Fractions

Addition

In order to add fractions you must find a common denominator: $\frac{\text{numerator}}{\text{denominator}}$ which means the bottom part of the fraction must be the same for both numbers. We can do this by finding the LCM of both numbers.

For example if we want to find the answer to: $\frac{10}{205} + \frac{5}{20}$

we first work out the lowest common multiple which is 820.

We then multiply the numerator and denominator by the same amount to get to 820 as the numerator. For example:

$\begin{aligned} 820 \div 205 &= \color{red}4, 820 \div 20 = \color{red}{41} \\[5mu] \dfrac{10 * \color{red}{4}}{205 * \color{red}{4}} + \dfrac{5 * \color{red}{41}}{20 * \color{red}{41}} &= \dfrac{40}{820} + \dfrac{205}{820} \end{aligned}$

Now that the we have the same denominator we can add the numerators together like a regular adding:

$\begin{aligned} \frac{40}{820} + \frac{205}{820} = \frac{245}{820} \end{aligned}$

However, this is not the simplest answer because we you can also simplify fractions

Simplifying

Provided you do the same thing to both top (numerator) and bottom (denominator) of the fraction you can make larger number fractions into simpler ones. For example:

$\begin{aligned} \dfrac{5 \div \color{blue}{5}}{20 \div \color{blue}{5}} &= \dfrac{1}{4} \\[10mu] \dfrac{10 \div \color{blue}{5}}{205 \div \color{blue}{5}} &= \dfrac{2}{41} \end{aligned}$

So for the above fractions we could instead find the LCM of 4 and 41 which is 164 so instead we would end up with:

$\begin{aligned} \dfrac{1 * \color{blue}{41}}{\color{red}{4} * \color{blue}{41}} + \dfrac{2 * \color{blue}{4}}{\color{red}{41} * \color{blue}{4}} = \dfrac{49}{164} \end{aligned}$

It's up to you whether you want to simplify both fractions first and then find a LCM or simplify the final fraction like this:

$\begin{aligned} \dfrac{245 \div \color{green}{5}}{820 \div \color{green}{5}} = \dfrac{49}{164} \end{aligned}$

Improper

Improper fractions are fractions with a numerator larger than the denominator. We need to know how to convert these into mixed fractions which have a whole and a fraction. For example: $\frac{20}{6}$ as an improper fraction. This can be easily converted to a mixed fraction by dividing 20 by 6 and leaving the remainder as a fraction:

$\dfrac{20}{6} = 20 \div 6 = 3 \frac{2 \div 2}{6 \div 2} = 3 \frac{1}{3}$

To convert a mixed fraction to an improper fraction all you need to do is multiply the number on the side by the denominator and at that to the numerator. For example converting $4 \frac{2}{10}$ to an improper fraction:

$\begin{aligned} 4 \dfrac{2 \div 2}{10 \div 2} = 4 \dfrac{1}{5} = \dfrac{4 * 5 + 1}{5} = \dfrac{21}{5} \end{aligned}$

Another example would be converting $5 \frac{2}{3}$ to an improper fraction:

$\begin{aligned} 5 \frac{2}{3} = \frac{5 * 3 + 2}{3} = \frac{17}{3} \end{aligned}$

Multiplying

Multiplying is pretty easy. You multiply the bottom numbers together and the top numbers together. For example:

$\begin{aligned} \frac{20}{5} * \frac{1}{5} = \frac{20 * 1}{5 * 5} = \frac{20}{25} \end{aligned}$

And then you can simplify the result. However, a faster way is to cross cancel or simplify like this:

$\begin{aligned} \frac{20 \div \color{blue}{5}}{5} * \frac{1}{5 \div \color{blue}{5}} = \frac{4 * 1}{5 * 1} = \frac{4}{5} \end{aligned}$

This is the same as simplifying but instead you are doing it diagonally with the numerator of 1 and the denominator of another fraction. The same process can be applied to harder problems which multiply more than 2 fractions such as:

$\begin{aligned} \frac{5}{4} * \frac{2}{5} * 1 \frac{3}{4} &= \frac{5 \div \color{green}{5}}{4 \div \color{blue}2} * \frac{2 \div \color{blue}2}{5\div \color{green}{5}} * \frac{4 * 1 + 3}{4} \\[10mu] &= \frac{1}{2} * \frac{1}{1} * \frac{7}{4} = \frac{7}{8} \end{aligned}$

Note that you can only cross simplify in pairs of 2. Even though 4 divides by 2 to give 2 we can't do that. If you get a question requiring you to multiply more than 2 fractions it may be easier to multiply 2 first to get an answer then multiply it by the final fraction.

