Convert the following decimals into `p//q` form. `(p,q in Z)`
`281.bar(31)`

Answer

To convert the repeating decimal \( 281.\overline{31} \) into the form \( \frac{p}{q} \), we can follow these steps:

### Step 1: Define the repeating decimal as a variable
Let \( x = 281.\overline{31} \).

### Step 2: Eliminate the repeating part
To eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part "31" has 2 digits):
\[ 100x = 28131.\overline{31} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 281.\overline{31} \) (Equation 1)
2. \( 100x = 28131.\overline{31} \) (Equation 2)

### Step 4: Subtract the two equations
Now, subtract Equation 1 from Equation 2:
\[ 100x - x = 28131.\overline{31} - 281.\overline{31} \]
This simplifies to:
\[ 99x = 28131 - 281 \]

### Step 5: Calculate the right side
Now, calculate \( 28131 - 281 \):
\[ 28131 - 281 = 27850 \]
So, we have:
\[ 99x = 27850 \]

### Step 6: Solve for \( x \)
Now, divide both sides by 99 to solve for \( x \):
\[ x = \frac{27850}{99} \]

### Step 7: Simplify the fraction (if needed)
The fraction \( \frac{27850}{99} \) is already in its simplest form since 27850 and 99 have no common factors other than 1.

### Final Result
Thus, the decimal \( 281.\overline{31} \) can be expressed in the form \( \frac{p}{q} \) as:
\[ \frac{27850}{99} \]