Question

Use the given long division calculations to determine the decimal expansion of 5/11.

Answer 1

To find the decimal expansion of \( \frac{5}{11} \) using long division, we divide 5 by 11.

1. **Set up the long division**: 5 is less than 11, so we start with 0. We put a decimal point and add a zero to make it 50.

\[
11 \) 5.00
\]

2. **Divide**: 11 goes into 50 four times (since \( 11 \times 4 = 44 \)). Write 4 in the quotient.

\[
11 \) 5.00
- 44
--------
6
\]

3. **Bring down another 0**: Now we have 60.

4. **Repeat the division**: 11 goes into 60 five times (since \( 11 \times 5 = 55 \)). Write 5 in the quotient.

\[
11 \) 5.00
- 44
--------
60
- 55
--------
5
\]

5. **Bring down another 0**: Now we have 50 again.

6. **Repeat the process**: 11 goes into 50 four times again. You will notice that this creates a repeating cycle.

\[
11 \) 5.00
- 44
--------
60
- 55
--------
50
- 44
--------
60
\]

Continuing this process will show that:

- The decimal representation of \( \frac{5}{11} \) is \( 0.454545\ldots \), where "45" is the repeating part.

Thus, the decimal expansion of \( \frac{5}{11} \) is:

\[
0.\overline{45}
\]