The purpose of this note is to teach you how to calculate the fractions of numbers. We first talk about what a fraction is and then discuss the addition, multiplication, and division of fractions. This part of mathematics is very important for the other chapters of mathematics. So definitely, you have to master the calculation of fractions.

## What is a fraction of two numbers?

Let $a$ and $b$ be two relative numbers, $a$ and $b$ in $\mathbb{Z}$, such that $b$ is non null,$b\neq 0$.  A fraction of $a$ et $b$ is a number of the form $\frac{a}{b},$ we say $a$ over $b$. The number $a$ is called the numerator, while the number $b$ is called the denominator of the fraction. Then the form of a fraction, numerator/denominator.

Examples of fractions: $\frac{1}{2}$, one half,  $\frac{1}{4}$, one quarter,  $\frac{1}{4},$ three quarter, $\frac{1}{3},$ one third.

### Multiplication of fractions

The multiplication of numbers with a fraction is easy, just multiply this number with the nominator. Here an example $3\times \frac{2}{5}=\frac{3\times 2}{5}=\frac{6}{5}$. The general rule is \begin{align*} c\times \frac{a}{b}=\frac{c\times a}{b}.\end{align*}The multiplication of two fractions is so simple, what you have to do is just multiply the nominators and denominators of these fractions. An an example: $\frac{3}{7}\times \frac{2}{5}=\frac{3\times 2}{7\times 5}=\frac{6}{35}$. Here comes the general rule\begin{align*}\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}.\end{align*}

I. The addition of fractions having the same denominators is very simple because we keep the same denominator and we only add numerators. Here is the rule: \begin{align*} \frac{a}{b}+\frac{c}{d}=\frac{a+c}{b}.\end{align*}Example:$\frac{3}{6}+\frac{1}{6}=\frac{3+1}{6}=\frac{4}{6}.$

II.The calculation of fractions with different denominators can be calculated using the first technique. In fact, what you need to do is modify each fraction so that the new fractions have the same denominators and then use the method explained in part I. Let’s give details of how to do this. Suppose we want to add the fractions $\frac{a}{b}$ and $\frac{c}{d}$. The technique consists in multiplying the numerator and the denominator of $\frac{a}{b}$ by $d$ and those of $\frac{c}{d}$ by $b$. We obtain $$\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd}.$$ Finally, we have the following rule:$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.$$ Example: \begin{align*}\frac{2}{5}+\frac{3}{7}&=\frac{2\times 7}{5\times 7}+\frac{5\times 3}{5\times 7}\cr & =\frac{14}{35}+\frac{15}{35}=\frac{14+15}{35}\cr &=\frac{29}{35}.\end{align*}

### Equivalent fractions and reductions

A fraction is unchanged if we multiply both its numerator and denominator by the same number. This means that $\frac{a}{b}=\frac{a\times c}{b\times c}$. This means that $\frac{1}{3}$ is the same as $\frac{2}{6}$ the same as $\frac{3}{9}$……But in these equalities $\frac{1}{3}$ is the better one because it is the result of simplification of reductions of the other fractions.

Example: simplify the fraction $\frac{35}{10}$. Remark that $35=5\times 7$ and $10=5\times 2$. Then $\frac{35}{10}=\frac{5\times 7}{5\times 2}=\frac{7}{2}$. We stop in this stage because we can not reduce $\frac{7}{2}$ as there is no common number in the decomposition of 7 and 2.

### The inverse of a number with respect to the multiplication operation

By definition the inverse of a non null number $a$ is $\frac{1}{a}$. Sometimes this inverse is denoted by the symbol $a^{-1}$.

The inverse of a fraction is$$\displaystyle\frac{1}{\displaystyle\frac{a}{b}}=\frac{b}{a}.$$

###### Exercises on how to find a fraction of the number

In this section, we propose exercises with detailed solutions to show you how to calculate the fractions of numbers.

Exercise: Calculate $\frac{8}{24}+\frac{10}{45}$.

Proof: The best way to answer this question is to first reduce the fractions $\frac{8}{24}$ and $\frac{10}{45}$ each.This will allow us to work with small numbers that we know well. As $24=3\times 8$ then $\frac{8}{24}=\frac{1}{3}$. On the other hand, as $10=2\times 5$ and $45=9\times 5$, then $\frac{10}{45}$ is equal to $\frac{2}{9}$. Then instead of calculating $\frac{8}{24}+\frac{10}{45},$ is the same to calculate $\frac{1}{3}+\frac{2}{9}$ which is much more simple. Thus \begin{align*}\frac{8}{24}+\frac{10}{45}&=\frac{1}{3}+\frac{2}{9}\cr & = \frac{3}{3\times 3}+\frac{2}{9}\cr & = \frac{3}{9}+\frac{2}{9}\cr &=\frac{3+2}{9}\cr &=\frac{5}{9}.\end{align*}

Exercise: Simplify the fraction $$\frac{1}{1+\frac{1}{2}}.$$

Proof: The first step to do is to calculate the number in the denominator. Using the argument cited above we have $$1+\frac{1}{2}=\frac{2+1}{2}=\frac{3}{2}.$$ Now we replace this quantity in the first fraction, and we obtain
$$\frac{1}{1+\frac{1}{2}}=\frac{1}{\frac{3}{2}}=\frac{2}{3}.$$

You can also consult the resolution of rational inequalities.