# Fractions

## Free Fraction Calculator (by Radu Turcan)

Operator: + - * /Solution:

#### Definition of a fraction

A number written as $\frac{a}{b}$ or a/b, where $a$ is an integer and $b$ is a non-zero integer, is called a **fraction**.

The number $a$ is **numerator**, and $b$ is the **denominator**.
A fraction represents either a part of a whole or any number of equal parts.

The denominator shows how many equal parts make up a whole, and the numerator shows how many of these parts we have in mind.

#### Examples of fractions

**Example 1:** Becky, Merry and John want to share a chocolate bar evenly.

What part of the bar will each of them take?

What part of the bar will Becky and Merry have together?

The children need to divide the bar into three pieces. So everyone will take $\frac{1}{3}$ of the chocolate bar.

Two girls together will have
two pieces, hence, mathematically speaking, they will have $\frac{2}{3}$ of the bar.

**Example 2:**
What part of the soldiers are yellow?

**Example 3:** What part of the apples is missing?

#### Fraction Rules

**Addition:(same denominators)**

$\frac{A}{B} +\frac{C}{B} = \frac{A + C}{B}$

**Subtraction:(same denominators)**

$\frac{A}{B} -\frac{C}{B} = \frac{A - C}{B}$

**Addition:(different denominators)**

$\frac{A}{B} +\frac{C}{D} = \frac{A\cdot D}{B\cdot D} +\frac{B\cdot C}{B\cdot D} = \frac{A\cdot D + B\cdot C}{B\cdot D}$

**Subtraction:(different denominators)**

$\frac{A}{B} -\frac{C}{D} = \frac{A\cdot D}{B\cdot D} -\frac{B\cdot C}{B\cdot D} = \frac{A\cdot D - B\cdot C}{B\cdot D}$

**Multiplication:**

$\frac{A}{B}\times\frac{C}{D} = \frac{A\cdot C}{B\cdot D}$

**Division:**

$\frac{A}{B}\div\frac{C}{D} = \frac{A}{B}\times\frac{D}{C}= \frac{A\cdot D}{B\cdot C}$

#### Properties of fractions

**Property I:**
All hatched parts of the circles represent one half $\frac{1}{2}, \frac{2}{4}$ and $\frac{3}{6}$, hence $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$

We get $\frac{2}{4}$ when we multiply the numerator and the denominator of the fraction $\frac{1}{2}$ by $2$.

We obtain $\frac{3}{6}$ by multiplying the numerator and the denominator of $\frac{1}{2}$ by $3$.

Let $a$ be an integer and $b$ and $c$ be non-zero integers.

Then:

$\frac{a}{b}=\frac{a\cdot c}{b\cdot c}$ and $\frac{a}{b}=\frac{a:c}{b:c}$

**Property II:** If two fractions have equal denominators, the fraction with the greater numerator is greater.

If $a$, $b$ and $c$ are integers and $c \ne 0$ then:

$\frac{a}{c}>\frac{b}{c}$, if $a>b$

Example: $\frac{4}{5} > \frac{3}{5} > \frac{2}{5}$

**Property III:**
If two fractions have equal numerators, the fraction with the smaller denominator is greater.

If $a$, $b$ and $c$ are integers, and $b$ and $c$ are non-zero then:

$\frac{a}{b}>\frac{a}{c}$, if $b< c$

Example: $\frac{3}{4} > \frac{3}{5} > \frac{3}{20}$