Express 8//11 as a repeating decimal. Mark the cycle of repeating digits with a bar

Answer 1

To express \( \frac{8}{11} \) as a repeating decimal, we can perform long division:

1. Dividing 8 by 11 gives us \( 0 \) with a remainder of 8.
2. Now we consider 80 (by adding a decimal point and a zero).
3. Dividing 80 by 11 gives us \( 7 \) (since \( 11 \times 7 = 77 \)), with a remainder of \( 3 \) (because \( 80 - 77 = 3 \)).
4. Next, we bring down another 0 to make it 30.
5. Dividing 30 by 11 gives us \( 2 \) (since \( 11 \times 2 = 22 \)), with a remainder of \( 8 \) (because \( 30 - 22 = 8 \)).
6. Now we can see we have returned to 80, which will repeat the previous steps.

Thus, the division leads us to the repeated cycle of \( 72 \). Therefore, we can write:

\[ \frac{8}{11} = 0.\overline{72} \]

This shows that \( 72 \) is the repeating part of the decimal.