Question

What is the fraction form of the following decimal expansion?
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:
\[
\boxed{\frac{16}{9}}
\]