Question

Find the decimal expansion of 4/7.
571.4285¯¯¯¯¯¯¯
57.1428¯¯¯¯¯¯¯
5.71428¯¯¯¯¯¯¯¯
0.571428¯¯¯¯¯¯¯¯¯¯

Answer 1

To find the decimal expansion of \( \frac{4}{7} \), we can perform the division.

When you divide 4 by 7, you get:

1. \( 7 \) goes into \( 40 \) (as \( 4.000 \)) five times (since \( 7 \times 5 = 35 \)).
- Subtracting \( 35 \) from \( 40 \) leaves \( 5 \).

2. Bring down a \( 0 \) to make it \( 50 \).
- \( 7 \) goes into \( 50 \) seven times (since \( 7 \times 7 = 49 \)).
- Subtracting \( 49 \) leaves \( 1 \).

3. Bring down another \( 0 \) to make it \( 10 \).
- \( 7 \) goes into \( 10 \) once (since \( 7 \times 1 = 7 \)).
- Subtracting \( 7 \) leaves \( 3 \).

4. Bring down another \( 0 \) to make it \( 30 \).
- \( 7 \) goes into \( 30 \) four times (since \( 7 \times 4 = 28 \)).
- Subtracting \( 28 \) leaves \( 2 \).

5. Bring down another \( 0 \) to make it \( 20 \).
- \( 7 \) goes into \( 20 \) two times (since \( 7 \times 2 = 14 \)).
- Subtracting \( 14 \) leaves \( 6 \).

6. Bring down another \( 0 \) to make it \( 60 \).
- \( 7 \) goes into \( 60 \) eight times (since \( 7 \times 8 = 56 \)).
- Subtracting \( 56 \) leaves \( 4 \).

At this point, we are back to \( 4 \), confirming that the decimal repeats. Thus, the repetitive decimal for \( \frac{4}{7} \) is \( 0.571428 \) with a repeating section of \( 571428 \).

Therefore, the decimal expansion of \( \frac{4}{7} \) is \( 0.571428 \overline{571428} \).

Among the given choices, the correct answer is:
**0.571428¯¯¯¯¯¯¯¯¯¯**.