Write `(3)/(13)` in decimal form and say what kind of decimal representating it has.

Options

A. terminating
B. non terminating non recurring
C. non terminating recurring
D. none of the above

Explanation

To convert \( \frac{3}{13} \) into decimal form and determine the type of decimal representation, we will perform long division.

### Step-by-Step Solution:

1. **Set up the division**: We need to divide 3 by 13. Since 3 is less than 13, we start by adding a decimal point and a zero to 3, making it 30.

2. **Divide 30 by 13**:
- 13 goes into 30 two times (since \( 13 \times 2 = 26 \)).
- Write down 2 after the decimal point.
- Subtract \( 26 \) from \( 30 \) to get a remainder of \( 4 \).

**Current result**: \( 0.2 \)

3. **Bring down another zero**: Now we have 40 (from the remainder 4).
- 13 goes into 40 three times (since \( 13 \times 3 = 39 \)).
- Write down 3.
- Subtract \( 39 \) from \( 40 \) to get a remainder of \( 1 \).

**Current result**: \( 0.23 \)

4. **Bring down another zero**: Now we have 10 (from the remainder 1).
- 13 goes into 10 zero times.
- Write down 0.
- The remainder is still 10.

**Current result**: \( 0.230 \)

5. **Bring down another zero**: Now we have 100.
- 13 goes into 100 seven times (since \( 13 \times 7 = 91 \)).
- Write down 7.
- Subtract \( 91 \) from \( 100 \) to get a remainder of \( 9 \).

**Current result**: \( 0.2307 \)

6. **Bring down another zero**: Now we have 90.
- 13 goes into 90 six times (since \( 13 \times 6 = 78 \)).
- Write down 6.
- Subtract \( 78 \) from \( 90 \) to get a remainder of \( 12 \).

**Current result**: \( 0.23076 \)

7. **Bring down another zero**: Now we have 120.
- 13 goes into 120 nine times (since \( 13 \times 9 = 117 \)).
- Write down 9.
- Subtract \( 117 \) from \( 120 \) to get a remainder of \( 3 \).

**Current result**: \( 0.230769 \)

8. **Notice the pattern**: When we bring down another zero, we are back to dividing 30 again, which we already did. This indicates that the decimal will start repeating.

### Final Result:
The decimal representation of \( \frac{3}{13} \) is \( 0.230769230769... \), which can be written as \( 0.\overline{230769} \).

### Type of Decimal:
The decimal representation is **non-terminating** and **recurring**.