The value of `2.bar(357)`, is

Options

A. `(2355)/(1001)`
B. `(2355)/(999)`
C. `(2355)/(1111)`
D. `(2354)/(1111)`

Answer

To find the value of \(2.\overline{357}\), we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 2.\overline{357} \). This means:
\[ x = 2.357357357\ldots \]

### Step 2: Multiply by a power of 10
Since the repeating part has 3 digits (357), we multiply \( x \) by \( 1000 \) to shift the decimal point three places to the right:
\[ 1000x = 2357.357357357\ldots \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 2.357357357\ldots \) (Equation 1)
2. \( 1000x = 2357.357357357\ldots \) (Equation 2)

### Step 4: Subtract the first equation from the second
Now, subtract Equation 1 from Equation 2:
\[ 1000x - x = 2357.357357357\ldots - 2.357357357\ldots \]
This simplifies to:
\[ 999x = 2357 - 2 \]
\[ 999x = 2355 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 999 to solve for \( x \):
\[ x = \frac{2355}{999} \]

### Step 6: Simplify the fraction
To simplify \( \frac{2355}{999} \), we can find the greatest common divisor (GCD) of 2355 and 999. After simplification, we find:
\[ x = \frac{2355 \div 3}{999 \div 3} = \frac{785}{333} \]

Thus, the value of \( 2.\overline{357} \) is:
\[ \frac{785}{333} \]