Algebra

UK Secondary (7-11)

LCD with rational expressions

Lesson

When we looked at how to find the lowest common denominator we learnt how to add and subtract fractions. To find the lowest common denominator (LCD), we need to find the lowest common multiple between the denominators.

For example, let's say we want to find the LCD between $\frac{1}{4}$14 and $\frac{3}{10}$310. We could multiply the denominators together to get a denominator of $40$40. However, this is not the *lowest* common denominator.

We can look at factors that the terms already have in common, then multiply the terms

$\frac{1}{4}=\frac{1}{2\times2}$14=12×2 and $\frac{3}{10}=\frac{3}{2\times5}$310=32×5

So $2$2 is a common factor. Now we can take this common factor and multiply it by any uncommon factors from the terms to find our LCD.

$2\times2\times5=20$2×2×5=20, so $20$20 is our lowest common denominator.

We can use this same process to simplify algebraic fractions. Let's look at an example:

Find the LCD for $\frac{12}{4x}$124`x` and $\frac{6}{20x^3}$620`x`3.

1. Find the LCM between the coefficients.

Our coefficients are $4$4 and $20$20. $4\times5=20$4×5=20 and $20\times1=20$20×1=20, so the LCM is $20$20.

2. Find the LCM of the variables.

Our variables are $x$`x` and $x^3$`x`3. $x\times x^2=x^3$`x`×`x`2=`x`3 and $x^3\times1=x^3$`x`3×1=`x`3, so the LCM is $x^3$`x`3.

3. Multiply the LCMs together.

The lowest common multiple is $20x^3$20`x`3.

Handy Hint

It may be helpful to factorise algebraic terms first before you look to find the LCD,

so make sure you're familiar with how to factorise different kinds of algebraic expressions.

Find the lowest common denominator for this pair of algebraic fractions:

$\frac{1}{18h}$118`h` and $\frac{1}{9h}$19`h`

In an upcoming election, it is anticipated that the number of men who vote, $x$`x`, will be greater than the number of women who vote, $y$`y`.

Each expression below represents the expected number of people who will NOT vote in two different counties.

Which county expects more people to not vote?

$\frac{x-y}{2}$

`x`−`y`2A$x-\frac{y}{2}$

`x`−`y`2B$\frac{x-y}{2}$

`x`−`y`2A$x-\frac{y}{2}$

`x`−`y`2B

Find the lowest common denominator of this pair of fractions:

$\frac{1}{q\left(q+5\right)}$1`q`(`q`+5) and $\frac{1}{q\left(q-2\right)}$1`q`(`q`−2)