We are going to look at how to do division but we are not just going to look at one method we will look at a total of four different methods of division.
The first is short division, a very fast method of division, most suited to dividing by a single digit divisor. Then we will show how to do long division step by step using the traditional long division method. Then we will have a look at the Vedic Math division method, well one form of it anyway as I have seen a couple of variations and finally we will look at the Trachtenberg System division method.
We will work through several division examples using each method so you can compare the merits of each system. The Vedic Math and Trachtenberg system both claim that long division is made easy when compared to the traditional long division method.
We will limit ourselves here to just dividing by one or two digit divisors when examining the division methods.
How to do Short Division
When looking at how to do division we have to look at short division as it is the fastest way to do division when using a single digit divisor. All you write down is the answer and any remainders as you go through the calculation, the rest you do mentally. The calculations you are doing mentally I will put in the colored box.
478 divided by 6 using Short Division
First up we will look at 478 divided by 6.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r } & && & && && \\ \cline{3-8} & 6 &\big)& 4 &\textcolor{White}{_0}&\textcolor{Black}{7}&\textcolor{White}{_0} &\textcolor{Black}{8}& \\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-023df19becc4af241774ace695158288_l3.png "Rendered by QuickLaTeX.com")
As 6 is larger than 4 we have to use 47 so our first step is find the largest multiple of 6 which is less than or equal to 47.
We put 7 up over the 7 of 47. Then we put the 5 remainder as a subscript in front of the next digit in the dividend, in this case the 8.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r } & && & &\textcolor{Red}{7}& && \\ \cline{3-8} & 6 &\big)& 4 &\textcolor{White}{_0}&\textcolor{Black}{7}&\textcolor{Blue}{_5} &\textcolor{Black}{8}& \\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-cafccefa79931b78a2691e791e8f779a_l3.png "Rendered by QuickLaTeX.com")
The next step is we read the "
" as 58 so have to now find the largest multiple of 6 less than or equal to 58.
The 9 is written up over the eight and we write R4 next to the 9 to indicate the remainder. That is it! We are finished.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r } & && & &7& &\textcolor{Red}{9}&\text{R}\textcolor{Blue}{4} \\ \cline{3-8} & 6 &\big)& 4 &\textcolor{White}{_0}&\textcolor{Black}{7}&_5&8& \\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-8625865fceb412ef41ee8409edb869f0_l3.png "Rendered by QuickLaTeX.com")
Short Division with larger numbers
As mentioned before the short division method is best suited to single digit divisors but can also be used for larger numbers, like 10, 12, 15, 20, 50 etc. that are easy to calculate multiples for.
As an example we will divide 7458 by 25 using the short division method.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r@{}*1r } & && & && && & \\ \cline{3-10} & 25 &\big)& 7 &\textcolor{White}{_{00}}&\textcolor{Black}{4}&\textcolor{White}{_{00}} &\textcolor{Black}{5}&\textcolor{White}{_{00}}&\textcolor{Black}{8}\\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-cfbf53e4cb38a629e61e121e0dc797e2_l3.png "Rendered by QuickLaTeX.com")
Firstly how many times does 25 go into 74?
We write 2 up over the 4 and then write 24 as a subscript in front of the 5.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r@{}*11 } & && & &\textcolor{Red}{2}& && & \\ \cline{3-10} & 25 &\big)& 7 &\textcolor{White}{_{00}}&\textcolor{Black}{4}&\textcolor{Blue}{_{24}} &\textcolor{Black}{5}&\textcolor{White}{_{00}}&\textcolor{Black}{8}\\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-a2d63dc2b6b2c25d77fdc37d9112d660_l3.png "Rendered by QuickLaTeX.com")
The next step is to find out how many times 25 goes into 245.
We write the 9 up over the 5 of the dividen and the remainder is 20 so we write 20 as a subscript in front of the 8 of the dividend.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r@{}*11 } & && & &\textcolor{Black}{2}& &\textcolor{Red}{9}& & \\ \cline{3-10} & 25 &\big)& 7 &\textcolor{White}{_{00}}&\textcolor{Black}{4}&\textcolor{Black}{_{24}} &\textcolor{Black}{5}&\textcolor{Blue}{_{20}}&\textcolor{Black}{8}\\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-1d1e8d1e8f8d316ec4a6ccfc6bd3b0d7_l3.png "Rendered by QuickLaTeX.com")
The question now is how many times does 25 go into 208?
We write the 8 up over the 8 of the dividend. We have 8 remaining so we put R8 next to our answer.
![\[ \renewcommand\arraycolsep{1.5pt}\def\arraystretch{1.2} \begin{array}{*2r @{\hskip\arraycolsep}c *2r@{}*1l *1r@{}*1l *1r@{}*1l } & && & &2& &9& & \textcolor{Red}{8}\text{ R}\textcolor{Blue}{8} \\ \cline{3-10} & 25 &\big)& 7 &\textcolor{White}{_{00}}&\textcolor{Black}{4}&\textcolor{Black}{_{24}} &\textcolor{Black}{5}&\textcolor{Black}{_{20}}&\textcolor{Black}{8}\\ \end{array} \]](https://trachtenbergspeedmath.com/wp-content/ql-cache/quicklatex.com-daa2cca592fd0919b6a54a1701ba1bce_l3.png "Rendered by QuickLaTeX.com")
So our answer is 298 with 8 remainder.
That is short division and as the name indicates the method is very short but is a useful method to know especially when dividing by numbers you know or can easily work out the multiples of.
Next we look at long division.
Since you picked two divisions that are difficult to do using the Vedic ‘crowning gem’ a.k.a. ‘flag’ method, it would be fair to do the same problems using the Trachtenberg system. That would be more convincing that the Trachtenberg method does not run into the same complications.
It is true division is not always as straightforward as multiplication and there are times you may need to adjust the last calculated digit of your answer up or down using the Trachtenberg method.