13.06 Adding and subtracting algebraic fractions

The most important things to remember when adding and subtracting fractions are 

• we need like denominators
• we need to keep our fractions equivalent

Now we are going to build on this knowledge and look at how to add and subtract algebraic fractions.

Worked example

Example 1 

Simplify $\frac{4m}{5}-\frac{2}{5}$4m5​−25​.

Think: The first thing we need to do is check that the denominators are the same. In this case, both denominators are $5$5, so then we move on to the next step.

Do: Because our denominators are the same, we can write the numerators together as a single expression over the common denominator. 

$\frac{4m}{5}-\frac{2}{5}=\frac{4m-2}{5}$4m5​−25​=4m−25​

Reflect: Because we cannot simplify the numerator $4m-2$4m−2 any further, this means this is fully simplified.

 

Example 2

Let's look at a very similar example, but where the denominators are not initially the same.

Simplify $\frac{2y}{3}+\frac{5}{6}$2y3​+56​.

Think: We cannot add or subtract fractions unless the denominators are the same. In this case we have one denominator of $3$3, and the other is $6$6. This means we need to find a common denominator between them.  We can see that $6$6 is a multiple of $3$3, and in fact $6$6 is the lowest common multiple of $3$3 and $6$6.  

Do:  Rewrite the first fraction to have a denominator of $6$6. 

$\frac{2y}{3}$2y3​	$=$=	$\frac{2y\times2}{3\times2}$2y×23×2​
	$=$=	$\frac{4y}{6}$4y6​

This means our expression becomes:

$\frac{2y}{3}+\frac{5}{6}=\frac{4y}{6}+\frac{5}{6}$2y3​+56​=4y6​+56​

Now that we have a common denominator, we can add the fractions. 

We can write the numerators as a single expression above the common denominator.

$\frac{4y}{6}+\frac{5}{6}=\frac{4y+5}{6}$4y6​+56​=4y+56​

Reflect:  Is this simplified enough?  As the terms $4y$4y and $5$5 are not like terms and the highest common factor between $4y$4y, $5$5 and $6$6 is $1$1, this is now fully simplified.

 

Example 3

Our final example involves subtraction. We have different denominators, and some simplification to perform at the final step. 

Simplify $\frac{7x}{4}-\frac{3x}{2}$7x4​−3x2​.

Think: Our first goal is to have common denominators.  Looking at the denominators we have, $4$4 and $2$2, we can see that $4$4 is a common multiple.  So let's write both fractions with a denominator of $4$4. 

Do:

$\frac{7x}{4}-\frac{3x}{2}$7x4​−3x2​	$=$=	$\frac{7x}{4}-\frac{3x\times2}{2\times2}$7x4​−3x×22×2​
	$=$=	$\frac{7x}{4}-\frac{6x}{4}$7x4​−6x4​

 

Now we have a common denominator, we write the fraction as a single expression over the common denominator and then simplify where we can:

$\frac{7x}{4}-\frac{6x}{4}$7x4​−6x4​	$=$=	$\frac{7x-6x}{4}$7x−6x4​
	$=$=	$\frac{x}{4}$x4​

 

Reflect: We collected the like terms of $7x$7x and $-6x$−6x.  Are there any other common terms?  Are they any common factors with the $x$x and $4$4?  No, so this is now fully simplified. 

 

Practice questions

Question 1

Question 2

Question 3

Sum and difference of algebraic fractions (Extended)

The same rules apply to the sum and difference of algebraic fractions with algebraic terms in the denominator. A common denominator is still required to add or subtract fractions. 

Question 4