Writing Fractions

We have $b \times \frac{a}{b} = a$ , for example:
2 $\frac{14}{2}$ = 14
In writing the fraction as $a : b$ , $a$ is the dividend and $b$ the divisor. For example when we write $6 : 11$ , the dividend is $6$ and the divisor is $11$ .
In writing the fraction as $\frac{a}{b}$ , $a$ is the numerator (originally as Latin numerator : the one that counts , $b$ the denominator (originally as Latin denominator : the one that denominates , as it denominates, it determines the unit).
For example in the fraction $\frac{11}{2}$ , the numerator is $11$ and the denominator is $2$ .
The fraction is a quotient; that is, the result of division, by a number .

Definition [Numerator]

This is the part of a fraction which counts how many units are contained in this fraction.

Definition [Denominator]

This is the part of a fraction which indicates how many equal parts the unit is divided into.

Writing Fractions → I Definitions and representations → I-1 Numerator and denominator

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

First Idea

Given the fraction

\frac{8}{3}

, it can be considered as

8 \times \frac{1}{3}

.
Consider the unit interval:

As the denominator is

3

, we divide the interval into

3

equal parts:

and we have to take

8

such parts:

it has the distance

\frac{8}{3}

Second idea

Take the fraction

\frac{8}{3}

, that can be considered as one third of 8. Take the unit interval:

Since the numerator is 8, it has 8 times the length of :

The length is divided into 3 parts.

we place our number there:

Third idea

The fraction

\frac{8}{3}

is the number which is an approximate value is

0

because the division does not terminate (you can see the accuracy). It is then easy to place this number, even approximately, on a graduated axis .

Remark

In all the above methods, the distance from 0 to

\frac{8}{3}

is always the same (note the change in the choice of the unit anyway!)

Writing Fractions → I Definitions and representations → I-2 Some ideas

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

II-1 A First approach

II-2 Theorem and exercises

II-3 Interesting extensions

Writing Fractions → II Different fractions for the same number

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

Here is an illustration:

We note that the two rectangles are superimopsable, so also the the colored areas.
Now in the first case les 4 fifths of the rectangle were colored, whereas in the second case, 8 were colored.
We can deduce that the fractions

\frac{4}{5}

and

\frac{8}{10}

are equal.

We note:

\frac{4}{5} = \frac{8}{10}

. The same number (it is a rational number) can be written in many different ways.
We also note that:

\frac{4}{5} = 0, 8

and

\frac{8}{10} = 0, 8

The two ratios are the same!

Writing Fractions → II Different fractions for the same number → II-1 A First approach

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

Theorem [Equal fractions]

The ratio

\frac{a}{b}

does not change when we multiply (or we divide) the numerator and denominator by the same non-zero number. When

a

,

b \neq 0

, and

k \neq 0

the numbers are:

\begin{matrix} \frac{a}{b} & = & \frac{a \times k}{b \times k} \\ \frac{a}{b} & = & \frac{a \div k}{b \div k} \end{matrix}

Example [Simplifying form of fractions ]

Using the above rule, write down the steps and simplify the fraction:

\frac{2}{6}

.
Can be simplified in two ways:

\frac{2}{6} = \frac{1 \times 2}{3 \times 2} = \frac{1}{3}

ou

\frac{2}{6} = \frac{2 \div 2}{6 \div 2} = \frac{1}{3}

Example [Find a decimal ]

Consider the following fractions:

\frac{78}{25}

. Write as a decimal fraction (fraction whose denominator is 10,100, 1000, ...) showing the steps in writing decimal.
We can write:

\frac{78}{25} = \frac{78 \times 4}{25 \times 4} = \frac{312}{100} = 3, 12

Writing Fractions → II Different fractions for the same number → II-2 Theorem and exercises

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

First extension

Given a fraction

\frac{2}{24}

. One can write

\frac{2}{24} = \frac{2 \div 4}{24 \div 4} = \frac{0, 5}{6}

. These entries represent the same number. In contrast, when the numerator or denominator are not integers, we no longer speak of arational number , but a quotient number.

Second extension

Given the fraction

\frac{0, 93}{0, 84}

, we can write

\frac{0, 93}{0, 84} = \frac{0, 93 \times 100}{0, 84 \times 100} = \frac{93}{84}

. That's why we had to learn how to make divisions with decimals! We've just written a fractional number as a rational number.

Decimals to fractions

All decimal numbers (a fortiori the integers) always admit a (and therefore in many ways) rational form. For example:

2, 53 = \frac{2530}{1000} =

.

Remark

It is important to note that the reverse is false. There are fractions that have no decimal notation, for example:

$\frac{10}{21}$

is a fraction, but is not a decimal, because if the division is performed, it does not stop.

Writing Fractions → II Different fractions for the same number → II-3 Interesting extensions

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

III-1 Point Method for Finding a Fractional Number

Writing Fractions → III Value of a Fractional Number

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

Theorem [Multiplying a decimal number by a fraction]

In order to multiply a decimal number

a

by a fraction

\frac{b}{c}

(with

c \neq 0

), that is, to calculate

a \times \frac{b}{c}

or

\frac{b}{c} \times a

, since multiplication is commutative, we can use either of the following methods:

Method 1 One first finds the ratio $b \div c$ and then multiplies the result by $a$ , which can be written as: $a \times \frac{b}{c} = a \times (b \div c)$ .
Method 2 One calculates the product $a \times b$ , which is then divided by $c$ , which can be written as: $a \times \frac{b}{c} = (a \times b) \div c$ .
M�thode 3 One calculates the ratio $a \div c$ , then the multiples the result by $b$ , which can be written as: $a \times \frac{b}{c} = (a \div c) \times b$ .

