17.06 Multiplying and dividing algebraic fractions (Extended)
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.
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.