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 \(\begin{align*} \frac{8x \;+\;3}{15} \end{align*}\).

2. 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:

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: 

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

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