Study S2 Mathematics Maths - Algebraic Fractions - Geniebook

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 learnt 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
Resources - Academic Topics
Primary
Primary 1
Primary 2
Primary 3
Primary 4
Primary 5
Primary 6
Secondary
Secondary 1
Secondary 2
English
+ More
Maths
Algebraic Fractions
Direct & Inverse Proportion
Congruence And Similarity
Factorising Quadratic Expressions
Further Expansion And Factorisation
Quadratic Equations And Graphs
Simultaneous Equation
+ More
Science
+ More
Secondary 3
Secondary 4
+ More
Sign up for a free demo
(P1 to S4 levels)
Our Education Consultants will get in touch to offer a complimentary product demo and Strength Analysis to your child.
Ready to power up your
child's academic success?
Let our Education Consultants show you how.
*By submitting your phone number, we have your permission to contact
you regarding Geniebook. See our Privacy Policy.