Fractions

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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 equal pieces, so each of them will take $\frac{1}{3}$ of the chocolate bar.
The two girls together will have two pieces, so, mathematically speaking, they will have $\frac{2}{3}$ of the bar.
All three pieces together make the whole bar: $\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=\frac{3}{3}=1$.

Example 2: What part of the soldiers are yellow?

Example 3: What part of the apples is missing?

Fraction Rules

Addition:(same denominators)
$\frac{A}{B} +\frac{C}{B} = \frac{A + C}{B}$
Example

Example:
$\frac{1}{5} +\frac{2}{5} = \frac{1 + 2}{5} = \frac{3}{5}$

Subtraction:(same denominators)
$\frac{A}{B} -\frac{C}{B} = \frac{A - C}{B}$
Example

Example:
$\frac{4}{5} - \frac{2}{5} = \frac{4 - 2}{5} = \frac{2}{5}$

Addition:(different denominators)
$\frac{A}{B} +\frac{C}{D} = \frac{A\cdot D}{B\cdot D} +\frac{B\cdot C}{B\cdot D} = \frac{A\cdot D + B\cdot C}{B\cdot D}$
Example

Example:
$\frac{3}{5} +\frac{1}{2} = \frac{3\cdot 2}{5\cdot 2} + \frac{5 \cdot 1}{5 \cdot 2}= \frac{3\cdot 2 + 5 \cdot 1}{5 \cdot 2}=\frac{11}{10}$

Subtraction:(different denominators)
$\frac{A}{B} -\frac{C}{D} = \frac{A\cdot D}{B\cdot D} -\frac{B\cdot C}{B\cdot D} = \frac{A\cdot D - B\cdot C}{B\cdot D}$
Example

Example:
$\frac{3}{5} -\frac{1}{2} = \frac{3\cdot 2}{5\cdot 2} - \frac{5 \cdot 1}{5\cdot 2}= \frac{3\cdot 2 - 5 \cdot 1}{5\cdot 2}=\frac{1}{10}$

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

Example:
$\frac{3}{5} \times \frac{7}{8} = \frac{3\cdot 7}{5\cdot 8} = \frac{21}{40}$

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

Example:
$\frac{3}{5} \div \frac{2}{9} = \frac{3}{5} \times \frac{9}{2} = \frac{3\times 9}{5\times 2} = \frac{27}{10}$

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 $c > 0$ 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 positive integers then:

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

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

Fraction Test

More about fractions in the math forum

Fractions forum

Worksheets

Fraction Circles Worksheet

Games

Create a Flag Using Fractions

Problems

Equivalent Fractions
Fraction Addition and Subtraction

Tests

Fractions Quiz