Example

Let us perform various calculations with the three methods described above:

First method:

a \times \frac{b}{c} = a \times (b \div c)

$A$	=	$180 \times \frac{2}{3}$
$A$	=	$180 \times$ non calculable
$A$	=	non calculable

Second method:

a \times \frac{b}{c} = (a \times b) \div c

$A$	=	$180 \times \frac{2}{3}$
$A$	=	$\frac{180 \times 2}{3}$
$A$	=	$\frac{360}{3}$
$A$	=	$120$

Third method:

a \times \frac{b}{c} = (a \div c) \times b

$A$	=	$180 \times \frac{2}{3}$
$A$	=	$\frac{180}{3} \times 2$
$A$	=	$60 \times 2$
$A$	=	$120$

Remark

The first method is not always possible.
The second method can handle more number of cases.
The third method often seems fast for calculations that can make take longer.

Writing Fractions → III Value of a Fractional Number → III-1 Point Method for Finding a Fractional Number

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

Theorem [Comparison of fractions with same denominator]

If two given fractional numbers have the same denominator, then the smallest is the one with the smallest numerator.

Example [ ]

The relation

\frac{2}{9} < \frac{5}{9}

can also be written as

\frac{5}{9} > \frac{2}{9}

.
Similarly we have:

\frac{19}{1, 1} > \frac{9}{1, 1}

can also be written as

\frac{9}{1, 1} < \frac{19}{1, 1}

.

Theorem [Compare fractional numbers with same numerator]

If two numbers have fractional forms with the same numerator, then the smallest is the one that has the highest denominator.

Example [ ]

The relation

\frac{6}{8} < \frac{6}{5}

can be written as

\frac{6}{5} > \frac{6}{8}

.
Similarly we find:

\frac{1, 5}{10} > \frac{1, 5}{13}

can be written as

\frac{1, 5}{13} < \frac{1, 5}{10}

.

Theorem [Comparing fractions with different denominators]

In order to compare fractions with different denominators, we convert them to a form with the same denominator.

Example [ ]

Compare the following fractions:

\frac{15}{7}

and

\frac{13}{42}

.
Now,

\frac{15}{7} = \frac{90}{42}

, it is like comparing

\frac{13}{42}

and

\frac{90}{42}

as 90>13 then

\frac{90}{42} > \frac{13}{42}

and thus finally

\frac{15}{7} > \frac{13}{42}

Writing Fractions → IV Comparison of Numbers in Fractional Form

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

Let us calculate the following expressions:

$a + \frac{b}{c}$

$\frac{a}{b + c}$

$\frac{a + c}{b}$

as shown by the following examples.

\begin{matrix} A & = & 46 + \frac{16}{4} \\ A & = & 46 + 4 \\ A & = & 50 \end{matrix}

\begin{matrix} B & = & \frac{46}{16 + 4} \\ B & = & \frac{46}{20} \\ B & = & 2, 3 \end{matrix}

\begin{matrix} C & = & \frac{46 + 4}{16} \\ C & = & \frac{50}{16} \\ C & = & 3,125 \end{matrix}

Above, we were applying the rules of precedence of the arithmetic operations for fractions. We can also write (as you do in a calculator):

$A = 46 + 16 \div 4$ , $B = 46 \div (16 + 4)$ et $C = (46 + 4) \div 16$

which clearly shows the priorities.

Remark

The same method of calculation is applied with subtraction instead of addition.

However, it would not go through for calculations with decimal numbers...

Writing Fractions → V Precedence Rules in Calculation with Ratios

I Definitions and representations
- I-1 Numerator and denominator
- I-2 Some ideas
II Different fractions for the same number
III Value of a Fractional Number
- III-1 Point Method for Finding a Fractional Number
IV Comparison of Numbers in Fractional Form
V Precedence Rules in Calculation with Ratios
VI Addition and Subtraction of Fractions

Theorem [Addition and subtraction of fractions]

For adding (or for subtracting) of two fractional numbers same denominator :

for adding ( or for subtracting ) of two numerators;
fractions with the same denominator.

In other words, if

a

,

b

and

d

are numbers (

d

non-zero),on a:

$\frac{a}{d} + \frac{b}{d} = \frac{a + b}{d}$

$\frac{a}{d} - \frac{b}{d} = \frac{a - b}{d}$

Example [ ]

\begin{matrix} A & = & \frac{11, 7}{20} - \frac{4, 8}{20} \\ A & = & \frac{11, 7 - 4, 8}{20} \\ A & = & \frac{6, 9}{20} \end{matrix}

and the simplified result is:

Remark

If the denominators are different, we start with fractions having same denominators.

Example [ ]

\begin{matrix} B & = & \frac{296}{154} + \frac{35}{77} \\ B & = & \frac{296}{154} + \frac{35 \times 2}{77 \times 2} \\ B & = & \frac{296}{154} + \frac{70}{154} \\ B & = & \frac{296 + 70}{154} \\ B & = & \frac{366}{154} \end{matrix}

and the simplified result is:

Writing Fractions → VI Addition and Subtraction of Fractions