Functional Skills: Fractions

Skill 1: Equivalent Fractions

Equivalent fractions are fractions that have different numbers on the top and bottom, but are equal in size.

Example: Write \dfrac{3}{4} as an equivalent fraction with 12 on the bottom.

To go from one fraction to an equivalent fraction, simply multiply or divide the top AND bottom by the same number:

So, \dfrac{3}{4} and \dfrac{9}{12} are equivalent fractions.

Skill 2: Simplifying Fractions

To simplify a fraction, repeatedly divide the top of the fraction and the bottom of the fraction by the same amount, until they can no longer be divided to make whole numbers.

Note: A simplified fraction is just an equivalent fraction to the original one, so they are both equal in size.

Example: Write the fraction \dfrac{18}{30} in its simplest form.

\textcolor{red}{18} and \textcolor{red}{30} are both divisible by 6.

\dfrac{18}{30}= \dfrac{18\div6}{30\div6} = \textcolor{black}{\dfrac{3}{5}}

3 and 5 cannot both be divided, so the fraction is in its simplest form.

Skill 3: Writing Numbers as a Fraction of Another

Example: Write \textcolor{red}{40} as a fraction of \textcolor{blue}{50}. Give your answer in its simplest form.

Write the first number, \textcolor{red}{40}, on the top and the second number, \textcolor{blue}{50}, on  the bottom:

\dfrac{\textcolor{red}{40}}{\textcolor{blue}{50}}

Then, simplify the fraction fully:

\dfrac{\textcolor{red}{40} \div 10}{\textcolor{blue}{50} \div 10} = \dfrac{4}{5}

Skill 4: Fractions of Amounts

To calculate a fraction of an amount, divide the amount by the bottom of the fraction and then multiply it by the top, or the other way around.

Example: Calculate \dfrac{\textcolor{blue}{2}}{\textcolor{red}{5}} of \textcolor{limegreen}{£240}.

‘Of‘ means that we multiply \dfrac{2}{5} by £240

First divide \textcolor{limegreen}{£240} by \textcolor{red}{5} and then multiply it by \textcolor{blue}{2}:

 \begin{aligned} \dfrac{2}{5} \text{ of } £240 &= (\textcolor{limegreen}{£240} \div \textcolor{red}{5}) \times \textcolor{blue}{2} \\ &= £48 \times \textcolor{blue}{2} \\ & = £96 \end{aligned}

Skill 5: Adding and Subtracting Fractions

To add or subtract fractions, they need to have the same bottom number (they need to have a common denominator). If the fractions do not have the same bottom number, then you need to turn them into equivalent fractions with the same number on the bottom. Then you simply add or subtract the top numbers.

Example: Solve \dfrac{2}{5} + \dfrac{1}{4}

Turn them into equivalent fractions with the same number on the bottom:

\dfrac{2}{5}=\dfrac{2\times 4}{5\times 4}=\dfrac{8}{20}

\dfrac{1}{4}=\dfrac{1\times5}{4\times 5}=\dfrac{5}{20}

Now, add the tops of the fractions together:

\dfrac{2}{5} + \dfrac{1}{4}=\dfrac{8}{20} + \dfrac{5}{20}= {\dfrac{13}{20}}

Note: A subtraction is done in the same way, except you subtract the top numbers at the end.

Skill 6: Converting between Improper Fractions and Mixed Numbers

An improper fraction is a fraction where the top number is bigger than the bottom number, e.g. \dfrac{7}{4}

A mixed number is the mix of a whole number and a fraction, e.g. 3 \dfrac{2}{5}

Improper fractions and mixed numbers can be converted between.

 

Example 1: Write \dfrac{\textcolor{red}{25}}{\textcolor{limegreen}{4}} as a mixed number.

A fraction is just a division. So,

\dfrac{\textcolor{red}{25}}{\textcolor{limegreen}{4}} = \textcolor{red}{25} \div \textcolor{limegreen}{4}

You can use the bus stop method, to find that

\textcolor{red}{25} \div \textcolor{limegreen}{4} = \textcolor{blue}{6} \text{ r } \textcolor{orange}{1}

Then, the number on top of the “bus stop” (the whole number part) is the number before the fraction, and the remainder goes on top with the dividing number on the bottom:

\dfrac{\textcolor{red}{25}}{\textcolor{limegreen}{4}} = \textcolor{blue}{6} \dfrac{\textcolor{orange}{1}}{\textcolor{limegreen}{4}}

 

Example 2: Write \textcolor{red}{5} \dfrac{\textcolor{orange}{2}}{\textcolor{purple}{3}} as an improper fraction.

When converting from a mixed number to an improper fraction, we need to multiply the whole number by the bottom and add it to the top.

Whole number multiplied by the bottom:

\textcolor{red}{5} \times \textcolor{purple}{3} = 15

Add this to the top:

15 + \textcolor{orange}{2} = \textcolor{blue}{17}

Then, this number goes above the bottom number of the original fraction part:

\textcolor{red}{5} \dfrac{\textcolor{orange}{2}}{\textcolor{purple}{3}} = \dfrac{\textcolor{blue}{17}}{\textcolor{purple}{3}}

Skill 7: Adding and Subtracting Mixed Numbers

You can add and subtract mixed numbers using the skills you have already learned to add and subtract fractions and convert between mixed numbers and improper fractions.

Example: Calculate 2 \dfrac{3}{4} + 3\dfrac{1}{2}

Step 1: Split up the whole number parts and fraction parts separately:

2 \dfrac{3}{4} + 3\dfrac{1}{2} = \textcolor{limegreen}{2 + 3} + \textcolor{red}{\dfrac{3}{4} + \dfrac{1}{2}}

Step 2: Add the numbers, then add the fractions:

\textcolor{limegreen}{2 + 3} = \textcolor{limegreen}{5}      and      \textcolor{red}{\dfrac{3}{4} + \dfrac{1}{2}} = \dfrac{3}{4} + \dfrac{2}{4} = \dfrac{5}{4} = \textcolor{red}{1 \dfrac{1}{4}}

Step 3: Finally, add the parts together:

\textcolor{limegreen}{5} + \textcolor{red}{1 \dfrac{1}{4}} = 6 \dfrac{1}{4}

 

Note: A subtraction is done in the same way, except you need to subtract the parts instead of adding. You still add the parts back together at the end though!

Skill 8: Comparing Fractions

All types of fractions can be compared, whether they have different numbers on the bottom, or they are improper fractions or mixed numbers. We can determine which ones are bigger and put them in order of size by converting between mixed numbers and improper fractions and using equivalent fractions.

Example: What is bigger, \dfrac{7}{5} or 1 \dfrac{1}{3}?

Step 1: The fractions need to be in the same form, so convert 1 \dfrac{1}{3} into an improper fraction, as these are easier to compare than mixed numbers:

1 \dfrac{1}{3} = \dfrac{4}{3}

Step 2: The bottom numbers are different, so you need to find equivalent fractions:

3 \times 5 = \textcolor{limegreen}{15} so multiply the top and bottom of each fraction to get \textcolor{limegreen}{15} on the bottom.

\dfrac{7}{5} = \dfrac{7 \times 3}{5 \times 3} = \dfrac{\textcolor{red}{21}}{\textcolor{limegreen}{15}}      and     \dfrac{4}{3} = \dfrac{4 \times 5}{3 \times 5} = \dfrac{\textcolor{blue}{20}}{\textcolor{limegreen}{15}}

Step 3: Finally, compare the size of the top of the fractions:

\textcolor{red}{21} is bigger than \textcolor{blue}{20}, so \dfrac{7}{5} is bigger than 1 \dfrac{1}{3}