Algebraic Fractions
Simplifying an Algebraic Fraction
Example
Simplify:
\[
\frac{4x + 8}{2}
\]
The task here is to break down the fraction so it can be divided by a common factor.
Here, a common factor of 4 exists:
\[
\begin{aligned}
\frac{4x + 8}{2}
&= \frac{4(x + 2)}{2}
\end{aligned}
\]
Top and bottom are now divided by 2:
\[
\begin{aligned}
\frac{4x + 8}{2}
&= \frac{4(x + 2)}{2} \\[6pt]
&= 2(x + 2)
\end{aligned}
\]
Leaving 2(x + 2) as the answer.
Example
\[ \frac{10a + 8b}{2} \]
\[
\begin{aligned}
\frac{10a + 8b}{2}
&= \frac{2(5a + 4b)}{2} \\[6pt]
&= \frac{\cancel{2}(5a + 4b)}{\cancel{2}} \\[6pt]
&= 5a + 4b
\end{aligned}
\]
Sometimes a line must be factorised first, using a difference of two squares if necessary:
Examples
\[ \frac{x^2 - 4}{x + 2} \]
\[
\begin{aligned}
\frac{x^2 - 4}{x + 2}
&= \frac{(x - 2)(x + 2)}{(x + 2)} \\[6pt]
&= \frac{(x - 2)(x + 2)}{(x + 2)} \\[6pt]
&= x - 2 \qquad (x \ne -2)
\end{aligned}
\]
\[ \frac{6x^2 - 24}{3(x + 2)} \]
\[
\begin{aligned}
\frac{6x^{2} - 24}{3(x+2)}
&= \frac{6(x^{2} - 4)}{3(x+2)} \\[6pt]
&= \frac{6(x - 2)(x + 2)}{3(x + 2)} \\[6pt]
&= \frac{6(x - 2)}{3} \\[6pt]
&= 2(x - 2)
\end{aligned}
\]
Solving Equations with Algebraic Fractions
Get rid of the denominators by multiplying through on both sides:
Example
\[
\begin{aligned}
\frac{2x + 4}{3} &= 10 \\[6pt]
2x + 4 &= 30 \\[6pt]
2x &= 26 \\[6pt]
x &= 13
\end{aligned}
\]
Example
\[
\begin{aligned}
\frac{x + 4}{3} &= \frac{3x + 5}{2} \\[10pt]
x + 4 &= 3\left(\frac{3x + 5}{2}\right)
\quad\text{Multiply both sides by 3} \\[10pt]
2(x + 4) &= 3(3x + 5)
\quad\text{Multiply both sides by 2} \\[10pt]
2x + 8 &= 9x + 15
\quad\text{Multiply out brackets} \\[10pt]
2x + 8 - 2x &= 9x + 15 - 2x
\quad\text{Subtract }2x\text{ from both sides} \\[10pt]
8 &= 7x + 15 \\[10pt]
8 - 15 &= 7x - 15
\quad\text{Subtract }15\text{ from both sides} \\[10pt]
-7 &= 7x \\[10pt]
x &= \frac{-7}{7}
\quad\text{Divide both sides by 7} \\[10pt]
x &= -1
\end{aligned}
\]
Operations on Algebraic Fractions
Use the normal rules for fractions.
To add or subtract fractions, the denominators must be the same.
Example:
\[
\frac{1}{x} + \frac{2}{3x}
\]
Normal way:
\[
\begin{aligned}
\frac{1}{x} + \frac{2}{3x}
&= \frac{1}{x}\times\frac{3x}{3x}
+ \frac{2}{3x}\times\frac{x}{x} \\[10pt]
&= \frac{3x}{3x^{2}} + \frac{2x}{3x^{2}} \\[10pt]
&= \frac{3x + 2x}{3x^{2}} \\[10pt]
&= \frac{5x}{3x^{2}} \\[10pt]
&= \frac{5\cancel{x}}{3\cancel{x}^2} \\[10pt]
&= \frac{5}{3x}
\end{aligned}
\]
Vedic Fractions
Vedic Fractions
Vedic fraction methods come from ancient India, where mathematicians developed fast, pattern‑based techniques for working with numbers. Instead of long written steps, the Vedic approach uses quick cross‑multiplying ideas to simplify fractions mentally.
Vedic WayThese methods are thousands of years old, yet still offer a fresh, intuitive way to understand fraction arithmetic.
