### What is a fraction of two numbers?

**numenator**, while the number $b$ is called the

**denomunator**of the fraction. Then the form of a fraction (numenator/denomunator).

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

\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 nemunators and denomunators 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}

### Addition of fractions

**I.**The addition of fractions having the same denominators is very simple because we keep the same demomunator and we only add numerators. Here is the rule:

\begin{align}\frac{a}{b}+\frac{c}{b}=\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 denomutaor of $\frac{a}{b}$ by $d$ and those of $\frac{c}{d}$ by $b$. We obtain

\begin{align}\frac{a}{b}+\frac{c}{d}=\frac{ad}{bd}+\frac{bc}{bd}=\frac{ad+bc}{bd}.\end{align}

Finally, we have the following rule:

\begin{align}\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.\end{align}

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

**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 commum numbre in the decomposition of 7 and 2.

**Exercise of application:**Calculate $\frac{8}{24}+\frac{10}{45}$. The best way to answer this question is to first reduce (simplify) 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 instaed 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}

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

\begin{align} \displaystyle\frac{1}{\displaystyle\frac{a}{b}}=\frac{b}{a}.\end{align}

**Exercise:** Simplify the fraction \begin{align}\frac{1}{1+\frac{1}{2}}.\end{align}

Solution: The first step to do is to calculate the number in the denomunator. Using the argument cited above we have \begin{align}1+\frac{1}{2}=\frac{2+1}{2}=\frac{3}{2}.\end{align}

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