Expression of `1. bar7` as a rational number in the form of `(p)/(q)` is

Options

A. `16/9`
B. `4/3`
C. `2/3`
D. `1/3`

Answer

To express the repeating decimal \( 1.\overline{7} \) as a rational number in the form of \( \frac{p}{q} \), we can follow these steps:

### Step 1: Let \( x = 1.\overline{7} \)
We start by defining the repeating decimal as a variable:
\[ x = 1.77777\ldots \]

### Step 2: Multiply both sides by 10
To eliminate the repeating part, we multiply both sides of the equation by 10:
\[ 10x = 17.77777\ldots \]

### Step 3: Set up the equation
Now we have two equations:
1. \( x = 1.77777\ldots \)
2. \( 10x = 17.77777\ldots \)

### Step 4: Subtract the first equation from the second
Subtract the first equation from the second to eliminate the repeating decimal:
\[ 10x - x = 17.77777\ldots - 1.77777\ldots \]
This simplifies to:
\[ 9x = 16 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 9 to solve for \( x \):
\[ x = \frac{16}{9} \]

### Conclusion
Thus, the expression of \( 1.\overline{7} \) as a rational number in the form of \( \frac{p}{q} \) is:
\[ \frac{16}{9} \]