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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