# Fractions

## In a nutshell

Fractions are a very intuitive way of understanding parts of a whole. It is important to know the different types of fractions as well as when two fractions are equivalent.

## Definitions

**Numerator **- The numerator of a fraction is the top number.

**Denominator **- The denominator of a fraction is the bottom number.

**Improper fractions **- An improper fraction is a fraction whose numerator is bigger than its denominator. An example of an improper fraction is $\dfrac{12}{15}$.

**Mixed numbers **- A mixed number is a number that is made up of both an integer and a fraction. An example of a mixed number is $2\dfrac{2}{5}$.

## Equivalent fractions

Multiplying or dividing both the top and bottom (or *numerator *and *denominator*) of a fraction by the same number doesn't change its value. These types of fractions are called **equivalent fractions**.

##### Examples

- $\frac{4}{5}=\frac{20}{25}$
*because $\frac{4}{5}\begin{array}{c}\overset{\times5}{\rightarrow}\\\underset{\times5}{\rightarrow}\end{array} \frac{20}{25}$*

- $\frac{30}{50}=\frac{3}{5}$
*because $\frac{30}{50}\begin{array}{c}\overset{\div10}{\rightarrow}\\\underset{\div10}{\rightarrow}\end{array} \frac{3}{5}$*

### Simplifying fractions

Simplifying a fraction means to divide the top and bottom of a fraction by the same number. A fraction in its **simplest form** is a fraction that cannot be simplified further.

##### Example 1

*Simplify the fraction $\frac{12}{18}$*.

*First, think of a number that divides both $12$ and $18$*. *Both numbers are even, so $2$ goes into both of them. Divide both the numbers by $2$:*

$12\div2=6$

$18\div2=9$

*Therefore, $\frac{12}{18}\begin{array}{c}\overset{\div2}{\rightarrow}\\\underset{\div2}{\rightarrow}\end{array} \frac{6}{9}$.*

*Can $\frac{6}{9}$ be simplified? Yes it can, because both $6$* *and $9$* *are divisible by $3$. So, divide both numbers by $3$:*

$6\div3=2$

$9\div3=3$

*Therefore, $\frac{6}{9}\begin{array}{c}\overset{\div3}{\rightarrow}\\\underset{\div3}{\rightarrow}\end{array} \frac{2}{3}$*

*Can $\frac{2}{3}$ be simplified further? No. There is no number that divides both $2$ and $3$.*

*$\underline{\frac{12}{18}=\frac{2}{3}}$*

## Converting between improper fractions and mixed numbers

### Improper fraction to a mixed number

- Divide the numerator by the denominator. The answer is the integer part.

- The remainder is the numerator of the fraction.

- The denominator of the improper fraction is the same as the denominator of the mixed number.

### Mixed number to an improper fraction

- Multiply the integer part by the denominator and add the numerator. This is the numerator of the improper fraction.

- The denominator of the improper fraction is the denominator of the mixed number.

##### Example 2

*1. Convert* $\dfrac{17}{3}$ *to a mixed number.*

*2. Convert *$4\dfrac{3}{7}$ *to an improper fraction.*

*Part 1:*

*Divide the numerator by the denominator:*

$17\div3=5$ *remainder* $2$.

*Therefore, the integer part is* $5$. *The numerator is* $2$. *The denominator stays the same, so it is* $3$. *Hence:*

$\underline{\frac{17}{3}=5\frac{2}{3}}$

*Part 2:*

*Multiply the integer by the denominator and add the numerator:*

$4\times7+3=28+3=31$

*Therefore, the numerator is* $31$. *The denominator stays the same, so it is* $7$.* Hence:*

$\underline{4\frac{3}{7}=\frac{31}{7}}$

## Ordering fractions

To compare two fractions, first make sure they both have the same denominator. This can be done by multiplying the top and bottom of one fraction by the denominator of the other fraction. Then, compare the two numerators.

##### Example 3

*Which is bigger:* $\frac{4}{9}$ *or* $\frac{9}{20}$*?*

*Multiply the top and bottom of* $\frac{4}{9}$ *by the denominator of the other fraction - which is* $20$:

$\frac{4}{9}=\frac{4\times20}{9\times20}=\frac{80}{180}$

*Multiply the top and bottom of* $\frac{9}{20}$ *by the denominator of the other fraction - which is* $9$:

$\frac{9}{20}=\frac{9\times9}{20\times9}=\frac{81}{180}$

*Compare the two new fractions by comparing their numerators:*

$81>80$, *so*

$\frac{81}{180}>\frac{80}{180}$

*Answer the question using the original fractions:*

$\underline{\frac{9}{20}}$ *is bigger than* $\underline{\frac{4}{9}}$