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Stage 5.1-3

2.06 Multiplying and dividing algebraic fractions

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Multiplication of 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.

Examples

Example 1

Simplify the following: \dfrac{b}{q}\times \dfrac{k}{u}

Worked Solution
Create a strategy

Multiply the numerators together and multiply the denominators together.

Apply the idea
\displaystyle \frac{b}{q}\times \frac{k}{u}\displaystyle =\displaystyle \frac{b\times k}{q\times u}Multiply the numerators and denominators
\displaystyle =\displaystyle \frac{bk}{qu}Evaluate

Example 2

Simplify \dfrac{3y}{8}\times \dfrac{4y}{9}.

Worked Solution
Create a strategy

Multiply the numerators together and multiply the denominators together.

Apply the idea
\displaystyle \frac{3y}{8}\times \frac{4y}{9}\displaystyle =\displaystyle \frac{3y \times 4y}{8\times 9}Multiply the numerators and denominators
\displaystyle =\displaystyle \frac{12y^2}{72}Evaluate
\displaystyle =\displaystyle \frac{12 \times y^2}{12 \times 6}Factorise 12 out
\displaystyle =\displaystyle \frac{y^2}{6}Simplify
Reflect and check

We could also have cancelled out any common factors before we performed the multiplication:

\displaystyle \frac{3y}{8}\times \frac{4y}{9}\displaystyle =\displaystyle \frac{3y}{4 \times 2}\times \frac{4y}{3 \times 3}Factorise the terms
\displaystyle =\displaystyle \frac{y}{ 2}\times \frac{y}{ 3}Cancel out the pairs of 3s and 4s
\displaystyle =\displaystyle \frac{y^2}{6}Simplify
Idea summary

The product of two fractions is the product of their numerators divided by the product of their denominators. \frac{A}{B}\times \frac{C}{D}= \frac{A\times C}{B\times D}

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.

Division of algebraic fractions

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

Examples

Example 3

Simplify \dfrac{9u}{36v}\div \dfrac{7v}{36u}.

Worked Solution
Create a strategy

Multiply the first fraction by the reciprocal of the second fraction.

Apply the idea
\displaystyle \frac{9u}{36v}\div \frac{7v}{36u}\displaystyle =\displaystyle \frac{9u}{36v}\times \frac{36u}{7v}Multiplying by the reciprocal
\displaystyle =\displaystyle \frac{9u}{4\times9 v}\times \frac{4\times9u}{7v}Factorise the terms
\displaystyle =\displaystyle \frac{u}{v}\times \frac{9u}{7v}Cancel out the pairs of 4s and 9s
\displaystyle =\displaystyle \frac{9u^2}{7v^2}Evaluate the products
Idea summary

To divide by a fraction, multiply by the reciprocal. \dfrac{A}{B} \div \dfrac{C}{D}=\dfrac{A}{B} \times \dfrac{D}{C}

Outcomes

MA5.2-6NA

simplifies algebraic fractions, and expands and factorises quadratic expressions

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