 QUESTION

# The rational form of $2.74\overline{35}$ is a.$\dfrac{27161}{9999}$b.$\dfrac{27}{99}$c.$\dfrac{27161}{9900}$d.$\dfrac{27161}{9000}$

Hint: Here, in the number $2.74\overline{35}$, the bar is placed above two numbers 3 and 5 which means they keep on repeating. Now, take $x=2.74\overline{35}$. To convert it into a rational number first we have to multiply by 100, we will get $100x=274.\overline{35}$. Now, again multiplied by 100 to take the numbers 3 and 5 to the left side, we get $10000x=27435.\overline{35}$. From the two equations we can solve for $x$ to get the rational number.

We are given the number $2.74\overline{35}$. Now, we have to find the rational form of $2.74\overline{35}$
Here, the bar is placed above the numbers which are repeating.

We know that the rational numbers can be converted to both terminating and non-terminating decimal numbers.
The non-terminating decimal numbers are classified as recurring decimal numbers and non-recurring decimal numbers.
The recurring decimal numbers are those numbers which keep on repeating the same value after decimal point. These numbers are also known as repeating decimals.
So here, the number $2.74\overline{35}$ is a recurring decimal number. Here, the bar is above two numbers 3 and 5 which means they keep on repeating.

First let us take $x=2.74\overline{35}$.
Since, the numbers 3 and 5 are repeating which means the bar is above two digits, so we have to multiply the number by 100. Thus we obtain:
$100x=274.\overline{35}$
Next, we have to bring the repeating numbers 3 and 5 to the left side, so we have to again multiply the above equation by 100. Hence, we get:
\begin{align} & 100x\times 100=274.\overline{35}\times 100 \\ & 10000x=27435.\overline{35} \\ \end{align}

Consider the two equations:
$10000x=27435.\overline{35}$ …… (1)
$100x=274.\overline{35}$ …… (2)
In the next step, we have to do subtraction. That is, equation (1) - equation (2) we obtain:
\begin{align} & 10000x-100x=27435.\overline{35}-274.\overline{35} \\ & 9900x=27161 \\ \end{align}
Next, by cross multiplication we get:
$x=\dfrac{27161}{9900}$
Therefore, the required rational number is $\dfrac{27161}{9900}$.
Hence, the correct answer for the question is option (d).

Note: Here, first we have to identify how many digits are repeating which you can find with the help of the bar symbol. Depending on the numbers which keep on repeating we have to do the multiplication. Here, there are two numbers, so we are multiplying by 100. If you don’t multiply then you can’t take the repeating decimals to the left side which will then be difficult to remove the decimal places.