Question

If the sum of the repeating decimals 0.353535...+0.252525... is written as a fraction in lowest terms, the product of the numerator and the denominator is:
A. 3465
B. 2475
C. 680
D. 670
E. 660

Answer 1

Hint: We will first write 0.353535... and 0.252525... in the form of $\dfrac{p}{q}$ where $q\ne 0$ separately. For doing that, we will express the first term as x = 0.353535… , then multiply with 100 as 100x = 35.353535 and then express it as 100x = 35 + 0.353535. Now, again this can be written as 100x = 35 + x. Now, we can get x in the fraction form. Similarly, for the second term also we will repeat this process. Then add the fraction form of both of these numbers. The fraction we then obtain will be our required fraction. Then we can multiply its numerator and denominator to get the required answer.

Complete step-by-step solution:
Now, we know that 0.353535... and 0.252525... are rational numbers as they have non-terminating but repeating decimal expansion. Hence, we can write to them in the form of $\dfrac{p}{q}$ where $q\ne 0$.
We will first covert 0.353535... into the form of $\dfrac{p}{q}$
Let 0.353535... be x
Therefore, we can say that:
$x=0.353535...$ ……………......(1)
This decimal expansion continues till infinity. Hence, if we will multiply this number by 100, its decimal expansion will still remain the same.
Therefore, multiplying equation (1) by 100, we get:
$\begin{align}
  & 100x=35.353535... \\
 & \Rightarrow 100x=35+0.353535... \\
\end{align}$
 Now, we know that 0.35353...=x
Thus, the above equation can be written as:
$\begin{align}
  & 100x=35+x \\
 & \Rightarrow 100x-x=35 \\
 & \Rightarrow 99x=35 \\
 & \Rightarrow x=\dfrac{35}{99} \\
\end{align}$
Therefore, 0.353535... can be written in the form of $\dfrac{p}{q}$ as $\dfrac{35}{99}$.
Now, we will convert 0.252525... into the form of $\dfrac{p}{q}$
Let 0.252525... be y
Therefore, we can say that:
$y=0.252525...$ ………………...(2)
This decimal expansion continues till infinity. Hence, if we will multiply this number by 100, its decimal expansion will still remain the same.
Therefore, multiplying equation (2) by 100, we get:
$100y=25.252525...$
Now, we know that 0.252525...=y
Thus the above equation can be written as:
$\begin{align}
  & 100y=25+y \\
 & \Rightarrow 100y-y=25 \\
 & \Rightarrow 99y=25 \\
 & \Rightarrow y=\dfrac{25}{99} \\
\end{align}$
Therefore, 0.252525... can be written in the form of $\dfrac{p}{q}$ as $\dfrac{25}{99}$
Now we will the $\dfrac{p}{q}$ forms of 0.353535.. and 0.252525..
Adding them we get:
$\begin{align}
  & \Rightarrow \dfrac{35}{99}+\dfrac{25}{99} \\
 & \Rightarrow \dfrac{60}{99} \\
 & \Rightarrow \dfrac{20}{33} \\
\end{align}$
Hence, there sum as a fraction in lowest terms is written as $\dfrac{20}{33}$
Thus our numerator is 20 and denominator is 33.
By multiplying the two of them we will get our answer.
Multiplying them we get:
$\begin{align}
  & \Rightarrow \operatorname{numerator}\times \operatorname{denominator} \\
 & \Rightarrow 20\times 33 \\
 & \Rightarrow 660 \\
\end{align}$
Therefore, the required answer is 660.
Thus, option (E) is the correct option.

Note: Here, we have multiplied both these numbers by 100 because the repeating expansion had 2 digits. If it would have had 3 digits, we would have multiplied it by 1000 and if 1 digit then by 10. Basically, to change these types of rational numbers into their fractional form, we multiply them by 10 raised to power the number of digits in the repeating decimal expansion.