Question

What is 0.583333333……. (repeating 3) as a fraction?

Answer 1

Hint: For solving this question you should know about the calculating of repeated 3’s. And it is done by a simple step of multiply by \[{{10}^{n}}\] and here the n is the no. of repeated n’s which we want to calculate and then subtract both from one another and then simplify you can get the value of x as a fraction form.

Complete step by step answer:
So, according to our question we have to calculate the fraction of (repeating 3’s) of 0.5833333……. Here are different ways to solve this question and the easiest and accurate way where no chance of mistakes are possible is method -1 because that is a much accurate method.
Let \[x=0.583333333.....\]
So, here we have to calculate the repeated 3’s.
So, we multiply x with 1000 and 100.
Therefore, by multiplying with 1000 and 100, we get
\[\begin{align}
  & 1000x=583.3\overline{3}\,-\left( 1 \right) \\
 & 100x=58.3\overline{3}\,-\left( 2 \right) \\
\end{align}\]
Now for getting the value of x, subtract equation (2) from equation (1).
\[\begin{align}
  & \Rightarrow 900x=525 \\
 & \Rightarrow x=\dfrac{525}{900} \\
\end{align}\]
By solving this,
\[\begin{align}
  & x=\dfrac{25\times 21}{25\times 36}=\dfrac{21}{36} \\
 & x=\dfrac{7}{12} \\
\end{align}\]
Or we can also solve this by another way:
We know that \[0.\overline{3}=\dfrac{1}{3}\],
Now subtract first:
\[0.58\overline{3}-0.\overline{3}=0.58\overline{3}-0.33\overline{3}=0.25=\dfrac{1}{4}\]
So, \[0.58\overline{3}=\dfrac{1}{3}+\dfrac{1}{4}=\dfrac{4}{12}+\dfrac{3}{12}=\dfrac{7}{12}\]

Note: While solving this question, the possible mistake one can make is not multiplying \[x=0.583333333.....\] by 100 and then ending up with the wrong answer. Also, we need to remember that while solving such types of questions, we will first keep only repeating digits after decimal and then solving.