## Tuesday, 29 May 2007

### The Relationship between Variables (Part II)

Inverse Proportion

Two sets of readings for the quantities x and y are given in the table below.

 x 3 4 6 12 y 4 3 2 1

Here, when x is doubled, y is halved; when x is trebled, y is one-third its previous value and so on.

We say that y is inversely proportional to x, or y varies inversely as x, that is

The graph of y versus x is as shown:

Notice that the product of x and y is the same for every point (12 in this case). Hence this product is known as the constant of proportionality or variation and it is given a symbol such as k. Therefore, we can write the exact relationship between x and y as

Assuming that there are points (x1, y1), (x2, y2), (x3, y3), …, we can say

xy = x1y1 = x2y2 = x3y3 … = k

Notes:
1. When a variable is proportional to two or more variables, we can represent this relationship as a joint variation. For example,
When
y x and y ∝ z
therefore
y xz or y = kxz

When

therefore
1. In every proportional relationship, the converse is also true. For example,
If y x, then x ∝ y.

Examples of physical quantities that vary inversely with each other are:
• Pressure and volume
• Frequency and periodic time
• Gravitational force and the distance squared

To be continued…