Joint Variation

In this lesson, we shall study how a quantity varies in relation to two or more other quantities.

For example, the area of a rectangle varies whenever its length or its width varies. We say that , where A is the area, l is the length and w is the width. This is an example of a joint variation.

There are 3 main types of joint variations:

Type 1: One quantity varies directly as two or more other quantities.

Example:

The mass of a solid varies directly as its volume and density. Express the mass, m, as a joint variation in terms of the volume v and density d.

Solution:

In other words, the bigger the volume or the higher the density, the heavier is the mass.

Type 2: A quantity varies inversely as two or more other quantities.

Example:

The figure below shows a rectangular solid with a fixed volume. Express its width, w, as a joint variation in terms of its length, l, and height, h.

Solution:

In other words, the longer the length l or the height h, the narrower is the width w.

Type 3: A quantity varies directly as one quantity and inversely as another.

Example:

The speed, s, of a moving object varies directly as the distance travelled, d, and varies inversely as the time taken, t. Express s as a joint variation in terms of d and t.

Solution:

In other words, the longer the distance or the shorter the time, the faster is the speed.

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Joint Variation

Joint Variation
In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have joint variation or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation.

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