Question

What is the rational number equivalent to 3.12121212121212...?

Answer 1

take the 12, can put it on top with two 9s. so 3 12/99

Answer 2

if it was 3.123 it would be 3 123/999

Answer 3

Thanks

Answer 4

in other words \[3 \frac{ 12 }{ 99 }\]

Answer 5

\[\begin{align*}3.\overline{12}&=3+\frac{12}{100}+\frac{12}{10000}+\frac{12}{1000000}+\cdots\\[1ex]
&=3+12\sum_{i=1}^\infty\left(\frac{1}{100}\right)^i\\[1ex]
&=3+\frac{\frac{12}{100}}{1-\frac{1}{100}}\\[1ex]
&=3+\frac{12}{99}\\[1ex]
&=\frac{209}{99}\\[1ex]
&=\frac{103}{33}
\end{align*}\]Another way to get the same answer:
\[\begin{align*}
3.\overline{12}&=x\\[1ex]
312.\overline{12}&=100x\\[1ex]
209&=99x\\
x&=\frac{103}{33}
\end{align*}\]