Question

How do you convert \[0.\overline{39}\] (\[39\] repeating) to a fraction?

Answer 1

Hint: We will first denote the recurring decimal by a variable and form an equation. Then we will check the number of recurring digits and multiply by a suitable power of \[10\] to get the second equation. Finally, we will find the difference between the resulting equations to get the required fraction.

Complete step by step solution:
Let us denote the given recurring decimal by a variable \[x\]. So,
\[x = 0.3939...\] ……….\[(1)\]
In the decimal part of \[0.3939...\], there are two digits that are recurring i.e., 3 and 9. In this case, we will multiply both sides of equation \[(1)\] by 100 to get a new recurring decimal. Thus, we get,
\[100x = 39.3939...\] ............ \[(2)\]
Now, we will subtract equation \[(1)\] from equation \[(2)\]. In doing so, we will subtract the LHS of both equations on the LHS, and the RHS of both equations on the RHS. Hence,
\[100x - x = 39.3939... - 0.3939...\]
Subtracting the terms, we get
\[99x = 39\]
Dividing both sides by 90, we get
\[x = \dfrac{{39}}{{99}}\] ……….\[(3)\]
We observe from equation \[(3)\] that on the RHS, there is a common factor in the numerator and the denominator, which is 3.
Dividing the numerator and denominator by 3, we get
\[x = \dfrac{{13}}{{33}}\] ………\[(4)\]
We see from equation \[(1)\] and equation \[(4)\] that the LHS is the same. So,
\[x = 0.3939... = \dfrac{{13}}{{33}}\]

Therefore, the recurring decimal \[0.3939...\] is represented as \[\dfrac{{13}}{{33}}\] in the fractional form.

Note:
While solving this problem, we must carefully place the decimal point right before the repeating digits. If the number of recurring digits is \[n\], then the recurring decimal should be multiplied by \[{10^n}\]. In the above problem, there were two recurring digits. Hence, we multiplied the recurring decimal by \[{10^2} = 100\]. Here, we can make a mistake by just multiplying and dividing the given number by 100 and convert it into a fraction. This will give us the wrong answer because the given number has two recurring digits, so it needs to be solved accordingly.