When we add two decimal numbers, we write the numbers such that the decimal points are one below the other. Now we can add going from right to left as we do for integers. For example, when we add $13.7$ to $6.832$, we will write it the following way:

$\phantom{00.}6.832 \\ \underline{+13.7\phantom{00}} \phantom{000} \Leftarrow \hbox{we aligned the decimal point one below the other} \\ \phantom{..} 20.532 \phantom{000} \Leftarrow \hbox{now we add both numbers like normal integers}$

You may already know this from your school or your friends. But do you know why we do this by writing the decimal points below one another? It will be good to understand why.

You know that in normal integer addition when you write two numbers, you write in such a way that the unit's digit is below the unit's digit, ten's digit below the ten's digit, and so on. Then we add the digits starting with the lowest place. We cannot add two digits having different place values. This is exactly the reason, why we write the decimal point one below the other. If we write the decimal point one below the other, all the place values for the two numbers will be in correct position. If the decimal points are not one below the other, the place values of the top digits will not be same as the bottom digits, and we will get a wrong result.

Subtraction is done the same way. Let us take an example to see this. Let us subtract $4.3$ from $21.3892$.

$\phantom{.}21.3892 \\ \underline{-4.3\phantom{000}} \phantom{000} \Leftarrow \hbox{we aligned the decimal point one below the other} \\ \phantom{.} 17.0892 \phantom{000} \Leftarrow \hbox{now we subtract the number like normal integers}$

We just need to take care of one thing in subtraction, if the top number has lesser number of digits after the decimal in the top number (also called the $Minuend$) we need to add as many $0$s to the right (called $trailing\ zeros$) to make them same. Let us try this with an example. We will subtract $7.5749$ from $18.4$

$\phantom{.}18.4000 \phantom{000} \Leftarrow \hbox{we added three 0s at the end} \\ \underline{-7.5749\phantom{}} \phantom{000} \Leftarrow \hbox{we aligned the decimal point one below the other} \\ \phantom{.} 10.8251 \phantom{000} \Leftarrow \hbox{now we subtract both numbers like normal integers}$

When we multiply two decimal numbers we do not have to align the decimal points. We multiply the decimal numbers like integers and then in the result place the decimal points as many places to the left as there are digits in the multiplicand and the multiplier after decimal points.

We will understand this with an example. Let us multiply $2.34$ by $9.1$, but before we do that let us count the number of digits to the right of the decimal point for both numbers. $2.34$ has digits to the right of the decimal place and $9.1$ has digit to the right of the decimal point. So totally there are to the right of the decimal place. Now let us do the multiplication.

$ \phantom{00..}2.34 \\ \underline{\phantom{0.}\times 9.1\phantom{}} \phantom{000} \Leftarrow \hbox{notice we did not align the decimal points} \\ \phantom{000.} 234 \phantom{000} \Leftarrow \hbox{we multiplied 234 by 1} \\ \underline{\phantom{0.} 2106\times} \phantom{00} \Leftarrow \hbox{we multiplied 234 by 9} \\ \phantom{0} 21.294 \phantom{000} \Leftarrow \hbox{after adding all the results, we put the decimal point 3 digits to the left}$

Therefore, $2.34 \times 9.1 = 21.294$

When we divide a decimal number by another decimal number, we need to convert the divisor to an integer by multiplying both the dividend and the divisor by $10$ as many times as required. Let us try this with an example, we will divide $20.382$ by $1.25$

We know that $20.382 \div 1.25 = \dfrac{20.382}{1.25}$

Multiplying the numerator and the denominator by the same number does not change the value of a fraction, and we can also see that if we multiply $1.25$ by $10$ twice (that is same as multiplying by ) we will get an integer which is .

Therefore,

$20.382 \div 1.25 = \dfrac{20.382}{1.25} = \dfrac{20.382 \times 100}{1.25 \times 100} = \dfrac{2038.2}{125}$

Now we can do our division using the normal long division method.

$ \begin{array}{rl} 16.3056 \\ 125 \enclose{longdiv}{2038.2 \phantom{00}} \phantom{0}\\[-3pt] \underline{125} \phantom{000000} & \Leftarrow & {1 \times 125 = 125}\\[-3pt] 788 \phantom{00000} & \Leftarrow & \hbox{remainder 78, bring down 8}\\[-3pt] \underline{750} \phantom{00000} & \Leftarrow & {6 \times 125 = 750}\\[-3pt] 382 \phantom{0000} & \Leftarrow & \hbox{remainder 38, bring down 2, also add the decimal point to the quotient}\\[-3pt] \underline{375} \phantom{0000} & \Leftarrow & {3 \times 125 = 375}\\[-3pt] 70 \phantom{000} & \Leftarrow & \hbox{remainder 7, bring down 0}\\[-3pt] \underline{\phantom{0}0} \phantom{000} & \Leftarrow & {0 \times 125 = 0}\\[-3pt] 700 \phantom{00} & \Leftarrow & \hbox{remainder 70, bring down 0}\\[-3pt] \underline{625} \phantom{00} & \Leftarrow & {5 \times 125 = 625}\\[-3pt] 750 \phantom{0} & \Leftarrow & \hbox{remainder 75, bring down 0}\\[-3pt] \underline{750} \phantom{0} & \Leftarrow & {6 \times 125 = 750}\\[-3pt] 0 \phantom{0} & \Leftarrow & \hbox{remainder 0}\\ \end{array} $

Therefore,

$20.382 \div 1.25 = 16.3056$