Rule of Separation of Division

Here we will learn the rule of separation of division of algebraic fractions with the help of some problems.

(i) $$\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$$

(ii) $$\frac{x - y}{k} = \frac{x}{k} - \frac{y}{k}$$, but $$\frac{k}{x + y} \neq \frac{k}{x} + \frac{k}{y}$$

By transposing the above two quantities we get;

(i) $$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$

(ii) $$\frac{x}{k} - \frac{y}{k} = \frac{x - y}{k}$$

These means, if two fractions are with same denominator then taking that common denominator as the ‘denominator’ and the sum of the numerators as ’numerator’, we get the sum of the two fractions. Similarly, taking the common denominator as the ‘denominator’ if the difference of the numerators is taken, we get the difference of two fractions.

Now we will learn how to solve the problems by using the rule of separation of division to determine the sum or difference of two algebraic fractions by taking common denominator.

1. Find the sum by taking common denominator:

$$\frac{m}{xy} + \frac{n}{yz}$$

Solution:

We observe the two denominators are xy and yz and their L.C.M. is xyz, so xyz is the least quantity which is divisible by xy and yz. So, keeping the value of $$\frac{m}{xy}$$ and $$\frac{n}{yz}$$ unchanged xyz should be made their common denominator. So, both the numerator and denominator is to be multiplied by xyz ÷ xy = z in case of $$\frac{m}{xy}$$ and xyz ÷ yz = x in case of $$\frac{n}{yz}$$.

Therefore, we can write

$$\frac{m}{xy} + \frac{n}{yz}$$

= $$\frac{m ∙ z}{xy ∙ z} + \frac{n ∙ x}{yz ∙ x}$$

= $$\frac{mz}{xyz} + \frac{nx}{xyz}$$

= $$\frac{mz + nx}{xyz}$$

2. Find the difference by taking common denominator:

$$\frac{a}{xy} - \frac{b}{yz}$$

Solution:

There are the two denominators xy and yz and their L.C.M. is xyz. To make both the fractions with the common denominator, both the numerator and denominator of these are to be multiplied by xyz ÷ xy = z in case of $$\frac{a}{xy}$$ and by xyz ÷ yz = x in case of $$\frac{b}{yz}$$.

Therefore, we can write

$$\frac{a}{xy} - \frac{b}{yz}$$

= $$\frac{a ∙ z}{xy ∙ z} - \frac{b ∙ x}{yz ∙ x}$$

= $$\frac{az}{xyz} - \frac{bx}{xyz}$$

= $$\frac{az - bx}{xyz}$$

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