# Algebraic Fractions

In this chapter, we will be discussing the below-mentioned topics in detail:

• Addition and Subtraction of Algebraic Fractions
• Number in denominator
• Variable in denominator

## Equivalent Fractions

At the P3 level, we have already learned about Equivalent Fractions.

### What are Equivalent Fractions?

The value of the fraction remains unchanged if the numerator and denominator are multiplied by or divided by the same non-zero number or expression.

## Addition and subtraction of Algebraic Fractions

Let’s understand this with the help of some examples:

Question 1:

Solve  \begin{align*} \frac{1}{4} + \frac {2}{3} \end{align*}

Solution:

Firstly, we need to make the denominator the same.

The lowest common multiple(LCM) of \begin{align*} 4 \end{align*} and \begin{align*} 3 \end{align*} is \begin{align*} 12 \end{align*}

So, multiply \begin{align*} 3 \end{align*} to the numerator and denominator of \begin{align*} \frac{1}{4} \end{align*} and multiply \begin{align*} 4 \end{align*} to the numerator and denominator of \begin{align*} \frac{2}{3} \end{align*}.

\begin{align*} \frac{1}{4} + \frac {2}{3} &= \frac{3}{12} + \frac{8}{12} \\ &= \frac{11}{12} \end{align*}

## Join, expand and simplify Linear Expressions with Fractional Coefficients

Question 2:

1. Express \begin{align*} \frac{x}{3} + \frac{x+1}{5} \end{align*} as a single fraction in the simplest form.

Solution:

Step 1: Join to get \begin{align*} \frac{5 \;(x)\;+\;3\;(x+1)}{15} \end{align*}.

Step 2: Expand the terms in the numerator to get \begin{align*} \frac{5x \;+\;3x \;+\;3}{15} \end{align*}.

Step 3: Simplify the numerator to get \begin{align*} \frac{8x \;+\;3}{15} \end{align*}.

1. Express  \begin{align*} \frac{3y}4- \frac{y\;-\;1}6 \end{align*} as a single fraction in the simplest form.

Solution:

\begin{align*} \frac{3y}{4} - \frac{y-1}{6} &= \frac{3(3y)- 2(y-1)}{12} && \text{..... Join} \\ &= \frac{9y-2y+2}{12} && \text{..... Expand} \\ &= \frac{7y+2}{12} && \text{..... Simplify} \end{align*}

Question 3:

Express each of the following as a fraction in its simplest form.

1. \begin{align*} \\ \frac{3}{8x} + \frac{11}{12x} \\ \\ \end{align*}
2. \begin{align*} \\ \frac{10}{9m} - \frac{7}{3m} + \frac{1}{4m} \\ \\ \end{align*}

Solution:

\begin{align*} A. \quad \frac{3}{8x} + \frac{11}{12x} &= \frac{3 (3)+2 (11)}{24x} &&&&&&&& \text{..... Join} \\ &= \frac{9+22}{24x} &&&&&&&& \text{..... Expand} \\ &= \frac{31}{24x} &&&&&&&& \text{..... Simplify} \\ \end{align*}

\begin{align*} B. \quad \frac{10}{9m} - \frac{7}{3m} + \frac{1}{4m} &= \frac{4 (10)-12 (7)+9 (1)}{36m} && \text{..... Join} \\ &= \frac{40-84+9}{36m} && \text{..... Expand} \\ &= \frac{-35}{36m} && \text{..... Simplify} \end{align*}

## Factorisation by extracting common factors

Let’s understand this with the help of some examples:

 $5x+10y$ $-6x + 15y$ $5x + 10y = 5 \;(x + 2y)$ \begin{align*} -6x + 15y &= 3 \;(-2x + 5y)\\ &= 3 \;( 5y \;– 2x)\\ &= -3 \;(2x - 5y) \end{align*} $2xy\; –\;6xz$ $-12x^2 \;–\; 20xy$ $2xy \;– \;6xz = 2x \;(y \;– 3z)$ $-12x^2 – 20xy = -4x \;( 3x + 5y)$

Question 4:

Express each of the following as a fraction in its simplest form.

1. \begin{align*} \\ \frac{2q\;-\;p}{9p\;-\;18} + \frac{2p\;-\;q}{12p\;-\;24 }\\ \\ \end{align*}
2. \begin{align*} \\ \frac{2y}{x\;-\;4y} - \frac{5x}{4y\;-\;x} \\\\ \end{align*}

Solution:

\begin{align*} A. \quad \frac{2q-p}{9(p -2)} + \frac{2p-q}{12(p-2) }&= \frac{4(2q-p) +3(2p-q)}{36(p-2)} &&\text{..... Join}\\ &= \frac{8q-4p + 6p-3q}{36 (p-2)} && \text{..... Expand} \\ &= \frac{5q+2p}{36 (p-2)} && \text{..... Simplify} \end{align*}

\begin{align*} B. \quad \frac{2y}{x-4y} - \frac{5x}{4y-x} &= \frac{2y}{x-4y} - \frac{5x}{- (x-4y)} \\ &= \frac{2y}{x-4y} + \frac{5x}{x-4y} \\ &= \frac{2y+5x}{x-4y} \end{align*}

Continue Learning
Algebraic Fractions Direct & Inverse Proportion
Congruence And Similarity Factorising Quadratic Expressions
Further Expansion And Factorisation Quadratic Equations And Graphs
Simultaneous Equation
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