# Decomposing, adding and subtracting fractions

Decomposing a fraction means to break him into smaller pieces. For example, if you have one whole that is divided into five parts you can also get that by adding five fifths. This means that every fraction can be separated by his numerator, you can do this, but you have to be careful for your denominator must be the same as the given fraction, and the sum of the numerators of new fractions must be equal to the numerator of given fraction. The decomposition of a fraction is not unique. You can decompose a fraction in any way you like.

$\displaystyle{\frac{5}{6} = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6} + \frac{1}{6}}$

$\displaystyle{\frac{5}{6} = \frac{2}{6} + \frac{2}{6} + \frac{1}{6}}$

$\displaystyle{\frac{5}{6}= \frac{4}{6} + \frac{1}{6}}$ and so on…

The decomposition of fraction leads us to very important operations with fractions – addition and subtraction.

These operations expect from you to have high skills in finding lowest common multiple.

To quickly solve these tasks requires a lot of work and practice so take it slow and don’t be discouraged if at first you don’t succeed.

Example 1. Add fractions $\displaystyle{\frac{1}{2} + \frac{1}{3}}$.

1. Step is finding the denominator.

First you find their least common multiple which is number $6$. That number will be the denominator of our sum.

$\displaystyle{\frac{1}{2} + \frac{1}{3} = \frac{x}{6}}$

2. Now you divide your new denominator with the first denominator and multiply with the first numerator.

$6 : 2 = 3$

$\displaystyle{\frac{1}{2} + \frac{1}{3} = \frac{(3 \cdot 1 + ?)}{6}}$
3. You do the same with the second fraction. Divide the new denominator with the second denominator, multiply that with second numerator and add to everything we did so far.

$\displaystyle{\frac{6}{3} = 2}$

$\displaystyle{\frac{1}{2} + \frac{(1)}{3} = \frac{(3 \cdot 1 + 1)}{6}}$

$\displaystyle{\frac{1}{2} + \frac{1}{3} = \frac{5}{6}}$

$\displaystyle{\frac{5}{6} = \frac{3}{6} + \frac{2}{6} = \frac{1}{2} + \frac{1}{3}}$.

Example 2. Add fracions $\displaystyle{\frac{2}{5} + \frac{4}{15}}$.

Again the first thing you do is finding the denominator. The least common multiple of $5$ and $15$ is $15$.

This means that the denominator of our sum will be number $15$.

Second, the first addend in a numerator of the sum will be $6$, because we divide $15$ with the first denominator, number $5$ which is number $3$, and then multiply it with its numerator which is number $2$.

The second addend in a numerator of the sum will be $4$, because number $15$ divided by $15$ is number $1$, and times $4$ is $4$.

This leads us to our solution:

$\displaystyle{\frac{2}{5} + \frac{4}{15} = \frac{10}{15} = \frac{2}{3}}$.

Try to make a habit in shortening the fractions. This will really come in handy when you’re up against much more complicated expressions.

What if you get more addends? The procedure is the same, only now you have three fractions, and do the procedure three times, and of course, the denominator of that sum will be the least common multiple of all three.

Example 3. Add fractions $\displaystyle{\frac{1}{5} + \frac{7}{30} + \frac{2}{3}}$

The least common multiple of $5$, $30$ and $3$ is number $30$.

$\displaystyle{\frac{30}{5} = 6}$, $\displaystyle{\frac{30}{30} = 1}$, $\displaystyle{\frac{30}{3} = 10}$

$\displaystyle{\frac{1}{5} + \frac{7}{30} + \frac{2}{3} = \frac{(1 \cdot 6 + 7 \cdot 1 + 2 \cdot 10)}{30} = \frac{33}{30} = \frac{11}{10}}$

If you have to add a whole number and a fraction, the procedure is the same, a whole number $a$ is a fraction  with the denominator $1$, $\displaystyle{\frac{a}{1}}$.

What about subtracting? The procedure is the same.

Example 4. Subtract fractions $\displaystyle{\frac{5}{8} – \frac{1}{2}}$.

Least common multiple of $8$ and $2$ is number $8$ and thats is the denominator of the difference.

$\displaystyle{\frac{8}{8} = 1; 1 \cdot 5 = 5, \frac{8}{2}= 4; 4 \cdot 1 = 4}$

$\displaystyle{\frac{5}{8} – \frac{1}{2} = \frac{(5-4)}{8} = \frac{1}{8}}$

Example 5. Subtract fractions $\displaystyle{\frac{1}{2} – \frac{5}{8}}$.

$\displaystyle{\frac{1}{2} – \frac{5}{8} = \frac{(4-5)}{8} = – \frac{1}{8}}$

Reminder: if you are unsure of which number is the least common multiple of your denominators, you can put any multiple, and shorten the fraction afterwards:

$\displaystyle{\frac{1}{2} – \frac{5}{8} = \frac{(8-10)}{16} = \frac{(8-10)}{16} = – \frac{2}{16}}$.