Express 0.\overline{212}_{30} as a base 10 fraction in reduced form.

Answer 1

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.