The most important things to remember when adding and subtracting fractions are
Now we are going to build on this knowledge and look at how to add and subtract algebraic fractions.
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.
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.
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.
Fully simplify the following expression:
$\frac{2x}{3}+\frac{10x}{3}$2x3+10x3
Consider the algebraic fractions $\frac{2m}{5}$2m5 and $\frac{3m}{6}$3m6.
Find the lowest common denominator of the two fractions.
Hence simplify $\frac{2m}{5}+\frac{3m}{6}$2m5+3m6.
Simplify the following: $\frac{4x-10}{3}+\frac{-5x+10}{3}$4x−103+−5x+103
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.
Simplify the following expression, giving your answer in fully factorised form:
$\frac{x}{x^2-16}-\frac{12}{x+4}$xx2−16−12x+4