Question

How do you convert $\dfrac{4}{{13}}$ to a decimal?

Answer 1

Hint: We will put 4 as dividend and 13 as a divisor and then just perform the long division and wherever it repeats itself, we will get the bar according to that.

Complete step-by-step answer:
We are given that we are required to convert $\dfrac{4}{{13}}$ to a decimal.
We here will take 4 as a dividend and 13 as a divisor.
Now, we will perform the long division as follows:-
$ \Rightarrow 13\mathop{\left){\vphantom{14}}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{4}}}
\limits^{\displaystyle \,\,\, {}}$
We see that we will have to take a decimal and a 0 to perform the division and then we will multiply 13 by 3 to get 39 to get the following:-
$ \Rightarrow 13\mathop{\left){\vphantom{1\begin{gathered}
  4.0 \\
  \underline {39} \\
  01 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  4.0 \\
  \underline {39} \\
  01 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {0.3}}$
We see that we will have to take two more 0’s to perform the division and then we will multiply 13 by 7 to get 91 to get the following:-
$ \Rightarrow 13\mathop{\left){\vphantom{1\begin{gathered}
  4.000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  0009 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  4.000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  0009 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {0.307}}$
We see that we will have to take one more 0 to perform the division and then we will multiply 13 by 6 to get 78 to get the following:-
$ \Rightarrow 13\mathop{\left){\vphantom{1\begin{gathered}
  4.0000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  90 \\
  \underline {78} \\
  12 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  4.0000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  90 \\
  \underline {78} \\
  12 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {0.3076}}$
We see that we will have to take one more 0 to perform the division and then we will multiply 13 by 9 to get 117 to get the following:-
$ \Rightarrow 13\mathop{\left){\vphantom{1\begin{gathered}
  4.00000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  90 \\
  \underline {78} \\
  120 \\
  \underline {117} \\
  3 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  4.00000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  90 \\
  \underline {78} \\
  120 \\
  \underline {117} \\
  3 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {0.30769}}$
We see that we will have to take one more 0 to perform the division and then we will multiply 13 by 2 to get 26 to get the following:-
$ \Rightarrow 13\mathop{\left){\vphantom{1\begin{gathered}
  4.000000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  90 \\
  \underline {78} \\
  120 \\
  \underline {117} \\
  30 \\
  \underline {26} \\
  4 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  4.000000 \\
  \underline {39} \\
  0100 \\
  \underline {0091} \\
  90 \\
  \underline {78} \\
  120 \\
  \underline {117} \\
  30 \\
  \underline {26} \\
  4 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {0.307692}}$
Now, we have again got a 4 and we will just now repeat all the numbers, therefore, we have:-

$ \Rightarrow \dfrac{4}{{13}} = 0.\overline {307692} $

Note:
The students must know that we have a prime number 13 as the denominator, therefore, we will never be able to get the terminating decimals.
If the denominator of a fraction can be written in the form of ${2^m} \times {5^n}$, where m and n are whole numbers, then the fraction is always terminating decimal. Here we had 13, which can never be written in the multiples of 2 and 5. So, we have got a non – terminating decimal as the answer.