Express $3.\overline{4}_{13}$ as a base 10 fraction in reduced form.
3 1/3 maybe?
The answer is 10/3
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)
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 $\frac{3\cdot 13 + 1}{12} = \frac{40}{12} = \frac{10}{3}.$
To express the repeating decimal $3.\overline{4}_{13}$ as a base 10 fraction in reduced form, we need to understand the base 13 number system.
In base 13, the digits used are 0 to 9, followed by A to C for the tenth, eleventh, and twelfth digits. The value of each digit is determined by its position, with the rightmost digit being multiplied by 13 raised to the power of 0, the next digit to the left being multiplied by 13 raised to the power of 1, and so on.
To convert the repeating decimal to a base 10 fraction, we can set up an equation using algebra. Let's represent $3.\overline{4}_{13}$ as the variable $x$:
$x = 3.44_{13}$
To remove the repeating part, we can multiply both sides of the equation by 13 raised to the number of digits in the repeating part. In this case, there is one digit repeating (the 4), so we multiply by 13:
$13x = 44.\overline{4}_{13}$
Now, let's subtract the original equation from this new equation:
$13x - x = 44.\overline{4}_{13} - 3.\overline{4}_{13}$
Simplifying the right side:
$13x - x = 44_{13} - 3_{13}$
$12x = 41_{13}$
To convert the base 13 numbers back to base 10:
$12x = 4 \cdot 13^1 + 1 \cdot 13^0$
$12x = 52 + 1 = 53$
Dividing both sides by 12 to solve for $x$:
$x = \frac{53}{12}$
Therefore, $3.\overline{4}_{13}$ as a base 10 fraction in reduced form is $\frac{53}{12}$.