Algebra
UK Secondary (7-11)
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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.

 

 

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