Multiply and divide algebraic fractions
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.