Convert the following decimals into `p//q` form.
`108.bar(001)`

Answer

To convert the decimal \( 108.\overline{001} \) into the form \( \frac{p}{q} \), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 108.\overline{001} \). This means:
\[ x = 108.001001001\ldots \]

### Step 2: Eliminate the repeating part
Since the repeating part "001" has 3 digits, we multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right:
\[ 1000x = 108001.001001001\ldots \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 108.001001001\ldots \) (Equation 1)
2. \( 1000x = 108001.001001001\ldots \) (Equation 2)

### Step 4: Subtract the two equations
Now, we subtract Equation 1 from Equation 2 to eliminate the repeating part:
\[ 1000x - x = 108001.001001001\ldots - 108.001001001\ldots \]
This simplifies to:
\[ 999x = 108001 - 108 \]

### Step 5: Calculate the right side
Now, we perform the subtraction:
\[ 108001 - 108 = 107893 \]
So we have:
\[ 999x = 107893 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{107893}{999} \]

### Final Result
Thus, the decimal \( 108.\overline{001} \) in the form \( \frac{p}{q} \) is:
\[ \frac{107893}{999} \]

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