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iGCSE (2021 Edition)

17.06 Multiplying and dividing algebraic fractions (Extended)

Lesson

Multiplying algebraic fractions

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.

Worked example

Example 1

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

Think: To multiply fractions we can multiply the numerators together and also multiply the denominators together. We can then simplify if there is a common factor between the new numerator and denominator.

Do:

$\frac{3}{4}\times\frac{5}{7}$34×57 $=$= $\frac{3\times5}{4\times7}$3×54×7

Multiplying numerators and denominators

  $=$= $\frac{15}{28}$1528

Evaluating the products


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 2

Simplify $\frac{2y}{5}\times\frac{3}{m}$2y5×3m.

Think: Again, we want to multiply the numerators together and also multiply the denominators together.

Do:

$\frac{2y}{5}\times\frac{3}{m}$2y5×3m $=$= $\frac{2y\times3}{5\times m}$2y×35×m

Multiplying numerators and denominators

  $=$= $\frac{6y}{5m}$6y5m

Simplifying the algebraic product in the numerator

Again, since the numerator $6y$6y and the denominator $5m$5m don't have any common factors, $\frac{6y}{5m}$6y5m is the simplest form of our answer.

Practice questions

Question 1

Simplify the following:

$\frac{b}{q}\times\frac{k}{u}$bq×ku

Question 2

Simplify the following:

$\frac{3y}{8}\times\frac{4y}{9}$3y8×4y9

 

Dividing algebraic fractions

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

Example 3

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

Think: Dividing by a fraction is the same as multiplying by the fraction's reciprocal, so we want to change the division to a multiplication and 'flip' the fraction we are going to divide by.

Do:

$\frac{2}{3}\div\frac{3}{5}$23÷​35  $=$= $\frac{2}{3}\times\frac{5}{3}$23×53   

Flip the second fraction and change the division to multiplication

  $=$= $\frac{2\times5}{3\times3}$2×53×3 

Multiply numerators and denominators respectively

  $=$= $\frac{10}{9}$109

Evaluate the products


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 4

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

Think: Dividing by a fraction is the same as multiplying by the fraction's reciprocal, so we want to change the division to a multiplication and 'flip' the fraction we are going to divide by.

Do:

$\frac{m}{3}\div\frac{5}{x}$m3÷​5x  $=$= $\frac{m}{3}\times\frac{x}{5}$m3×x5  

Dividing by a fraction is the same as multiplying by its reciprocal

  $=$= $\frac{m\times x}{3\times5}$m×x3×5

Multiply numerators and denominators respectively

  $=$= $\frac{mx}{15}$mx15

Evaluate the products

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.

 

Remember!

Before multiplying two fractions together, look for common factors that you can cancel out first. This will make the resulting multiplication easier in most cases, as there will be less factors to deal with.

 

Practice questions

Question 3

Simplify the following:

$\frac{u}{3}\div\frac{4}{v}$u3÷​4v

Question 4

Simplify $\frac{9u}{36v}\div\frac{7v}{36u}$9u36v÷​7v36u.

Outcomes

0580E2.3

Manipulate algebraic fractions. Factorise and simplify rational expressions.

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