Question

Express 711 as a repeating decimal. Mark the cycle of repeating digits with a bar.

Note: You will find the option to add a bar above text in the Geometry ( ∠ ) keyboard.

Show your work.

(1 point)

Answer 1

To express \( \frac{711}{999} \) as a repeating decimal, we first divide 711 by 999.

1. **Perform the Long Division:**

- 711 divided by 999 is 0 as 711 is less than 999.
- We find \( 7110 \div 999 \). 999 goes into 7110 about 7 times (since \( 999 \times 7 = 6993 \)).
- Subtract \( 6993 \) from \( 7110 \):
\[
7110 - 6993 = 117
\]
- Bring down a 0 to make it \( 1170 \).

- Next, \( 1170 \div 999 \) goes about 1 time (since \( 999 \times 1 = 999 \)).
- Subtract \( 999 \) from \( 1170 \):
\[
1170 - 999 = 171
\]
- Bring down another 0 to make it \( 1710 \).

- Now, \( 1710 \div 999 \) goes about 1 time (again, \( 999 \times 1 = 999 \)).
- Subtract \( 999 \) from \( 1710 \):
\[
1710 - 999 = 711
\]
- Bring down a 0 to make it \( 7110 \).

At this point, we can see that we have cycled back to \( 7110 \), indicating that the digits we obtained will repeat.

2. **Express the Result:**

The decimal part of our result is \( 0.7111711... \).

Thus, we can write it as:

\[
0.\overline{711}
\]

Where the bar indicates that "711" is the repeating part of the decimal.

So the final answer is:

\[
0.\overline{711}
\]