Multiply denominators, cross‑multiply and add:
\[
\begin{aligned}
\frac{1}{x} + \frac{2}{3x}
&= \frac{3x + 2x}{3x^{2}} \\[8pt]
&= \frac{5x}{3x^{2}} \\[8pt]
&= \frac{5\cancel{x}}{3\cancel{x}^{2}} \\[8pt]
&= \frac{5}{3x}
\end{aligned}
\]
Another example:
\[
\frac{3x + 2}{2x + 3}
\;+\;
\frac{x + 1}{x + 2}
\]
\[
\begin{aligned}
\frac{3x + 2}{2x + 3} + \frac{x + 1}{x + 2}
&= \frac{(3x + 2)(x + 2)}{(2x + 3)(x + 2)}
+ \frac{(x + 1)(2x + 3)}{(x + 2)(2x + 3)} \\[10pt]
&= \frac{(3x + 2)(x + 2) + (x + 1)(2x + 3)}
{(2x + 3)(x + 2)} \\[10pt]
&= \frac{(3x^{2} + 6x + 2x + 4) + (2x^{2} + 3x + 2x + 3)}
{(2x + 3)(x + 2)} \\[10pt]
&= \frac{5x^{2} + 13x + 7}{(2x + 3)(x + 2)}
\end{aligned}
\]
Same question done the Vedic way:
\[
\begin{aligned}
\frac{3x + 2}{2x + 3} + \frac{x + 1}{x + 2}
&= \frac{(3x + 2)(x + 2) + (x + 1)(2x + 3)}
{(2x + 3)(x + 2)} \\[10pt]
&= \frac{(3x^{2} + 6x + 2x + 4) + (2x^{2} + 3x + 2x + 3)}
{(2x + 3)(x + 2)} \\[10pt]
&= \frac{5x^{2} + 13x + 7}
{(2x + 3)(x + 2)}
\end{aligned}
\]
Multiplication & Division
To multiply fractions, multiply the numerators and then the denominators.
Don't forget to simplify if possible.
Example:
\[
\frac{3}{x} \times \frac{2}{7x}
\]
\[
\begin{aligned}
\frac{3}{x} \times \frac{2}{7x}
&= \frac{3 \times 2}{x \times 7x} \\[8pt]
&= \frac{6}{7x^{2}}
\end{aligned}
\]
Another example:
\[
\begin{aligned}
\frac{3x + 6}{2x + 14} \times \frac{2x - 8}{7x + 7}
&= \frac{(3x + 6)(2x - 8)}{(2x + 14)(7x + 7)} \\[10pt]
\text{Take out common factors} \\[6pt]
&= \frac{3(x + 2)\,(2x - 8)}{2(x + 7)\,(7x + 7)} \\[10pt]
&= \frac{3(x + 2)\,2(x - 4)}{2(x + 7)\,7(x + 1)} \\[10pt]
\text{simplify} \\[6pt]
&= \frac{3(x + 2)\,\cancel{2}(x - 4)}{\cancel{2}(x + 7)\,7(x + 1)} \\[10pt]
&= \frac{3(x + 2)(x - 4)}{7(x + 7)(x + 1)}
\end{aligned}
\]
When dividing fractions, turn the second fraction upside down and multiply.
