If you mean
3.4444... base 13, that would be
3 4/12 = 10/3
In base n, the fraction
.1111111... is 1/(n-1)
Express $3.\overline{4}_{13}$ as a base 10 fraction in reduced form.
6 answers
EDHIFHISDH
1/(n-1) is not correct..
3 1/3 maybe?
10/3 is correct. We can solve this with the geometric series, with $3 + \frac{4}{13} + \frac{4}{169} + \dots = 3 + 4 \cdot \frac{1}{12} = 3 + \frac{1}{3} = \frac{10}{3}$.
We can also use the methods we learned in class.
We can let $x = 3.4444\ldots_{13}.$
Since the repeat is one digit long, we multiply both sides of this equation by $10_{13}$, so we have $10_{13}x = 34.444\ldots_{13}$.
As we did before, we have
\begin{align*}
10_{13}x &= \phantom{-}34.444\ldots_{13} \\
- \phantom{10_{13}}x &= -\phantom{0}3.444\ldots_{13} \\
\hline\\
C_{13}x &= 31_{13} \\
\end{align*}
Thus, we have $x = \frac{31_{13}}{C_{13}}$. Converting this fraction to base 10, we get $\frac{3\cdot 13 + 1}{12} = \frac{40}{12} = \frac{10}{3}.$
We can also use the methods we learned in class.
We can let $x = 3.4444\ldots_{13}.$
Since the repeat is one digit long, we multiply both sides of this equation by $10_{13}$, so we have $10_{13}x = 34.444\ldots_{13}$.
As we did before, we have
\begin{align*}
10_{13}x &= \phantom{-}34.444\ldots_{13} \\
- \phantom{10_{13}}x &= -\phantom{0}3.444\ldots_{13} \\
\hline\\
C_{13}x &= 31_{13} \\
\end{align*}
Thus, we have $x = \frac{31_{13}}{C_{13}}$. Converting this fraction to base 10, we get $\frac{3\cdot 13 + 1}{12} = \frac{40}{12} = \frac{10}{3}.$
The answer is 10/3
5 answers