Question

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

Answer 1

Hint:
Consider the given relation as $x$. Multiply the given equation with $100$ and again multiply the equation with$100$, After that subtract the obtained equations to get the value of $x$.

Useful Formula: Multiply with $100$ until you get all the three digits $u\,,\,v\,$and $w$ in the equation. Do the normal calculation to find the value of $x$.

Complete step by step solution:
Given that, $u\,,\,v\,$and $w$ are the digits of decimal system, then the rational number represented by $0.uwuvuvuvuv.........$
We want to find the equivalent value of given rational number $0.uwuvuvuvuv.........$
Let us assume that the rational number as $x\, = \,0.uwuvuvuvuv.........$
That is $uwuvuvuvuv$ is in the dot operation of$0$.
$x\, = \,0.uwuvuvuvuv.........\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \to \,(\,1\,)$
Multiply the above equation by $100$ on both the sides:
$(100)x\, = \,(100)0.uwuvuvuvuv.........$
By multiplying with$100$, the result is as follows:
$100\,x\, = \,uw\,.\,uvuvuvuv..........\,\,\,\,\,\,\,\,\,\, \to \,(\,2\,)$
Again, multiply the above equation by $100$ on equation $(\,2\,)$as follows:
$(100)100\,x\, = \,(100)\,uw.uvuvuvuv...........$
By multiplying with$100$, the result is as follows:
$10000\,x\, = \,uwuv.uvuvuvuv.......\,\,\,\,\,\,\,\,\,\,\,\,\, \to (\,3\,)$
Now, subtract the equation $(\,2\,)$ from $(\,3\,)$ as follows:
$
  10000\,x\, = \,uwuv.uvuvuvuv.......\, \\
  \,\,\,\,\,100\,x\, = \,uw.uvuvuvuvuv.......\, \\
  \,\,9900\,x\, = \,uwuv\, - \,uw \\
 $
The above equation obtained by cancelling $100\,x$ from $10000\,x$ to get$9900\,x$. After that subtract $uwuv$ from $uw$.
Now, find the value of $x$ by getting $9900$on right side:
$x\, = \,\dfrac{{uwuv\, - \,uw}}{{9900}}$
This is the obtained value of $x$.
Now simplify the value of $x$ as follows:
$x\, = \,\dfrac{{uwuv\, - \,uw}}{{9900}}$
Now, split the $uwuv$ as $uw\, \times \,100\, + \,uv$ based on equation$(\,2\,)$:
$x\, = \,\dfrac{{uw\, \times \,100\, + \,uv\, - \,uw}}{{9900}}$
Thus, $uw$ should be subtracted from the $uw\, \times \,100\,$.
Because $uw\, \times \,100\,$ is the multiple of 100 and $uw$ is the multiple of$1$. Subtracting $uw$ from $uw\, \times \,100\,$becomes $uw$ with the multiple of$99$.
Thus, the value of $x$ becomes as follows:
$x\, = \,\dfrac{{99\,uw\, + \,uv}}{{9900}}$
The above equation is the final answer for the given equation:

Thus, the option $C\,)\,\dfrac{{99\,uw\, + \,uv}}{{9900}}$ is the correct answer for equivalent value of given rational number $0.uwuvuvuvuv.........$

Note:
Multiply the given equation by $100$and check the result if the result contains all the three parts on the right side of it. If not again multiply the equation with$100$ and again check the result. If all the three parts are contained in the result equation. After finding the value of $x$ simplify the equation based on the following condition $1225\, = \,1200\, + \,25\,$that is $uwuv\, = \,uw\, \times \,100\, + \,uv$.