Express `0.2bar(35)` in the form p/q where p and q are integers and `q ne 0`.

Answer

To express \( 0.2\overline{35} \) in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.2\overline{35} \). This means \( x = 0.235353535...\).

### Step 2: Eliminate the repeating part
To eliminate the repeating decimal, we can multiply \( x \) by a power of 10 that shifts the decimal point to the right, so that the repeating part aligns with itself. Since "35" has 2 digits, we multiply by \( 1000 \) (to account for the "2" before the repeating part and the two digits in the repeating part).

\[ 1000x = 235.353535... \]

### Step 3: Set up a second equation
Now, we also multiply \( x \) by \( 10 \) to shift the decimal point just past the non-repeating part:

\[ 10x = 2.353535... \]

### Step 4: Subtract the two equations
Now we subtract the second equation from the first:

\[ 1000x - 10x = 235.353535... - 2.353535... \]

This simplifies to:

\[ 990x = 233 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \):

\[ x = \frac{233}{990} \]

### Final Result
Thus, \( 0.2\overline{35} \) can be expressed as:

\[ \frac{233}{990} \]