The rational number which equals the number `2. 357 ` with recurring decimal is `(2355)/(1001)` b. `(2379)/(997)` c. `(2355)/(999)` d. none of these

Options

A. `(25)/(9)`
B. `(25)/(3)`
C. `(25)/(18)`
D. `(10)/(3)`

Explanation

To convert the recurring decimal \(2.357\) (where \(357\) is the repeating part) into a rational number, we can follow these steps:

### Step 1: Define the recurring decimal as a variable
Let \(X = 2.357357357...\)

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

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

### Step 4: Subtract the first equation from the second
Subtract the first equation from the second:
\[ 1000X - X = 2357.357357357... - 2.357357357... \]
This simplifies to:
\[ 999X = 2355 \]

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

### Step 6: Identify the answer
The rational number that equals the recurring decimal \(2.357\) is:
\[ \frac{2355}{999} \]

### Conclusion
Thus, the correct option is:
**c. \(\frac{2355}{999}\)**

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