 Mixed Operations with Algebraic Fractions

Lesson

We looked at how to add and subtract fractions in More and Less Parts.

The most important things to remember when adding and subtracting fractions (of any kind) are

• we need like denominators (bases)
• 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

$\frac{4m}{5}-\frac{2}{5}$4m525

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-2}{5}$4m25

Reflect: Because we cannot simplify the numerator $4m-2$4m2 any further, this means this is a simplified as we can get.

Example 2

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

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

Think: We cannot add or subtract fractions unless the denominators are common.  In this case we one denominator of $3$3, and the other with $6$6.  We need to find a common denominator.  Let's choose $6$6, as it is a common multiple of both $3$3 and $6$6.

Do:  Change 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​

So this means our expression now becomes:

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

Success, now we can add the fractions as the denominators are common.

Do: Now 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, yes, this is simplified as much as we can.

Example 3

Our final example is where we have different denominators, and some simplification to perform at the final step.

$\frac{3x}{4}+\frac{3x}{2}$3x4+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}{4}+\frac{3x}{2}$3x4​+3x2​ $=$= $\frac{3x}{4}+\frac{3x\times2}{2\times2}$3x4​+3x×22×2​ $=$= $\frac{3x}{4}+\frac{6x}{4}$3x4​+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}{4}+\frac{6x}{4}$3x4​+6x4​ $=$= $\frac{3x+6x}{4}$3x+6x4​ $=$= $\frac{9x}{4}$9x4​

Reflect: We collected the like terms of $3x$3x and $6x$6x.  Are there any other common terms?  Are they any common factors with the $9x$9x and $4$4?  No, so this is a simplified as this answer gets.

Practice Questions

Question 1

Simplify the following:

$\frac{6x}{2}-\frac{7x}{2}$6x27x2

Question 2

Simplify the following: $\frac{3x}{5}-\frac{x}{7}$3x5x7

Question 3

Simplify the expression $\frac{11x}{14}+\frac{7x}{21}$11x14+7x21.

Multiplying and Dividing

Multiplying

When it comes to working with algebraic fractions and applying the four operations, the process is exactly the same as when we worked with numeric fractions.

Let's have a look at a simple example of multiplying two numerical fractions.

Example 4

Simplify $\frac{3}{4}\times\frac{5}{7}$34×57

 $\frac{3}{4}\times\frac{2}{5}$34​×25​ $=$= $\frac{3\times5}{4\times7}$3×54×7​     Multiplying numerators and denominators $=$= $\frac{15}{28}$1528​                 Simplifying the numerator

Since $\frac{15}{28}$1528 doesn't have any common factors between the numerator and denominator, that is the most simplified form of our answer.

Now let's apply the same process to multiplying algebraic fractions.

Example 5

Simplify $\frac{y}{5}\times\frac{3}{m}$y5×3m

 $\frac{y}{5}\times\frac{3}{m}$y5​×3m​ $=$= $\frac{y\times3}{5m}$y×35m​    Multiplying numerator and denominators $=$= $\frac{3y}{5m}$3y5m​     Simplifying the numerator

Again, since the numerator $3y$3y and the denominator $5m$5m don't have any common factors, $\frac{3y}{5m}$3y5m is the simplest form of our answer.

Practice Questions

Question 4

Simplify the expression:

$\frac{a}{7}\times\frac{a}{12}$a7×a12

Question 5

Simplify the expression:

$\frac{8u}{3v}\times\frac{2v}{7u}$8u3v×2v7u

Dividing

Again, the process for dividing is the same as when we divided numeric fractions.

Example 6

Simplify $\frac{2}{3}\div\frac{3}{5}$23÷​35

 $\frac{2}{3}\div\frac{3}{5}$23​÷​35​ $=$= $\frac{2}{3}\times\frac{5}{3}$23​×53​ Dividing by a fraction is the same as multiplying by its reciprocal.  So invert and multiply. $=$= $\frac{2\times5}{3\times3}$2×53×3​ Multiply numerators and denominators respectively. $=$= $\frac{10}{9}$109​

Since $\frac{10}{9}$109 doesn't have any common factors between the numerator and denominator, that is the most simplified form of our answer.

Now let's apply the same process to dividing algebraic fractions.

Example 7

Simplify $\frac{m}{3}\div\frac{5}{x}$m3÷​5x

 $\frac{m}{3}\div\frac{5}{x}$m3​÷​5x​ $=$= $\frac{m}{3}\times\frac{x}{5}$m3​×x5​ Dividing by a fraction is the same as multiplying by its reciprocal.  So invert and multiply. $=$= $\frac{m\times x}{3\times5}$m×x3×5​ Multiply numerators and denominators respectively. "=" $\frac{mx}{15}$mx15​

Again, since the numerator $mx$mx and the denominator $15$15 don't have any common factors, $\frac{mx}{15}$mx15 is the simplest form of our answer.

Practice Questions

Question 6

Simplify the expression:

$\frac{m}{8}\div\frac{3}{n}$m8÷​3n

Question 7

Simplify the following: $\frac{-2x}{11}\div\frac{7y}{5}$2x11÷​7y5

Question 8

Simplify $\frac{-2x}{11}\div\frac{2x}{3}$2x11÷​2x3.