Fractions

Fractions are a fancier way of writing something divided by something else. Learning a few simple rules will help you when you need to add, subtract, multiply or divide with them. Once you master these tricks fractions turn out to be quite fun.

Requirements

• No requirement but having an understanding of what an integer is from the previous tutorial would be recommended.

Delving into fractions…

Fractions are just a different symbol for showing something divided by something else:

$$ 8 \div 60 = \dfrac{8}{60} $$

Another way to look at fractions is the top part (known as the numerator) shows the amount you have over the total amount (the bottom part known as the denominator). Integers show you something that is whole, a whole cake, 2 buses, 4 trees. But what if you want to show something that isn’t a whole number? Well you can’t really have 1 and then a bit of a tree… but what about if you had 6 pieces of a cake (equal in size) and someone was to eat one piece?

You’d be left with 5 pieces of a cake. We can represent how much of a cake you have left in maths by a fraction as we show the amount we have now compared to the total amount.

$$ 5 \div 6 = \dfrac{5}{6} $$

Another way to understand a fraction is the top part and the bottom part are a ratio (more on that in a later tutorial). In other words, if you were to multiply or divide the top and bottom bit by the same number you’d end up with the same result.

$$ \dfrac{1 \color{red}{\times 5}}{2\color{red}{\times 5}} = \dfrac{5}{10} $$

This is pretty cool because it lets us simplify fractions, so the top part and bottom part are smaller numbers (and easier to work with!). There are 2 ways to simplify fractions, dividing repeatedly and if the top bit and bottom bit gives a nice integer number trying it again until you can’t divide anymore or by finding the Highest Common Factor of both of the numbers.

$$ \dfrac{24 \color{red}{\div 6}}{36 \color{red}{\div 6}} = \dfrac{4 \color{red}{\div 2}}{6 \color{red}{\div 2}} = \dfrac{2}{3} $$

When simplifying fractions some people like to cross the numbers out and write the smaller numbers, however, for this we will use the operations as they show exactly what were doing.

It’s really important to remember this golden rule for simplifying fractions: you must divide or multiply the top or bottom number by the same amount to give a nice whole number (integer).

We can multiply 2 fractions together by multiplying the numbers on the top and the numbers on the bottom:

$$ \dfrac{5}{4} \times \dfrac{4}{5} = \dfrac{\color{red}{(5\times4)}}{\color{red}{(5\times4)}} = \dfrac{20 \color{red}{\div 20}}{20 \color{red}{\div 20}} = 1 $$

A cool trick we can do when multiplying fractions is cross cancelling (or cross simplifying). This is when we simplify fractions in a diagonalish way. This massively helps when dealing with large numbers in fractions.

$$ \dfrac{5 \color{red}{\div 5}}{4 \color{blue}{\div 4}} \times \dfrac{4 \color{blue}{\div 4}}{5 \color{red}{\div 5}} = \dfrac{1}{1} \times \dfrac{1}{1} = 1 $$

well that’s pretty cool but what do we do if say we have 3 fractions instead of 2 or even 4? We can still cross cancel diagonally nothing has changed (provided you have a diagonal pair your fine!). Let’s take this for example:

$$ \dfrac{4 \color{red}{\div 2}}{5} \times \dfrac{3 \color{blue}{\div 3}}{10 \color{red}{\div 2}} \times \dfrac{25}{30 \color{blue}{\div 3}} = \dfrac{2}{5} \times \dfrac{1}{5 \color{red}{\div 5}} \times \dfrac{25 \color{red}{\div 5}}{10} $$

$$ \dfrac{2 \color{red}{\div 2}}{5 \color{blue}{\div 5}} \times \dfrac{1}{1} \times \dfrac{5 \color{blue}{\div 5}}{10 \color{red}{\div 2}} = \dfrac{1}{1} \times \dfrac{1}{1} \times \dfrac{1}{5} = \dfrac{1}{5} $$

we can also divide fractions by turning the division operation into a multiplication operation BUT… flipping all of the fractions top bit and bottom bit after the first fraction. Here’s how we can divide 2 fractions:

$$ \dfrac{1}{5} \div \dfrac{5}{1} = \dfrac{1}{5} \color{red}{ \times \dfrac{1}{5}} = \dfrac{1}{25} $$

Here’s how we would divide 3 fractions (remember that the line means the same as divide):

$$ \dfrac{\tfrac{5}{1}}{\tfrac{5}{2}} \div \dfrac{1}{5} = \dfrac{5}{1} \div \dfrac{5}{2} \div \dfrac{1}{5} $$

$$ \dfrac{5 \color{red}{\div 5}}{1} \times \dfrac{2}{5 \color{red}{\div 5}} \times \dfrac{1}{5} = \dfrac{1}{1} \times \dfrac{2}{1} \times \dfrac{1}{5} = \dfrac{2}{5} $$

