1.3 Joint Variation - SPM Mathematics

1.3 Joint Variation

1.3a Representing a Joint Variation using the symbol ‘α’.

1. If one quantity is proportional to two or more other quantities, this relationship is known as joint variation.

2. ‘y varies directly as x and z’ is written as y α xz.

3. ‘y varies directly as x and inversely z’ is written as

y α \frac{x}{z} .

4. ‘y varies inversely as x and z’ is written as

y α \frac{1}{x z} .

Example 1:

State the relationship of each of the following variations using the symbol ‘α’.

(a) x varies jointly as y and z.

(b) x varies inversely as y and

\sqrt{z} .

(c) x varies directly as r³ and inversely as y.

Solution:

\begin{array}{l} (a) x α y z \\ (b) x α \frac{1}{y \sqrt{z}} \\ (c) x α \frac{r^{3}}{y} \end{array}

1.3b Solving Problems involving Joint Variation

1. If

y α x^{n} z^{n}, then y = k x^{n} z^{n}

, where k is a constant and n = 2, 3 and ½.

2. If

y α \frac{1}{x^{n} z^{n}}, then y = \frac{1}{k x^{n} z^{n}}

, where k is a constant and n = 2, 3 and ½.

3. If

y α \frac{x^{n}}{z^{n}}, then y = \frac{k x^{n}}{z^{n}}

, where k is a constant and n = 2, 3 and ½.

Example 2:

Given that

p α \frac{1}{q^{2} \sqrt{r}}

when p = 4, q = 2 and r = 16, calculate the value of r when p = 9 and q = 4.

Solution:

\begin{array}{l} Given that p α \frac{1}{q^{2} \sqrt{r}}, p = \frac{k}{q^{2} \sqrt{r}} \\ When p = 4, q = 2 and r = 16, \\ 4 = \frac{k}{2^{2} \sqrt{16}} \\ 4 = \frac{k}{16} \\ k = 64 \\ ∴ p = \frac{64}{q^{2} \sqrt{r}} \\ When p = 9 and q = 4, \\ 9 = \frac{64}{4^{2} \sqrt{r}} \\ 9 = \frac{4}{\sqrt{r}} \\ \sqrt{r} = \frac{4}{9} \\ r = {(\frac{4}{9})}^{2} = \frac{16}{81} \end{array}