Question

Convert \[1.3\bar{4}\bar{5}\] into fraction.

Answer 1

Hint: To solve this problem, we first assume \[1.3\bar{4}\bar{5}\] as a variable x. Since, x is recurring in two decimal places and after one decimal place, we multiply it with $1000$ and then with $10$ . After that, we subtract the two equations and eliminate the repeating part. The fraction that we get is \[\dfrac{1332}{990}\] . Dividing the numerator and denominator by $18$ , we receive the actual answer.

Complete step by step answer:
To convert the given number to a fraction form, we will first assume the number \[1.3\bar{4}\bar{5}\] as a variable x.
$\Rightarrow x=1.3\bar{4}\bar{5}$
It is given that $45$ is repeated which means that two decimal places after $3$ are repeated. Thus, we will first multiply it with $10$ .
$\Rightarrow 10x=13.\bar{4}\bar{5}$
Now, we will multiply with $1000$ .
$\Rightarrow 1000x=1345.\bar{4}\bar{5}$
Now, we will subtract the two equations.
$\begin{align}
  & \Rightarrow 1000x-10x=1345.\bar{4}\bar{5}-13.\bar{4}\bar{5} \\
 & \Rightarrow 990x=1332 \\
\end{align}$
We will now divide both of the sides by $990$ to obtain the fraction form.
\[\Rightarrow x=\dfrac{1332}{990}\]
We can see that both numerator and denominator are divisible by $18$ . So, dividing, we get,
 \[\Rightarrow x=\dfrac{74}{55}\]
Thus, we can conclude that the fraction form of \[1.3\bar{4}\bar{5}\] is \[\dfrac{74}{55}\] .

Note: As we have seen in this problem, all recurring decimals can be converted into fractions. To find these fractions, we need to generate two equations which have the same repeating part and subtract one from the other to eliminate it. For this problem, we tend to make a very common mistake. We try to subtract x from $1000x$ . This subtraction does not eliminate the repeating parts and we are left stuck with the problem. So, we should be careful while generating the equations.