# Fractions

## In a nutshell

Fractions are used to represent ratios or percentages. A fraction is displayed with a **numerator** (the top number) and **denominator** (the bottom number).

## Numerators and denominators

When using fractions, the numerator and the denominator should be whole numbers. The denominator of a fraction cannot equal zero.

## Equivalent fractions

Fractions that are equivalent are the same in value. They have different denominators and numerators, but when simplified they are the same.

To determine whether two fractions are equivalent, simply divide the top and bottom by the same value until they cannot be divided any further. Those that lead to the same result are equivalent.

##### Example 1

*Are *$\dfrac{9}{27}$* and *$\dfrac{3}{9}$* equivalent?*

*Simplify each fraction.*

$\dfrac{9}{27}=\dfrac{\cancel9^1}{\cancel{27}^3}=\dfrac{1}{3}$

$\dfrac{3}{9}=\dfrac{\cancel3^1}{\cancel9^3}=\dfrac{1}{3}$**

**

*Both fractions simplify to a third and so *__they are equivalent.__

## Finding fractions of quantities

Finding a fraction of a quantity involves multiplying the quantity by the fraction you need to find. There are also some simple fractions that are easy to calculate which can help you.

### Using simple fractions

You can use simple fractions to work out one part and then multiply by the numerator.

- To work out $\frac{1}{4}$, divide by four.

- To work out $\frac{1}{2}$, divide by two.

- To work out $\frac{1}{10}$, divide by ten.

- To work out $\frac{1}{100}$, divide by one hundred.

*Note:** To work out **one** out of any value, simply divide by the denominator!*

##### Example 2

*What is *$\dfrac{3}{10}$* of *$82$*?*

*Use simple fractions to work out one tenth.*

$82\div10=8.2$**

*Multiply by three to get three tenths.*

$8.2\times3=\underline{24.6}$

### Multiplying by the fraction

Multiplying a quantity by a fraction is another way of finding that fraction of the quantity. You may be asked to do this without a calculator.

##### Example 3

What is $\dfrac{7}{18}$ of $144$?

Multiply $\dfrac{7}{18}$ by $144$.

$\begin{aligned}\dfrac{7}{18}\times144&=\dfrac{7\times144}{18}\\ \ \\&=\dfrac{7\times\cancel{144}^8}{\cancel{18}^1}\\ \ \\&=7\times8\\ \ \\&=\underline{56}\end{aligned}$

## Converting between improper fractions and mixed numbers

Another useful skill is being able to convert between mixed numbers and improper fractions.

### Definitions

**Proper fraction **- A fraction where the denominator is greater than the numerator.

**Improper fraction **- A fraction where the numerator is greater than or equal to the denominator (also known as top-heavy).

**Mixed number **- A whole number and a proper fraction combined.

### Converting from an improper fraction to a mixed number

#### PROCEDURE

- Divide the numerator by the denominator.

- Write the whole number part as an integer.

- Write the remainder as a fraction: with the remainder as the numerator and the original denominator.

- Simplify the fraction (if required).

##### Example 4

*Convert *$\dfrac{92}{10}$* to a mixed number.*

*Divide *$92$* by *$10$*.*

$92\div10=9 \text{ remainder }2$

*Write $9$ as the integer and $2$ as the fraction part.*

$\dfrac{92}{10}=9\dfrac{2}{10}$**

*Simplify the fraction.*

$\dfrac{92}{10}=\underline{9\dfrac{1}{5}}$

### Converting a mixed number to an improper fraction

#### PROCEDURE

- Multiply the whole number by the denominator.

- Add the result to the numerator.

- Write the result as the numerator of the improper fraction over the original denominator.

##### Example 5

*Convert *$6\dfrac{2}{5}$ *to an improper fraction.*

*Multiply six by five.*

$6\times5=30$

*Add two and rewrite as a fraction.*

$30+2=32\\ \ \\6\dfrac{2}{5}=\underline{\dfrac{32}{5}}$

## Addition and subtraction

You can add and subtract fractions without using a calculator.

#### procedure

- Make the denominator of each fraction the same, by either finding the LCM of the denominators and multiplying, or just multiplying the two denominators together.

- Add or subtract the numerators of the fractions, keeping the denominator the same.

- Simplify the resulting fraction (if required).

##### Example 6

*Calculate $\dfrac{31}{24} - \dfrac{11}{16}$, writing your answer as a fraction in its simplest form.*

*Find the LCM.*

*The LCM of $24$ and $16$ is $48$.*

*Multiply to get the same denominators.*

*$\dfrac{31^{\times2}}{24^{\times2}}=\dfrac{62}{48}\\$*

*$\dfrac{11^{\times3}}{16^{\times3}}=\dfrac{33}{48}$*

**

*Take away the numerators.*

$\dfrac{62}{48} - \dfrac{33}{48} = \dfrac{62-33}{48} = \underline{\dfrac{29}{48}}$

## Multiplication and division

Multiplying and dividing fractions is often more simple than adding and subtracting as you don't need to change the denominator.

### Multiplication

To multiply two fractions, simply multiply the numerator by the numerator and the denominator by the denominator.

##### Example 7

*Calculate *$\dfrac{2}{16}\times\dfrac{13}{4}$*, writing your answer as a fraction in its simplest form.*

*Multiply the top and the bottom, simplifying as you go.*

$\dfrac{2}{16}\times\dfrac{13}{4}=\dfrac{\cancel2^1\times13}{16\times\cancel4^2}=\underline{\dfrac{13}{32}}$

### Division

#### Procedure

- Swap the numerator and denominator of the dividing fraction and change the sign from division to multiplication.

- Multiply the numerators of both fractions.

- Multiply the denominators of both fractions.

- Simplify the resulting fraction (if required).

##### Example 8

*Calculate $\dfrac{28}{46} \div \dfrac{7}{12}$*, *writing your answer as a fraction in its simplest form.*

*Flip the $\dfrac{7}{12}$ and change the sign.*

*$\dfrac{28}{46} \div \dfrac{7}{12} = \dfrac{28}{46} \times\dfrac{12}{7}$*

*Carry out the multiplication.*

$\dfrac{28}{46} \times \dfrac{12}{7} = \dfrac{\cancel{28}^4 \times\cancel{12}^6}{\cancel{46}^{23} \times \cancel7^1} = \underline{\dfrac{24}{23}}$