Question

Express 711%0D%0A7%0D%0A11%0D%0A as a repeating decimal. Mark the cycle of repeating digits with a bar.%0D%0A%0D%0ANote: You will find the option to add a bar above text in the Geometry (∠%0D%0A∠%0D%0A) keyboard.%0D%0A%0D%0AShow your work. %0D%0A%0D%0A(1 point)

Answer 1

To convert the fraction \(\frac{711}{7}\) into a decimal, we can perform long division.

1. **Long Division Calculation:**

- Divide 711 by 7.
- 7 goes into 71 ten times (7 * 10 = 70), so write down 10.
- Subtract 70 from 71, leaving 1.
- Bring down the next digit (1), making it 11.
- 7 goes into 11 once (7 * 1 = 7).
- Subtract 7 from 11, leaving 4.
- Bring down the next digit (0), making it 40.
- 7 goes into 40 five times (7 * 5 = 35).
- Subtract 35 from 40, leaving 5.
- Bring down the next digit (0), making it 50.
- 7 goes into 50 seven times (7 * 7 = 49).
- Subtract 49 from 50, leaving 1.
- Bring down another 0, making it 10.
- 7 goes into 10 once (7 * 1 = 7).
- Subtract 7 from 10, leaving 3.
- Bring down another 0, making it 30.
- 7 goes into 30 four times (7 * 4 = 28).
- Subtract 28 from 30, leaving 2.
- Bring down another 0, making it 20.
- 7 goes into 20 two times (7 * 2 = 14).
- Subtract 14 from 20, leaving 6.
- Bring down another 0, making it 60.
- 7 goes into 60 eight times (7 * 8 = 56).
- Subtract 56 from 60, leaving 4.

At this point, we recognize that we are repeating a previous calculation (with a remainder of 4). This indicates that the decimal will repeat.

2. **Summary of the Division Results:**
- The whole number part is 101.
- The decimal part that started repeating is \(142857\).

So, the entire division gives us:

\[
\frac{711}{7} = 101.142857\overline{142857}
\]

3. **Final Answer:**
The repeating decimal representation of \( \frac{711}{7} \) is:

\[
101. \overline{142857}
\]

This means that the digits \(142857\) repeat indefinitely.