Express the number as the ratio of two integers.
-$0.3131313 \ldots$
A) $\frac { 31 } { 99 }$
B) $\frac { 62 } { 99 }$
C) $\frac { 310 } { 99 }$
D) $\frac { 31 } { 33 }$

Question

Quizplus · Accepted Answer

Notice that $0.3131313 \ldots$ is a repeating decimal with a period of $2$. Thus, we can write $0.3131313 \ldots$ as $0.\overline{31}$. Let $x=0.\overline{31}$. Multiplying by $100$, we have $100x=31.\overline{31}$. Subtracting the first equation from the second equation, we get \begin{align*}
100x-x &= 31.\overline{31}-0.\overline{31}\
99x &=31\
x&=\frac{31}{99}.
\end{align*} Thus, $0.\overline{31}=\boxed{\frac{31}{99}}$.

Express the Number as the Ratio of Two Integers 0.3131313…0.3131313 \ldots0.3131313…

Express the Number as the Ratio of Two Integers $0.3131313 \ldots$