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 division step by step.

### Step 1: Set up the division
We want to divide 22 by 7.

### Step 2: Perform the division
1. **Divide 22 by 7**:
- 7 goes into 22 three times (since \( 7 \times 3 = 21 \)).
- Write down 3 and subtract \( 21 \) from \( 22 \), which gives us \( 1 \).

2. **Add a decimal point**:
- Since we have a remainder, we can add a decimal point and a zero to continue the division.
- Now we have \( 10 \) (from bringing down a zero).

3. **Divide 10 by 7**:
- 7 goes into 10 once (since \( 7 \times 1 = 7 \)).
- Write down 1 and subtract \( 7 \) from \( 10 \), which gives us \( 3 \).

4. **Bring down another zero**:
- Now we have \( 30 \).

5. **Divide 30 by 7**:
- 7 goes into 30 four times (since \( 7 \times 4 = 28 \)).
- Write down 4 and subtract \( 28 \) from \( 30 \), which gives us \( 2 \).

6. **Bring down another zero**:
- Now we have \( 20 \).

7. **Divide 20 by 7**:
- 7 goes into 20 two times (since \( 7 \times 2 = 14 \)).
- Write down 2 and subtract \( 14 \) from \( 20 \), which gives us \( 6 \).

8. **Bring down another zero**:
- Now we have \( 60 \).

9. **Divide 60 by 7**:
- 7 goes into 60 eight times (since \( 7 \times 8 = 56 \)).
- Write down 8 and subtract \( 56 \) from \( 60 \), which gives us \( 4 \).

10. **Bring down another zero**:
- Now we have \( 40 \).

11. **Divide 40 by 7**:
- 7 goes into 40 five times (since \( 7 \times 5 = 35 \)).
- Write down 5 and subtract \( 35 \) from \( 40 \), which gives us \( 5 \).

12. **Bring down another zero**:
- Now we have \( 50 \).

13. **Divide 50 by 7**:
- 7 goes into 50 seven times (since \( 7 \times 7 = 49 \)).
- Write down 7 and subtract \( 49 \) from \( 50 \), which gives us \( 1 \).

14. **Repeat the process**:
- Notice that we are back to \( 10 \), which means the digits will start repeating from here.

### Conclusion
The decimal expansion of \( \frac{22}{7} \) is \( 3.142857142857... \) which can be written as \( 3.\overline{142857} \). This shows that the decimal is non-terminating and repeating.

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

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