Express $0.\overline{21}_3$ as a base 10 fraction in reduced form.
or was that 0.21... a base-3 number?
Well, that would of course be 21/223 = 7/810
its 7/8 !
no 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
Well, well, well, let's have some math fun, shall we? To convert this pesky number into base 10, we need to apply some clown tricks!
First, let's call the number $x = 0.\overline{21}_3$. Now, let's multiply both sides of this equation by the base 3, which is 3, to get rid of the repeating part.
So, we have: $3x = 2.\overline{21}_3$. Now, let's subtract the original equation from this new one to eliminate the repeating part.
That gives us the equation: $3x - x = 2.\overline{21}_3 - 0.\overline{21}_3$. On the left side, we have $2x$. On the right side, the repeating part cancels out, leaving us with 2.
Therefore, $2x = 2$. If we divide both sides by 2, we find that $x = 1$. So, $0.\overline{21}_3$ in base 10 is equal to 1.
Now, don't be a sad clown, we've simplified the fraction as much as we can. And remember, math can be silly too!
To express the repeating decimal $0.\overline{21}_3$ as a base 10 fraction, we can use a simple method.
Let's denote $0.\overline{21}_3$ as $x$.
To remove the repeating decimal part, we can multiply both sides of the equation by the base 3 raised to the number of decimal places. Since there are two repeating decimal places after the decimal point, we multiply both sides by $3^2$:
$3^2 \cdot x = 0.\overline{21}_3 \cdot 3^2$
On the left side, $3^2 \cdot x$ is equal to $9x$.
On the right side, multiplying $0.\overline{21}_3$ by $3^2$ gives us $21.\overline{21}_3$.
Now we have:
$9x = 21.\overline{21}_3$
Next, we subtract $0.\overline{21}_3$ from both sides to isolate the repeating decimal part:
$9x - 0.\overline{21}_3 = 21.\overline{21}_3 - 0.\overline{21}_3$
Simplifying, we get:
$9x = 21$
To find $x$, we divide both sides by 9:
$\frac{9x}{9} = \frac{21}{9}$
This simplifies to:
$x = \frac{7}{3}$
Therefore, $0.\overline{21}_3$ as a base 10 fraction in reduced form is $\frac{7}{3}$.
0.21... = 21/99
I'm sure you can reduce that as needed
Not sure what the _3 contributes ...