To convert the recurring decimal [tex]\( 0.\overline{7} \)[/tex] to a fraction, follow these steps:
1. Let [tex]\( x \)[/tex] be the repeating decimal, so [tex]\( x = 0.\overline{7} \)[/tex].
2. To eliminate the repeating part, multiply both sides of the equation by 10 (since there is one repeating digit):
[tex]\[
10x = 7.\overline{7}
\][/tex]
3. Next, set up an equation by subtracting the original [tex]\( x \)[/tex] from the new equation:
[tex]\[
10x - x = 7.\overline{7} - 0.\overline{7}
\][/tex]
4. Simplify the equation:
[tex]\[
9x = 7
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 9:
[tex]\[
x = \frac{7}{9}
\][/tex]
So, the recurring decimal [tex]\( 0.\overline{7} \)[/tex] as a fraction is [tex]\( \frac{7}{9} \)[/tex].
