Question

How do you convert \[0.\overline{237}\] to a fraction?

Answer 1

Hint: In order to find the solution to this question, we will start it by assuming the given term as x and then we will multiply both sides by 1000, and then we will subtract the obtained equation from the assumed equation and then by necessary calculations, we will get our answer.

Complete step by step answer:
According to the question, we have been asked to convert \[0.\overline{237}\] into fractions.
To convert a repeating, non-terminating decimal value to a fraction, we need to start by assuming the desired fraction as a variable, say \[x\]. Therefore, we can say
\[x=0.\overline{237}\] ----(1)
Now, we will multiply both sides of equation (1) by 1000 because three of the digits are repeating. Therefore, we get
\[1000\times x=0.\overline{237}\times 1000\]
Now, we know that \[0.\overline{237}\] is the same as \[0.237\overline{237}\]. Therefore, we can write the above equation as
\[\Rightarrow 1000x=237.\overline{237}\] -----(2)
Now, we will subtract equation (1) from equation (2). Therefore, we get
\[\Rightarrow 1000x-x=237.\overline{237}-0.\overline{237}\]
Now, we can see that \[\overline{237}\] is the repeating part. Therefore, on subtraction, we can remove this part. Hence, we get
\[\Rightarrow 1000x-x=237\]
And to simplify it further we will take x common from both the terms of the left-hand side. Therefore, we get
\[\Rightarrow \left( 1000-1 \right)x=237\]
And we know that the above equation can also be written as
\[\Rightarrow 999x=237\]
Now, we will divide both sides of the equation by 999. So, we can write
\[\Rightarrow \dfrac{999x}{999}=\dfrac{237}{999}\]
Now, we know that common terms of numerator and denominator of the same fraction cancel out. Therefore, we get
\[\Rightarrow x=\dfrac{237}{999}\]
And on further simplifications, we get
\[\therefore x=\dfrac{79}{333}\]
Thus, we get the value of \[0.\overline{237}\] in fractions as \[\dfrac{79}{333}\], which is our required answer.

Note:
In these types of questions, the most common mistake is to multiply both sides of the equation by the wrong multiple of 10. One should write the equation and if required, write the terms completely, so as to visualize what all digits are subtracted. This approach can help to reduce the chances of making mistakes.