Question

$xy$ is a number that is divided by $ab$ where$xy<ab$ and gives a result $0.xyxyxy...$then$ab$equals to

• A. $11$
• B. $33$
• C. $99$
• D. $66$

Answer 1

The correct option is C

$99$

Solution:

Step 1: Convert $0.xyxyxyxyxy\dots$in expanded form:

$0.xyxyxyxyxy\dots$
$$\begin{gathered} =\left(\frac{xy}{100}\right)+\left(\frac{xy}{10000}\right)+\left(\frac{xy}{1000000}\right)+... \\ =\left(\frac{xy}{100}\right)\times \left(1+\frac{1}{100}+\frac{1}{1000000}+...\right)[Taking\frac{xy}{100}common] \end{gathered}$$
Step 2: Solve the expression:

Here, the second part is a geometric series with the first term and $1$ and the common ratio as $\frac{1}{100}$.
$$\begin{gathered} \Rightarrow \frac{xy}{ab}=\left(\frac{xy}{100}\right)\times \left(\frac{1}{1-{\displaystyle \frac{1}{100}}})\right)\left[\because \mathrm{Sum}\mathrm{of}\mathrm{an}\mathrm{infinite}\mathrm{GP}=\frac{a}{1-r}\right] \\ \Rightarrow \frac{xy}{ab}=\left(\frac{xy}{100}\right)\times \left(\frac{100}{99}\right) \\ \Rightarrow \frac{xy}{ab}=\frac{xy}{99} \end{gathered}$$

Step 3: Compare the value:

As, $\frac{xy}{ab}=0.xyxyxyxy\dots =\frac{xy}{99}$
Therefore,$ab=\mathbf{99}$

Final answer: Hence, the correct option is C.