4 Fractions
Fractions can be written in two ways:
- as decimals fractions, for example \(0.5\), \(0.25\) and \(0.\dot{3}\).
- as vulgar fractions, the following fractions have the same values as the examples above, \(\frac{1}{2}\), \(\frac{1}{4}\) and \(\frac{1}{3}\). Vulgar fractions consist of two parts. The top, or numerator, and the bottom, the denominator.
Vulgar fractions are useful in algebra. The next section looks at some techniques for dealing with them.
\[\require{cancel}\]
4.1 Simplifying
Fractions can be cancelled down or simplified by dividing the numerator and denominator by a common factor i.e. we look for a number that goes into both the top and the bottom of the fraction. For example:
\[ \begin{aligned} \frac{18}{24} &= \frac{3 \times 6}{4 \times 6} \\ &= \frac{3 \times \cancel{6}}{4 \times \cancel{6}} \\ &= \frac{3}{4} \end{aligned} \]
The same can be done with algebraic fractions.
\[ \begin{aligned} \frac{4xy}{6x} &= \frac{2y \times 2x}{3 \times 2x} \\ &= \frac{2y \times \cancel{2x}}{3 \times \cancel{2x}} \\ &= \frac{2y}{3} \end{aligned} \]
Sometimes you’ll need to factorise expressions in the fraction in order to cancel it down.
\[ \begin{aligned} \frac{10x^2 + 5x}{4x+2} &= \frac{5x \times 2x + 5x \times 1}{2 \times 2x + 2 \times 1} \\ &= \frac{5x(2x+1)}{2(2x+1)} \\ &= \frac{5x\cancel{(2x+1)}}{2\cancel{(2x+1)}} \\ &= \frac{5x}{2} \end{aligned} \]
Here are some practice questions.
It is tempting to want to make cancellations like this:
\[ \begin{aligned} \frac{2x^2}{3x+7} &= \frac{2x\cancel{x}}{3\cancel{x}+1} \\ &= \frac{2x}{3+7} \\ &= \frac{2x}{10} \\ &= \frac{x}{5} \end{aligned} \]
However, please don’t do it, as it’s just plain wrong! Lets let \(x=3\) and substitute it into the original \(\frac{2x}{3x+7}\) and into incorrectly simplified version \(\frac{x}{5}\). If the algebra is correct it should give the same answer.
We claim:
\[ \frac{2x^2}{3x+7} = \frac{x}{5} \]
but if we substitute \(x=2\) into both sides we get:
\[ \begin{aligned} \frac{2(3)^2}{3(3)+7} &= \frac{(3)}{5} \\ \frac{2 \times 9}{9+7} &= \frac{3}{5} \\ \frac{18}{16} &= \frac{3}{5} \\ \frac{9}{8} &= \frac{3}{5} \end{aligned} \]
Which is nonsense!
4.2 Multiplication and division
Multiplication and division of fractions is, thankfully, really easy!
4.2.1 Multiplication
For multiplication you simply multiply the different fractions numerators and denominators together. In other words the top of the first fraction with the top of the second one and so on. After the multiplication you may be able to cancel down the fraction. Just like this:
\[ \begin{aligned} \frac{2}{5} \times \frac{3}{4} &= \frac{2 \times 3}{5 \times 4} \\ &= \frac{6}{20} \\ &= \frac{3 \times 2}{10 \times 2} \\ &= \frac{3 \times \cancel{2}}{10 \times \cancel{2}} \\ &= \frac{3}{10} \end{aligned} \]
It is possible to cancel before multiplying. Here is the same example revisited:
\[ \begin{aligned} \frac{2}{5} \times \frac{3}{4} &= \frac{2 \times 3}{5 \times 4} \\ &= \frac{2 \times 3}{5 \times 2 \times 2} \\ &= \frac{\cancel{2} \times 3}{5 \times 2 \times \cancel{2}} \\ &= \frac{3}{10} \end{aligned} \]
This can be super useful when dealing with large numbers or complex algebraic fractions.
4.2.2 Division
We can change a division into a multiplication by remembering keep, change, flip. We keep the first fraction as it is. Change the division, \(\div\), symbol to a multiplication, \(\times\), and flip the last fraction - swap the places of the numerator and denominator. This is called taking the reciprocal of the fraction. For example:
\[ \begin{aligned} \frac{3}{7} \div \frac{5}{2} &= \frac{3}{7} \times \frac{2}{5} \\ &= \frac{3 \times 2}{7 \times 5} \\ &= \frac{6}{35} \end{aligned} \]
4.3 Addition and subtraction
Addition and subtraction is easy if the denominators are the same. We just add the numerators together and the denominators stays the same. For example:
\[ \begin{aligned} \frac{2}{5} + \frac{1}{5} &= \frac{2+1}{5} \\ &= \frac{3}{5} \end{aligned} \]
If the denominators are different we must make equivalent fractions with a common denominators first. Finding a common denominator is like simplification, or cancelling down, but in reverse.
If we want to add \(\frac{2}{3}\) and \(\frac{2}{9}\) for example, we want to rewrite the first fraction so that it has \(9\) as the denominator. To do this, we multiply the top and bottom of the fraction by \(3\) (Remember to multiply both the numerator and denominator by \(3\) to make sure the fractions are equivalent!) :
\[ \begin{aligned} \frac{2}{3} + \frac{2}{9} &= \frac{2 \times 3}{3 \times 3} + \frac{2}{9} \\ &= \frac{6}{9} + \frac{2}{9} \\ &= \frac{6 + 2}{9} \\ &= \frac{8}{9} \end{aligned} \]