Question

Find the decimal expansion of 10/13.
7.628
0.769230¯¯¯¯¯¯¯¯¯¯
0.257
0.258

Answer 1

To find the decimal expansion of \( \frac{10}{13} \), we can perform the division:

1. Divide 10 by 13.

- 13 goes into 10 zero times, so we start with 0.
- Then we can consider 10.0 (or 100 tenths), and see how many times 13 goes into 100.

2. \( 13 \times 7 = 91 \) — that works, and leaves a remainder:
- \( 100 - 91 = 9 \)

3. Bring down the next 0 (making it 90):
- \( 13 \times 6 = 78 \) — then,
- \( 90 - 78 = 12 \)

4. Bring down the next 0 (making it 120):
- \( 13 \times 9 = 117 \) — then,
- \( 120 - 117 = 3 \)

5. Bring down the next 0 (making it 30):
- \( 13 \times 2 = 26 \) — then,
- \( 30 - 26 = 4 \)

6. Bring down the next 0 (making it 40):
- \( 13 \times 3 = 39 \) — then,
- \( 40 - 39 = 1 \)

7. Bring down the next 0 (making it 10 again):
- \( 13 \times 0 = 0 \) — remaining 10.

At this point, we notice the sequence starts repeating itself, as we are back where we began, having decimal digits \( 769230 \).

Thus, the decimal expansion of \( \frac{10}{13} \) is:
\[
0.769230769230...
\]
This is a repeating decimal, often denoted as \( 0.769230\overline{769230} \).

Among the choices provided, the closest representation is:
\[
0.769230\overline{~}
\]

So the correct answer from your options is:
\[
0.769230\overline{~}
\]