What is the fraction form of the following decimal expansion?

Question

1.overline 7

Answer 1

To convert the decimal \( 1.\overline{7} \) (which represents \( 1.77777\ldots \)) into a fraction, we can follow these steps:

1. Let \( x = 1.\overline{7} \).

2. This implies that \( x = 1.77777\ldots \).

3. To isolate the repeating part, we multiply \( x \) by 10 (since the repeating part is one digit long): \[ 10x = 17.77777\ldots \]

4. Now, we have two equations: \[ x = 1.77777\ldots \quad (1) \] \[ 10x = 17.77777\ldots \quad (2) \]

5. Next, we subtract equation (1) from equation (2): \[ 10x - x = 17.77777\ldots - 1.77777\ldots \] \[ 9x = 16 \]

6. Now, solve for \( x \): \[ x = \frac{16}{9} \]

Thus, the fraction form of \( 1.\overline{7} \) is: \[ \boxed{\frac{16}{9}} \]