Examples:
\[
\begin{aligned}
\frac{x + 6}{5} \div \frac{3}{5x + 20}
&= \frac{x + 6}{5} \times \frac{5x + 20}{3} \\[10pt]
&= \frac{(x + 6)(5x + 20)}{5 \times 3} \\[10pt]
&= \frac{(x + 6)\,5(x + 4)}{5 \times 3} \\[10pt]
&= \frac{(x + 6)(x + 4)}{3}
\end{aligned}
\]
\[
\begin{aligned}
\frac{3x + 6}{2x + 14} \div \frac{2x - 8}{7x + 7}
&= \frac{3x + 6}{2x + 14} \times \frac{7x + 7}{2x - 8} \\[10pt]
&= \frac{(3x + 6)(7x + 7)}{(2x + 14)(2x - 8)} \\[10pt]
&= \frac{3(x + 2)\,7(x + 1)}{2(x + 7)\,2(x - 4)} \\[10pt]
&= \frac{21(x + 2)(x + 1)}{4(x + 7)(x - 4)}
\end{aligned}
\]
Alternative method:
\[
\begin{aligned}
\frac{x + 6}{5} \div \frac{3}{5x + 20}
&= \frac{\frac{x + 6}{5}}{\frac{3}{5x + 20}} \\[10pt]
&= \frac{(x + 6)(5x + 20)}{5 \times 3} \\[10pt]
&= \frac{(x + 6)\,5(x + 4)}{5 \times 3} \\[10pt]
&= \frac{(x + 6)(x + 4)}{3}
\end{aligned}
\]
\[
\begin{aligned}
\frac{3x + 6}{2x + 14} \div \frac{2x - 8}{7x + 7}
&= \frac{\frac{3x + 6}{2x + 14}}{\frac{2x - 8}{7x + 7}} \\[10pt]
\frac{\text{Top times bottom}}{\text{Middle times middle}} \\[10pt]
&= \frac{(3x + 6)(7x + 7)}{(2x + 14)(2x - 8)} \\[10pt]
&= \frac{3(x + 2)\,7(x + 1)}{2(x + 7)\,2(x - 4)} \\[10pt]
&= \frac{21(x + 2)(x + 1)}{4(x + 7)(x - 4)}
\end{aligned}
\]
Complex Fractions
Complex fractions such as:
\[
\frac{a + \frac{1}{b}}{a - \frac{1}{b}}
\]
Can be simplified using the multiplicative identity:
\[
\frac{a}{b}
= \frac{a}{b} \times 1
= \frac{a}{b} \times \frac{m}{m}
= \frac{am}{bm},
\quad m \ne 0
\]
- Multiply numerator and denominator by the LCM.
\[
\frac{a + \frac{1}{b}}{a - \frac{1}{b}}
=
\frac{b\left(a + \frac{1}{b}\right)}{b\left(a - \frac{1}{b}\right)}
\]
- Then expand brackets, factorise and simplify.
\[
\begin{aligned}
\frac{a + \frac{1}{b}}{a - \frac{1}{b}}
&= \frac{b\left(a + \frac{1}{b}\right)}{b\left(a - \frac{1}{b}\right)} \\[10pt]
&= \frac{ab + \frac{b}{b}}{ab - \frac{b}{b}} \\[10pt]
&= \frac{ab + 1}{ab - 1}
\end{aligned}
\]
Examples:
\[
\begin{aligned}
\frac{\frac{1}{y} - \frac{1}{x}}{\frac{x}{y} - \frac{y}{x}}
&= \frac{xy\left(\frac{1}{y} - \frac{1}{x}\right)}
{xy\left(\frac{x}{y} - \frac{y}{x}\right)} \\[10pt]
&= \frac{\frac{xy}{y} - \frac{xy}{x}}
{\frac{x^{2}y}{y} - \frac{xy^{2}}{x}} \\[10pt]
&= \frac{x - y}{x^{2} - y^{2}} \\[10pt]
&= \frac{(x - y)}{(x - y)(x + y)} \\[10pt]
&= \frac{1}{x + y}
\end{aligned}
\]
\[
\begin{aligned}
\frac{\frac{1}{a^{2}} - 4}{\frac{1}{a} - 2}
&= \frac{a^{2}\left(\frac{1}{a^{2}} - 4\right)}
{a^{2}\left(\frac{1}{a} - 2\right)} \\[10pt]
&= \frac{1 - 4a^{2}}{a - 2a^{2}} \\[10pt]
&= \frac{1 - (2a)^{2}}{a - 2a^{2}} \\[10pt]
&= \frac{(1 + 2a)(1 - 2a)}{a(1 - 2a)} \\[10pt]
&= \frac{1 + 2a}{a}
\end{aligned}
\]
\[
\begin{aligned}
\frac{x - \frac{2}{x} + 1}{x + \frac{1}{x} - 2}
&= \frac{x\left(x - \frac{2}{x} + 1\right)}
{x\left(x + \frac{1}{x} - 2\right)} \\[10pt]
&= \frac{x^{2} - 2 + x}{x^{2} + 1 - 2x} \\[10pt]
&= \frac{x^{2} + x - 2}{x^{2} - 2x + 1} \\[10pt]
&= \frac{(x + 2)(x - 1)}{(x - 1)(x - 1)} \\[10pt]
&= \frac{x + 2}{x - 1}
\end{aligned}
\]
Alternatively, using Vedic addition and the alternative division method:
