Question

What is the repeating decimal of $\dfrac{2}{3}$ ?

Answer 1

Hint: To solve this question, we will first divide 2 by 3 using a long division method and obtain the quotient value. Then we will observe the pattern in the quotient to obtain the repeating decimal.
A repeating decimal, also known as a recurring decimal is a number in which the decimal representation becomes periodic.i.e., it repeats in a particular sequence.
A repeating decimal is represented by a vinculum
For example,
$\dfrac{1}{7}=0.\overline{142857}$

Complete step by step solution:
Firstly, we will divide 2 by 3 using long division method,
$3\overset{0.66}{\overline{\left){\begin{align}
  & 20 \\
 & \underline{18} \\
 & \text{ }20 \\
 & \text{ }\underline{\text{18}} \\
 & \text{ 2} \\
\end{align}}\right.}}$
On dividing 2 by 3, since the dividend is smaller than the divisor, we will add a decimal point in the quotient. This would add a zero to the dividend and the number formed would be 20. Now, the value nearest to 20 in the table of 3 is 18, which is obtained by multiplying 3 by 6. So, the quotient will contain 6, and on subtracting 18 from 20, we will obtain 2 as a remainder.
On continuing the process further, we find that the process goes in repeating and we get the remainder 2 continuously. So, we stop here and observe the obtained quotient.
Here, the quotient is $0.66...$ and the pattern goes on repeating. So, we can write it as $0.\overline{6}$.

Hence, the repeating decimal for $\dfrac{2}{3}$ is $0.\overline{6}$.

Note: Usually, whenever we obtain a repeating decimal, we write it in approximate form. You would find that most of the time $\dfrac{2}{3}$ is written as $0.67$ or $0.667$ . So, this is done just to avoid repetition and make our calculations easier and convenient.