Add and subtract algebraic fractions with binomial numerators

Lesson

Worked Example

Example 1

The following example shows how to simplify algebraic fractions where we have different denominators, and some simplification to perform at the final step.

$\frac{3x+1}{4}+\frac{3x}{2}$3x+14+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 use $4$4

Do:

 $\frac{3x+1}{4}+\frac{3x}{2}$3x+14​+3x2​ $=$= $\frac{3x+1}{4}+\frac{3x\times2}{2\times2}$3x+14​+3x×22×2​ $=$= $\frac{3x+1}{4}+\frac{6x}{4}$3x+14​+6x4​

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

Do

 $\frac{3x+1}{4}+\frac{6x}{4}$3x+14​+6x4​ $=$= $\frac{3x+1+6x}{4}$3x+1+6x4​ $=$= $\frac{9x+1}{4}$9x+14​

Reflect: We collected the like terms of $3x$3x and $6x$6x.  Are there any other common terms?  No, so this is a simplified as this answer gets.

Example 2

What if the fraction we are changing to get a common denominator has multiple terms in the numerator?  This means we will have to do some extra manipulation.

$\frac{2m-4}{3}+\frac{m}{4}$2m43+m4

Think: Our first goal is to have common denominators.  Looking at the denominators we have, $3$3 and $4$4, we can see that $12$12 is a common multiple.  So we will multiply the numerator and denominator of the first fraction by $4$4, and multiply the top and bottom of the second fraction by $3$3.

Do:

 $\frac{2m-4}{3}+\frac{m}{4}$2m−43​+m4​ $=$= $\frac{\left(2m-4\right)\times4}{3\times4}+\frac{m\times3}{4\times3}$(2m−4)×43×4​+m×34×3​ $=$= $\frac{4\left(2m-4\right)}{12}+\frac{3m}{12}$4(2m−4)12​+3m12​

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

Do

 $\frac{4\left(2m-4\right)}{12}+\frac{3m}{12}$4(2m−4)12​+3m12​ $=$= $\frac{4\left(2m-4\right)+3m}{12}$4(2m−4)+3m12​ $=$= $\frac{8m-16+3m}{12}$8m−16+3m12​ $=$= $\frac{11m-16}{12}$11m−1612​

Reflect: See where we expanded the brackets?  It's important that when you are changing the denominators and the numerator has a binomial (expression with two parts), that you put it in brackets and expand it correctly.  This is a common mistake made by students.

Practice Questions

Question 1

Simplify $\frac{3x}{10}+\frac{5x+4}{10}$3x10+5x+410.

Question 2

Simplify $\frac{8x^2}{11}+\frac{5x^2-2x}{55}$8x211+5x22x55.

Question 3

Simplify the following:

$\frac{3x+5}{5}+\frac{5x+4}{3}$3x+55+5x+43