0.21... = 21/99
I'm sure you can reduce that as needed
Not sure what the _3 contributes ...
Express $0.\overline{21}_3$ as a base 10 fraction in reduced form.
5 answers
or was that 0.21... a base-3 number?
Well, that would of course be 21/223 = 7/810
Well, that would of course be 21/223 = 7/810
no its 7/8
its 7/8 !
To get the answer quickly we can use a geometric series .21212121... = 7/9+7/81+7/729..., = 7/8
A more simply but longer method is to make .21212121..._3 equal to X, and then multiply both sides of the equation by 100_3. This makes 100_3*X=21.212121..._3, so 100_3*X=21+X, because 0.212121..._3 = X. Then, we have 22_3X=21_3, after subtracting one equation from the other. So, X= (21_3)/(22_3). In base 10, this is (2*3+1)/(2*3+2)= 7/8
*Note that number_3 is that number is base 3
A more simply but longer method is to make .21212121..._3 equal to X, and then multiply both sides of the equation by 100_3. This makes 100_3*X=21.212121..._3, so 100_3*X=21+X, because 0.212121..._3 = X. Then, we have 22_3X=21_3, after subtracting one equation from the other. So, X= (21_3)/(22_3). In base 10, this is (2*3+1)/(2*3+2)= 7/8
*Note that number_3 is that number is base 3
6 answers