New Zealand
Level 6 - NCEA Level 1

# Multiply and divide algebraic fractions

Lesson

### 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 1

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 2

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 1

Simplify the expression:

$\frac{a}{7}\times\frac{a}{12}$a7×a12

##### Question 2

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 3

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 2

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 4

Simplify the expression:

$\frac{m}{8}\div\frac{3}{n}$m8÷​3n

##### Question 5

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

##### Question 6

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

### Outcomes

#### NA6-5

Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns

#### NA6-6

Generalise the properties of operations with rational numbers, including the properties of exponents

#### 91027

Apply algebraic procedures in solving problems