Another important thing to note is that we can write integers as fractions easily. So if we want to do an operation with a integer and fraction we just convert the integer to a fraction!

$$ 5 \times \dfrac{1}{5} = \dfrac{5 \color{red}{\div 5}}{1} \times \dfrac{1}{5 \color{red}{\div 5}} = 1 $$

To add fractions we need to make the denominators of all the numbers the same and then we can add the numerators together. This is known as a common denominator. We can get a common denominator by multiplying the top and bottom of each number by the denominator of the other number:

$$ \dfrac{1}{5} + \dfrac{2}{3} = \dfrac{1 \color{red}{\times 3}}{\color{blue}{5} \color{red}{\times 3}} + \dfrac{2 \color{blue}{\times 5}}{\color{red}{3} \color{blue}{\times 5}} = \dfrac{3}{15} + \dfrac{10}{15} = \dfrac{13}{15} $$

subtracting fractions is exactly the same instead we just subtract the numerators instead of adding them up. Another way to get our common denominator would be to find the lowest common multiple of the numbers:

$$ \dfrac{2}{7} - \dfrac{3}{4} - \dfrac{1}{2} = \dfrac{2 \color{red}{\times 4}}{7 \color{red}{\times 4}} - \dfrac{3 \color{blue}{\times 7}}{4 \color{blue}{\times 7}} - \dfrac{1 \color{green}{\times 14}}{2 \color{green}{\times 14}} $$ $$ \dfrac{8}{28} - \dfrac{21}{28} - \dfrac{14}{28} = \dfrac{8 - (21 + 14)}{28} = $$ The lowest common multiple of 7, 4 and 2 is 28 so we multiply the denominators to make that number.

Now what happens if we want to represent 1 and a bit of something or in other words a whole number with a fraction. Piece of cake…we either use vulgar fractions or mixed numbers (bring on the buzz words!)…

A vulgar fraction (also known as an improper fraction) is when the numerator (top number) is greater than the denominator (the bottom number) in a fraction:

$$ \dfrac{6}{5} = 6 \div 5 = 1.2 $$

Notice that if we change the number to a decimal with a calculator we get a number greater than 1.

Another way to write a fraction which gives a number greater than 1 is a mixed number where you have an integer bit and a leftover bit as a fraction. In order to convert between mixed numbers and improper fractions we need to multiply the integer bit by the denominator (bottom bit) and add it on to the top:

$$ 1 \dfrac{6}{5} = \dfrac{6 + \color{red}{(1 * 5)}}{5} = \dfrac{11}{5} $$

To convert an improper fraction into a mixed number we need to divide the numerator by the denominator and see how many whole amounts we get, this gives us the integer bit, we then leave the rest as a fraction:

$$ \dfrac{13}{4} = \color{red}{(12 \div 4)} \dfrac{1}{4} = 3 \dfrac{1}{4} $$

We can order fractions (like we ordered integers) by finding a common denominator and arranging the numbers by the size of their numerator. For example:

We can arrange these numbers in ascending order of size:

$$ 5, \dfrac{5}{2}, \dfrac{5}{3}, \dfrac{5}{4} $$ The lowest common multiple of the denominators here is 12. $$ \dfrac{5 \color{red}{\times 12}}{1 \color{red}{\times 12}}, \dfrac{5 \color{blue}{\times 6}}{2 \color{blue}{\times 6}}, \dfrac{5 \color{green}{\times 4}}{3 \color{green}{\times 4}}, \dfrac{5 \color{purple}{\times 3}}{4 \color{purple}{\times 3}} $$ $$ = \dfrac{60}{12}, \dfrac{30}{12}, \dfrac{20}{12}, \dfrac{15}{12} $$ So we now rearrange them into the correct order: $$ \dfrac{5}{4}, \dfrac{5}{3}, \dfrac{5}{2}, 5 $$

a percentage is just a number divided by 100. Percentages are often used to compare fractions or order them. For example the stock market (where people buy and fractions of ownership of companies) often uses percentages to compare how much a companies price has increased or decreased.

We can convert between fractions, percentages (fractions with 100 as the denominator) and decimals pretty easily. (Especially if you have a calculator! :)).

Let’s convert 50 divided by 60 to a decimal and then to a percentage.

$$ \dfrac{50 \color{red}{\div 10}}{60 \color{red}{\div 10}} = 5 \div 6 = \dfrac{0.83\dot{3} \color{red}{\times 100}}{1 \color{red}{\times 100}} = \dfrac{83.3\dot{3}}{100} = 83.3\dot{3} \% $$

Pretty easy stuff right? Just remember to use your calculator for converting fractions to decimals or a division method like bus stop.

Further reading

• Another list with links to other websites

• Try to include 2-3 or more, you can always link youtube videos.

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