Variation. Types of Variation - Revision Notes

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4.5Variation. Types of Variation


In these revision notes for Variation. Types of Variation, we cover the following key points:

  • What is variation?
  • What are the three types of variation?
  • Where does variation differ from proportion?
  • Where to use proportional and variation approach in graph problems?
  • What is the constant of proportionality used in variation?
  • What is the shape of the graph that illustrates direct/inverse variation?
  • Which are the properties of variation?

Variation. Types of Variation Revision Notes

Variation represents any change in a quantity due to the change in another if the two quantities have a direct relationship with each other.

Proportion and variation are similar concepts but not entirely identical. They differ in the following aspects:

  1. A proportion does not necessarily involve related terms, i.e. the existence of one term of proportion does not necessarily imply the existence of the other term.
  2. Proportion is limited to a finite number of terms while variation includes a much wider range of terms related through a formula. Hence, the graph of a variation usually contains an uninterrupted line.

There are numerous ways in which two quantities are related to each other but only three of them are considered as variations. They are direct, inverse and joint variation.

A direct variation is a variation that can be expressed through the general formula

y = k × x

where x and y represent possible values of the two quantities involved in the variation while k is the constant of proportionality that gives the value of each individual ratio or rate involved in variation. The graph that represents the direct variation is a straight line that passes through the origin geometrically.

Inverse variation is similar to inverse proportion. By definition, two quantities vary inversely when their product is always constant.

Mathematically, we express an inverse variation as

x × y = k

or

y = k/x

where x and y are the quantities involved in the variation and k is the constant of proportionality

Inverse proportion represents a special case of inverse variation, which shows the relationship between two quantities in two different instants (1) and (2). In this case, we can write

x1 × y1 = x2 × y2 = k

or

x1/x2 = y2/y1

Obviously, the graph of an inverse variation is an uninterrupted hyperbola, while that of inverse proportion shows a finite number of points in the graph - the points whose coordinates represent the quantities involved in the proportion.

The graph of an inverse variation is an uninterrupted hyperbola, unlike that of inverse proportion, which shows a finite number of points in the graph.

When we have to divide an amount into more than two shares where the criterion is determined by the inverse relationship between the quantities involved in the variation, we express the relationship as

a:b:c = k/d:k/e:k/f

This is because when we separately consider each quantity on the left to the corresponding quantity on the right, we obtain the following relations:

a × d = k
b × e = k
c × f = k

If there are more than two quantities involved in a variation where both types of variation are present, then we are dealing with a joint variation.

For example, the quantity x may vary inversely with a quantity y and directly with another quantity z to form a joint variation. In such cases, we can write for two different situations (1) and (2):

z1 = k × x1 × y1

and

z2 = k × x2 × y2

or

z1/z2 = x1 × y1/x2 × y2

The general rule of joint variation is that the two quantities that are at the same side of the formula vary inversely, while each of them varies directly with the quantity that is present at the other side of the formula.

The following rules are also true for variation:

  1. If x and y vary directly as a and b, then
    x/a = y/b = k
  2. If x and y vary inversely as a and b, then
    a × x = b × y = k
  3. In a joint variation where y and y vary inversely as a and b while z varies directly with either x or y as c with a or b, then
    z/c = x × y/a × b

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