Question

Write $ \dfrac{3}{13} $ in decimal form and say what kind of decimal expansion it has.
(a) 0.230769, terminating and non-repeating
(b) 0.230769, non-terminating and repeating
(c) 0.230769, non-terminating and non-repeating
(d) 0.230769, terminating and repeating

Answer 1

Hint: To solve this question, we will perform long division. We will divide 3 by 13 until we cannot divide it further, or until the numbers after decimal in the quotient starts repeating. If the number 3 is completely divided by 13, such that we can’t divide further, we say that it is terminating. If the division goes on infinitely, we say its non-terminating. If the number starts repeating themselves, we say its repeating. If the digits are decimal and don't follow any particular pattern of repetition, we say it is non-repeating.

Complete step-by-step answer:
We need to divide 3 by 13 through a long division method.
 $ 13\overline{\left){3}\right.} $
We can see that the dividend is smaller than the divisor. We will write a 0 in the quotient with a decimal point after 0 and add a zero to the dividend. Thus, dividend 30 and thus bigger than divisor.
 $ 13\overset{0.}{\overline{\left){30}\right.}} $
Now, 3 times 13 is 39, which is greater than 30. 2 times 13 is 26.
Thus, we add 2 in the quotient after decimal and subtract 26 from the dividend 30.
 $ 13\overset{0.2}{\overline{\left){\begin{align}
  & \ \ 30 \\
 & -26 \\
 & \overline{\ \ \ \ 4} \\
\end{align}}\right.}} $
The remainder is 4, which is less than 13. Since, we have added a decimal in the quotient, we don’t need to do it again and we can directly add a 0 after 4. Thus, the new dividend is 40.
3 times 13 is 39. So, we will add a 3 to the quotient.
 $ 13\overset{0.23}{\overline{\left){\begin{align}
  & \ \ 30 \\
 & -26 \\
 & \overline{\begin{align}
  & \ \ \ \ 40 \\
 & \ \ -39 \\
\end{align}} \\
 & \overline{\ \ \ \ \ \ \ 1} \\
\end{align}}\right.}} $
The remainder now is 1. Even after adding one 0, it will be less than 13. So, we will add another 0 to new dividend and add a zero to the quotient.
 $ 13\overset{0.2307}{\overline{\left){\begin{align}
  & \ \ 30 \\
 & -26 \\
 & \overline{\begin{align}
  & \ \ \ \ 40 \\
 & \ \ -39 \\
\end{align}} \\
 & \overline{\begin{align}
  & \ \ \ \ \ \ \ 100 \\
 & \ \ \ \ \ \ -91 \\
 & \overline{\ \ \ \ \ \ \ \ \ \ 9} \\
\end{align}} \\
\end{align}}\right.}} $
Now, in similar fashion, we continue adding 0’s in the dividend and keep on dividing, until we are able to completely divide the divisor or at least two numbers repeat themselves.
\[13\overset{0.23076923}{\overline{\left){\begin{align}
  & \ \ 30 \\
 & -26 \\
 & \overline{\begin{align}
  & \ \ \ \ 40 \\
 & \ \ -39 \\
\end{align}} \\
 & \overline{\begin{align}
  & \ \ \ \ \ \ \ 100 \\
 & \ \ \ \ \ \ -91 \\
 & \overline{\begin{align}
  & \ \ \ \ \ \ \ \ \ \ 90 \\
 & \ \ \ \ \ \ \ -78 \\
 & \overline{\begin{align}
  & \ \ \ \ \ \ \ \ \ \ 120 \\
 & \ \ \ \ \ \ \ -117 \\
 & \overline{\begin{align}
  & \ \ \ \ \ \ \ \ \ \ \ \ \ 30 \\
 & \ \ \ \ \ \ \ \ \ \ -26 \\
 & \overline{\begin{align}
  & \ \ \ \ \ \ \ \ \ \ \ \ \ \ 40 \\
 & \ \ \ \ \ \ \ \ \ \ \ -39 \\
 & \overline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}} \\
\end{align}}\right.}}\]
As we can see, the digits after 9 are 2 and then 3 and from the remainder we can guess it will be 0.
Thus, the digits after decimal in the quotient are repeating themselves and they will never end.
Thus, $ \dfrac{3}{13} $ in decimal form is 0.230769 and it is non-terminating and repeating.
So, the correct answer is “Option B”.

Note: A shortcut way to identify whether the proper fraction is terminating or not is to look at the denominator. If the denominator can be factorised as $ {{2}^{m}}\times {{5}^{n}} $ after dividing out the common factors from the numerator and denominator, where m and n are whole real numbers, the fraction is terminating. In all the other cases, it is non terminating.