The decimal expansion of `22/7` is:

Options

A. terminating
B. non-terminating and non-repeating
C. non-terminating and repeating
D. None of the above

Answer

To find the decimal expansion of \( \frac{22}{7} \), we will perform the long division of 22 by 7.

### Step-by-Step Solution:

1. **Set up the division**: We want to divide 22 by 7.
- 22 is the dividend, and 7 is the divisor.

2. **Divide 22 by 7**:
- 7 goes into 22 three times (since \( 7 \times 3 = 21 \)).
- Write down 3 as the first digit of the quotient.

3. **Subtract**:
- Subtract \( 21 \) from \( 22 \) to get a remainder of \( 1 \).
- So, \( 22 - 21 = 1 \).

4. **Bring down a decimal**:
- Add a decimal point to the quotient and bring down a 0 to make the new number \( 10 \).

5. **Divide 10 by 7**:
- 7 goes into 10 once (since \( 7 \times 1 = 7 \)).
- Write down 1 next to the 3 in the quotient.

6. **Subtract again**:
- Subtract \( 7 \) from \( 10 \) to get a remainder of \( 3 \).
- So, \( 10 - 7 = 3 \).

7. **Bring down another decimal**:
- Bring down another 0 to make it \( 30 \).

8. **Divide 30 by 7**:
- 7 goes into 30 four times (since \( 7 \times 4 = 28 \)).
- Write down 4 in the quotient.

9. **Subtract**:
- Subtract \( 28 \) from \( 30 \) to get a remainder of \( 2 \).
- So, \( 30 - 28 = 2 \).

10. **Bring down another decimal**:
- Bring down another 0 to make it \( 20 \).

11. **Divide 20 by 7**:
- 7 goes into 20 two times (since \( 7 \times 2 = 14 \)).
- Write down 2 in the quotient.

12. **Subtract**:
- Subtract \( 14 \) from \( 20 \) to get a remainder of \( 6 \).
- So, \( 20 - 14 = 6 \).

13. **Bring down another decimal**:
- Bring down another 0 to make it \( 60 \).

14. **Divide 60 by 7**:
- 7 goes into 60 eight times (since \( 7 \times 8 = 56 \)).
- Write down 8 in the quotient.

15. **Subtract**:
- Subtract \( 56 \) from \( 60 \) to get a remainder of \( 4 \).
- So, \( 60 - 56 = 4 \).

16. **Bring down another decimal**:
- Bring down another 0 to make it \( 40 \).

17. **Divide 40 by 7**:
- 7 goes into 40 five times (since \( 7 \times 5 = 35 \)).
- Write down 5 in the quotient.

18. **Subtract**:
- Subtract \( 35 \) from \( 40 \) to get a remainder of \( 5 \).
- So, \( 40 - 35 = 5 \).

19. **Bring down another decimal**:
- Bring down another 0 to make it \( 50 \).

20. **Divide 50 by 7**:
- 7 goes into 50 seven times (since \( 7 \times 7 = 49 \)).
- Write down 7 in the quotient.

21. **Subtract**:
- Subtract \( 49 \) from \( 50 \) to get a remainder of \( 1 \).
- So, \( 50 - 49 = 1 \).

22. **Notice the pattern**:
- We have returned to a remainder of \( 1 \), which means the decimal will start repeating from here.

### Final Result:
The decimal expansion of \( \frac{22}{7} \) is \( 3.142857142857... \) which is non-terminating and non-repeating.