Question

If $u$,$v$ and $w$ are the digits of decimal system, then the rational number represented by $0.uwuvuvuvuv.....$ is
(A)$\dfrac{{(100uw + 99uv)}}{{99}}$ (B) $\dfrac{{(99uw + uv)}}{{980}}$ (C)$\dfrac{{(99uw + uv)}}{{9900}}$ (D)$\dfrac{{(9uw + 99uv)}}{{900}}$

Answer 1

Hint-When converting repeating decimals to fractions, we just need to follow the five steps below carefully
-> First place the non-repeating digits after the decimal point to before the decimal point and name this equation.
->Then place the repeating digits to before the decimal point and name the equation
Using the two equations you found in step 3 and step 4, subtract both sides of the two equations.
Now, by simplifying the equation found in step-5, get the value of $x$.


Complete step by step solution-
Let $x = 0.uwuvuvuvuv.....$
We have to represent $x$ in the rational number form that is $\dfrac{p}{q}$ form, where $p$ and $q$ are the integer and $q \ne 0$.
Now we have to determine the number of repeating digits. Here the given decimal number has two non-repeating digits and two repeating digits after the decimal point.
 So first multiply by $100$ to shift the non-repeating digits to before the decimal point and then again multiply by $100$ to shift the one group of repeating digits to before the decimal point.
$100 \times x = 100 \times 0.uwuvuvuv...$
$ \Rightarrow 100x = uw.uvuvuv....$ equation (1)
$ \Rightarrow 10000x = uwuv.uvuv....$ equation (2)
On subtracting equation (1) from equation (2),
$ \Rightarrow 9900x = uwuv - uw$ equation (3)
As we know that $uwuv$is a four-digit number where $u$,$v$ and $w$ are the digits of decimal system, we can expand this number in many ways and one of them is
$uwuv = (100 \times uw) + uv$
On substituting this value in equation (3),
$ \Rightarrow 9900x = (100 \times uw) + uv - uw$
$ \Rightarrow 9900x = 99uw + uv$
On dividing by $9900$both sides
$ \Rightarrow x = \dfrac{{99uw + uv}}{{9900}}$
Hence the rational number represented by $0.uwuvuvuvuv.....$ is $\dfrac{{99uw + uv}}{{9900}}$.
Answer- C


Note:Errors can be avoided by keeping in mind some points like do not forget to put the decimal point right before the repeating digits and keep the good track of the number of places the decimal point should be moved.