More than two variables may be involved in a direct variation. Thus if z = kxy, we say that z is a joint (direct) variation of z with x and y. Similarly, if z = kx^{2}/y, we say that z varies directly with x^{2} and inversely with y.

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