$ ext{The rational form of } 0.\overline{001} ext{ is: }$

Q: $\text{The rational form of } 0.\overline{001} \text{ is: }$

To determine the rational form of the repeating decimal $0.\overline{001}$, we need to convert it into a fraction. The repeating part of the decimal is "001", which repeats indefinitely. Let's denote $x = 0.\overline{001}$. 1. **Multiply by 1000**: Since the repeating block "001" has three digits, we multiply $x$ by 1000 to shift the decimal point three places to the right: \[ 1000x = 1.001001001\ldots \] 2. **Subtract the original equation**: Subtract the original $x = 0.001001001\ldots$ from this new equation: \[ 1000x - x = 1.001001001\ldots - 0.001001001\ldots \] \[ 999x = 1 \] 3. **Solve for $x$**: Divide both sides by 999 to isolate $x$: \[ x = \frac{1}{999} \] Thus, the rational form of $0.\overline{001}$ is $\frac{1}{999}$, which corresponds to Option 3. ### Explanation for Other Options: - **Option 1: $\frac{1}{99}$**: This fraction represents a repeating decimal of $0.\overline{01}$, not $0.\overline{001}$. The repeating block "01" is only two digits long, which is different from the three-digit repeating block "001". - **Option 2: $\frac{100}{99}$**: This fraction simplifies to a repeating decimal of $1.0\overline{1}$, which is not equivalent to $0.\overline{001}$. The repeating part here is "1", not "001". - **Option 4: $\frac{100}{999}$**: This fraction simplifies to a repeating decimal of $0.\overline{100}$, which is not the same as $0.\overline{001}$. The repeating block "100" is different from "001". Therefore, the correct rational form of $0.\overline{001}$ is indeed $\frac{1}{999}$, making Option 3 the correct choice.

Question

$	ext{The rational form of } 0.\overline{001} 	ext{ is: }$

Accepted Answer

To determine the rational form of the repeating decimal $0.\overline{001}$, we need to convert it into a fraction. The repeating part of the decimal is "001", which repeats indefinitely.

Let's denote $x = 0.\overline{001}$.

1. **Multiply by 1000**: Since the repeating block "001" has three digits, we multiply $x$ by 1000 to shift the decimal point three places to the right:
   $$
   1000x = 1.001001001\ldots
   $$

2. **Subtract the original equation**: Subtract the original $x = 0.001001001\ldots$ from this new equation:
   $$
   1000x - x = 1.001001001\ldots - 0.001001001\ldots
   $$
   $$
   999x = 1
   $$

3. **Solve for $x$**: Divide both sides by 999 to isolate $x$:
   $$
   x = \frac{1}{999}
   $$

Thus, the rational form of $0.\overline{001}$ is $\frac{1}{999}$, which corresponds to Option 3.

### Explanation for Other Options:

- **Option 1: $\frac{1}{99}$**: This fraction represents a repeating decimal of $0.\overline{01}$, not $0.\overline{001}$. The repeating block "01" is only two digits long, which is different from the three-digit repeating block "001".

- **Option 2: $\frac{100}{99}$**: This fraction simplifies to a repeating decimal of $1.0\overline{1}$, which is not equivalent to $0.\overline{001}$. The repeating part here is "1", not "001".

- **Option 4: $\frac{100}{999}$**: This fraction simplifies to a repeating decimal of $0.\overline{100}$, which is not the same as $0.\overline{001}$. The repeating block "100" is different from "001".

Therefore, the correct rational form of $0.\overline{001}$ is indeed $\frac{1}{999}$, making Option 3 the correct choice.