This is called the decimal representation. Using this representation we separate out the whole number and the fraction part using a $.$ called the $decimal\ point$

When we write the fraction $3\dfrac{1}{2}$ it means $3 + \dfrac{1}{2}$. The whole number is $3$ and the fraction part is $\dfrac{1}{2}$. Using decimal notation we can write this as

$3.5$

. This means . We see soon, how $\dfrac{1}{2}$ became $0.5$, but before that let us understand a little bit more about decimals.

In fraction representation, the fraction part does not have any place value, but in decimal representation the digits in fraction part also have place values.

In integer we saw that as we move from left to right the place value gets divided by $10$.

For example in the number $2983$ the place value of $2$ is , the place value of $9$ is , and so on. The same process continues as we cross the decimal point and go further right.

When we write a decimal value like $938.2765$, the place values are as follows:

As we can see when we move from left to right the place value gets divided by .

The first digit to the left of the decimal point has a place value of $1$ and the first digit to the right of the decimal point has a place value of $\dfrac{1}{10}$

Another important point about decimal representation is that, when we add any number of zeros at the right end, after the decimal point, it does not change the value of the number.

For example we can write:

$23.85 = 23.850 = 23.8500 = 23.85000 = \ldots$

These zeros are called: $Trailing\ Zeros$

Now, we will take a look at how to convert fractions to decimal and the other way.

We will convert the fraction $\dfrac{36}{125}$ to decimal. Just remember that we can add a decimal point after $36$ and write as many zeros to get  $36.000\ldots$ . Let us look at the steps below:

$ \begin{array}{rl} 0.288 \phantom{00}\\ 125 \enclose{longdiv}{36 \phantom{0000}} \phantom{0}\\[-3pt] \underline{\phantom{0}0} \phantom{00000.} & \Leftarrow & {0 \times 125 = 0}\\[-3pt] 360 \phantom{0000.} & \Leftarrow & \hbox{remainder 36, bring down the first 0 after adding the decimal point to the quotient}\\[-3pt] \underline{250} \phantom{0000.} & \Leftarrow & {2 \times 125 = 250}\\[-3pt] 1100 \phantom{000.} & \Leftarrow & \hbox{remainder 110, bring down another 0}\\[-3pt] \underline{1000} \phantom{000.} & \Leftarrow & {8 \times 125 = 1000}\\[-3pt] 1000 \phantom{00.} & \Leftarrow & \hbox{remainder 100, bring down 0}\\[-3pt] \underline{\phantom{}1000} \phantom{00.} & \Leftarrow & {8 \times 125 = 1000}\\[-3pt] 0 \phantom{000} & \Leftarrow & \hbox{remainder 0, we can stop now}\\[-3pt] \end{array} $

Therefore,

$\dfrac{36}{125} = 0.288$

There is one more approach to for converting fractions to decimals which you may find working very fast in many cases.

Let us understand this by using some examples. Let us convert $\dfrac{14}{25}$ to decimal. Just by looking at the decimal we know that if we multiply $25$ by $4$ we get $100$.

So, let us multiply both numerator and denominator by $4$ and we get:

$\dfrac{14}{25} = \dfrac{14 \times 4}{25 \times 4} = \dfrac{56}{100}$

Now, we see that the denominator has $1$ followed by some zeros. We can get the corresponding decimal value by simply shifting the decimal point as many places to the left as there are zeros in the denominator.

$\dfrac{14}{25} = \dfrac{14 \times 4}{25 \times 4} = \dfrac{56}{100} = 0.56$

We will take one more example for this. We will convert $\dfrac{73}{125}$ to its decimal value. We can notice that if we multiply $125$ by $8$ we will get $1000$, which is $1$ followed by $3$ zeros. Therefore we can multiply the numerator by $8$ and shift the decimal point $3$ places to the left.

$\dfrac{73}{125} = \dfrac{73 \times 8}{125 \times 8} = \dfrac{584}{1000} = 0.584$

So, if you find an easy way of converting the denominator to some number, which is $1$ followed by some zeros, by multiplying by a number, you should do that instead of the long division. 

The decimal representation of some of the common fractions are as follows:

$\dfrac{1}{4} =$

$\dfrac{1}{2} =$

$\dfrac{3}{4} =$

You should try converting them yourself as a simple exercise.

$7.2765 = 7 + 0.2765$

$= 7 + \dfrac{0.2765}{1}$

$= 7 + \dfrac{2.765}{1 \times 10}$

$= 7 + \dfrac{27.65}{1 \times 10 \times 10}$

$= 7 + \dfrac{276.5}{1 \times 10 \times 10 \times 10}$

$= 7 + \dfrac{2765}{1 \times 10 \times 10 \times 10 \times 10}$

$= 7 + \dfrac{2765}{10000}$

Note that the above steps were shown just for explanation, you really do not need to multiply by $10$ so many times, just multiply by $1$ followed by as many zeros as there are digits to the right of the decimal point like below:

$7 + \dfrac{0.2765}{1} = 7 + \dfrac{2765}{1 \times 10000} = 7 + \dfrac{2765}{10000}$

Now we can reduce the fraction as usual

$7 + \dfrac{\cancel{2765} \raise{3pt}{553}}{\cancel{10000} \lower{3pt}{2000}}$

$= 7 + \dfrac{553}{2000}$

$= 7\dfrac{553}{2000}$