Convert the following repeating decimal to a fraction in simplest form. 0.01 (01 is repeated)

Answer 1

To convert the repeating decimal $0.01\overline{01}$ (where $01$ is the repeating part) into a fraction, we can use the following steps:

Let $x=0.01\overline{01}$.

This means: $$x=0.0101010101\dots$$

To eliminate the repeating decimals, we can multiply $x$ by 10000 (since there are 4 digits in each full cycle of the repetition): $$10000x=101.0101010101\dots$$

Now, we have two equations:

1. $x=0.0101010101\dots$ (the original equation)
2. $10000x=101.0101010101\dots$ (the equation after multiplying by 10000)

Next, we subtract the first equation from the second: $$10000x-x=101.0101010101\dots -0.0101010101\dots$$ This simplifies to: $$9999x=101$$

Now, solve for $x$: $$x=\frac{101}{9999}$$

To simplify this fraction, we can check if 101 and 9999 have any common factors. Since 101 is a prime number, we can check if 9999 is divisible by 101. Dividing $9999$ by $101$: $$9999÷101\approx 99$$ Calculating $101\times 99=9999$, which confirms $9999=101\times 99$.

Thus, we can simplify: $$\frac{101}{9999}=\frac{1}{99}$$

Therefore, the simplest form of the fraction that represents the repeating decimal $0.01\overline{01}$ is: $$\boxed{{\displaystyle \frac{1}{99}}}$$