To express 0.\overline{212}_{30} as a base 10 fraction in reduced form, we can set x = 0.\overline{212}_{30} and use the properties of repeating decimals to solve for x.
First, we can represent 0.\overline{212}_{30} as an infinite geometric series:
x = 0.212212212..._{30}
x = 0.2123_{30} + 0.00002123_{30} + 0.00000002123_{30} + ...
We can convert each term in this series to base 10:
x = \frac{2}{30} + \frac{1}{30^2} + \frac{2}{30^3} + ...
Now, we can use the formula for the sum of an infinite geometric series:
S = a / (1 - r)
where a is the first term and r is the common ratio. In this case, a = 2/30 and r = 1/30.
Plugging in the values, we get:
x = \frac{\frac{2}{30}}{1 - \frac{1}{30}}
x = \frac{\frac{2}{30}}{\frac{29}{30}}
x = \frac{2}{29}
Therefore, 0.\overline{212}_{30} as a base 10 fraction in reduced form is 2/29.
Express 0.\overline{212}_{30} as a base 10 fraction in reduced form.
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