Babbage

Keep it Simple

The first rule of fractions is to make them as simple as possible. If there is a common factor in the numerator and the denominator then cancel it from both. For example $\dfrac{5}{30} = \dfrac{1 \times\ 5}{6\ \times\ 5} = \dfrac{1}{6}$

Multiplication of Fractions

Let's look at multiplication of fractions.

Imagine a circular pizza. If we cut it in half and eat one of the pieces we are left with a half. Numerically we can express this as $1 \times 1/2 = 1/2$

one times a half equals a half — Figure 1: $1 \times 1/2 = 1/2$

Now, if we cut the half into half and eat one of the pieces we are left with a quarter. Numerically we can express this as $1/2 \times 1/2 = 1/4$

a half times a half equals a quater — Figure 2: $1/2 \times 1/2 = 1/4$

So the rule for multiplying fractions is multiply the numerators then multiply the denominators. For example $\dfrac{3}{4} \times \dfrac{2}{5} = \dfrac{3\ \times\ 2}{4\ \times\ 5} = \dfrac{3}{10}$

Division of Fractions

Now let's think about dividing fractions. Multiplying a number by a half is the same as dividing the number by 2. For example $10 \times 1/2 = 5$ and $10 \div 2 = 5$.

note: We can always turn an integer into a fraction by putting it over $1$.
$2$ is an integer, $2/1$ is a fraction.

Addition and Subtraction of Fractions

Next we need to look addition and subtraction of fractions. The key thing is make the denominators the same. If you want to add $1/2$ to $1/3$ you need to find the lowest common multiple of the denominators $2$ and $3$. The lowest common multiple of $2$ and $3$ is $6$ so we multiply $1/2$ by $3/3$ and we multiply $1/3$ by $2/2$.

As an example let's add $\dfrac{1}{2}$ and $\dfrac{2}{3}$

$\dfrac{1}{2} + \dfrac{2}{3}$	=	$\dfrac{1}{2} \times \dfrac{3}{3} + \dfrac{2}{3} \times \dfrac{2}{2}$
	=	$\dfrac{3}{6} + \dfrac{4}{6}$
	=	$\dfrac{3+4}{6}$
	=	$\dfrac{7}{6}$

The process is the same for subtraction. For example $\dfrac{3}{5} - \dfrac{1}{3}$. The denominators $5$ and $3$ are both prime so the lowest common multiple is $15$.

$\dfrac{3}{5} - \dfrac{1}{3}$	=	$\dfrac{3}{5} \times \dfrac{3}{3} - \dfrac{1}{3} \times \dfrac{5}{5}$
	=	$\dfrac{9}{15} - \dfrac{5}{15}$
	=	$\dfrac{9-5}{15}$
	=	$\dfrac{4}{15}$

Mixed Fractions

A mixed fraction has a whole number part and a fractional part. For example, $13/5$ can be written as the mixed fraction $2+ \frac{3}{5}$. Remember to put a + between the integer and the fraction.

Improper Fractions