`0.bar(16)=?`

Options

A. `16/99`
B. `16/90`
C. `4/25`
D. None of these

Answer

To solve the question \(0.\overline{16}\), we can follow these steps:

### Step 1: Understand the notation
The notation \(0.\overline{16}\) means that the digits "16" repeat indefinitely. This can be expressed as \(0.161616...\).

### Step 2: Set up the equation
Let \(x = 0.\overline{16}\). This means:
\[ x = 0.161616... \]

### Step 3: Eliminate the repeating decimal
To eliminate the repeating part, we can multiply \(x\) by 100 (since "16" has 2 digits):
\[ 100x = 16.161616... \]

### Step 4: Set up the equation to solve for \(x\)
Now we have two equations:
1. \(x = 0.161616...\)
2. \(100x = 16.161616...\)

We can subtract the first equation from the second:
\[ 100x - x = 16.161616... - 0.161616... \]
This simplifies to:
\[ 99x = 16 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 99:
\[ x = \frac{16}{99} \]

### Conclusion
Thus, the value of \(0.\overline{16}\) is:
\[ 0.\overline{16} = \frac{16}{99} \]