Dividing

All you need to do to divide a fraction by a fraction is turn the second fraction upside down (it must be improper not mixed though). For example:

$\begin{aligned} \dfrac{6}{20} \div 1 \dfrac{5}{4} &= \dfrac{6}{20} \div \dfrac{1 * 4 + 5}{4} \\[10mu] &= \dfrac{6 \div \color{purple} 3}{20 \div \color{orange} 4} * \dfrac{4 \div \color{orange} 4}{9 \div \color{purple} 3} = \dfrac{2}{5} * \dfrac{1}{3} = \dfrac{2}{15} \end{aligned}$

Because $5 \div 3$ is the same as $\frac{5}{3}$ you may get a hard question that looks impossible. However, provided you remember that they mean the same thing it becomes just as easy as the above one:

$\begin{aligned} \dfrac{6}{\dfrac{2}{3}} = 6 \div \dfrac{2}{3} = \frac{6}{1} * \frac{3}{2} = \frac{18 \div \color{black} 2}{2 \div \color{black} 2} = 9 \end{aligned}$

Note: it is also worthwhile remembering that an integer like 6 equals $\frac{6}{1}$ as a fraction.

"Fractions of something"

"of" means $*$. Provided you remember this rule then you can solve these problems easily. For example, "What is $\frac{9}{20}$ of $260?":

$\dfrac{9}{20 \div \color{red}{20}} * \dfrac{260 \div \color{red}{20}}{1} = 9 * 13 = 117$

Expressing a number as a fraction of another number

All you need to do is write the first number over the second number. And... then you simplify.

For example "Express 90 as a fraction of 180":

$\dfrac{90 \div \color{red}{30}}{180 \div \color{red}{30}} = \dfrac{3 \div \color{red}{3}}{6 \div \color{red}{3}} = \dfrac{1}{2}$

Note: How could you have made this quicker?

Percentages and Decimals

A percentage ($\%$) just means a number out of 100 (or a fraction where the denominator is 100). For example:
$35\% = \dfrac{35}{100}$ as a fraction.

A decimal is just an ordinary number which has a remainder (meaning that it doesn't divide exactly).

They all represent a certain amount (or quantity) of something and you can convert between them very easily.

Converting Fractions, Decimals and Percentages

Decimal to Fraction

Decimals which don't go on forever (terminating)

For numbers with a single decimal you could convert them to a fraction out of 10 and then simplify:
$0.8 = \dfrac{8 \div \color{red}{2}}{10 \div \color{red}{2}} = \dfrac{4}{5}$

For numbers with more digits after the decimal place you could add an extra power of 10. For example for $0.788$ you could do:

$0.788 = \dfrac{788 \div \color{red}{2}}{1000 \div \color{red}{2}} = \dfrac{394 \div \color{red}{2}}{500 \div \color{red}{2}} = \dfrac{197}{250}$

Decimals which repeat forever (non-terminating)

It is recommended to use some very simple algebra:

1. let the decimal equal $x$
2. multiply your decimal by a power of 10 until one full repetition is past the decimal place.
3. subtract to get rid of decimal part
4. divide both sides to get $x$ to 1.
5. Boom!! Answer!

For example "Write 0.17777... as a fraction":

$\begin{aligned} x &= 0.177... \\ 100 * x &= 17.777... \\ 10 * x &= 1.7777... \\ 100x - 10x &= 90x \div \color{red}{90} = 16 \div \color{red}{90} \\ x &= \dfrac{16}{90} \end{aligned}$

If you don't understand this you might want to visit the basics of algebra.

Fractions into Recurring decimals

The simplest way to do this is to try and make the denominator all nines. Then the numerator equals the recurring decimal.

$\begin{aligned} \dfrac{8 * 3}{33 * 3} = \dfrac{24}{99} = 0.242424... \end{aligned}$

Or just use your calculator :) (like what most people do!)