Find the decimal expansion of the rational number `(14587)/(1250)`.

Options

A. `11.6696`
B. `12.6182`
C. `9.3120`
D. `10.717`

Answer

To find the decimal expansion of the rational number \( \frac{14587}{1250} \), we will perform long division.

### Step-by-Step Solution:

1. **Set Up the Division**:
We need to divide 14587 by 1250. Here, 14587 is the dividend and 1250 is the divisor.

2. **Determine How Many Times 1250 Fits into 14587**:
- 1250 goes into 14587 approximately 11 times because \( 1250 \times 11 = 13750 \) and \( 1250 \times 12 = 15000 \) (which is too high).
- Write down 11 as the quotient.

3. **Subtract to Find the Remainder**:
- Subtract \( 13750 \) from \( 14587 \):
\[ 14587 - 13750 = 837 \]

4. **Add Decimal Point and Zero**:
- Since we have a remainder, we add a decimal point to the quotient and bring down a zero, making it \( 8370 \).

5. **Divide Again**:
- Now, see how many times 1250 fits into 8370.
- 1250 goes into 8370 approximately 6 times because \( 1250 \times 6 = 7500 \) and \( 1250 \times 7 = 8750 \) (which is too high).
- Write down 6 in the quotient.

6. **Subtract Again**:
- Subtract \( 7500 \) from \( 8370 \):
\[ 8370 - 7500 = 870 \]

7. **Bring Down Another Zero**:
- Bring down another zero, making it \( 8700 \).

8. **Divide Again**:
- Now, see how many times 1250 fits into 8700.
- 1250 goes into 8700 approximately 6 times again.
- Write down 6 in the quotient.

9. **Subtract Again**:
- Subtract \( 7500 \) from \( 8700 \):
\[ 8700 - 7500 = 1200 \]

10. **Bring Down Another Zero**:
- Bring down another zero, making it \( 12000 \).

11. **Divide Again**:
- Now, see how many times 1250 fits into 12000.
- 1250 goes into 12000 exactly 9 times because \( 1250 \times 9 = 11250 \).
- Write down 9 in the quotient.

12. **Final Subtraction**:
- Subtract \( 11250 \) from \( 12000 \):
\[ 12000 - 11250 = 750 \]

13. **Bring Down Another Zero**:
- Bring down another zero, making it \( 7500 \).

14. **Divide Again**:
- 1250 goes into 7500 exactly 6 times.
- Write down 6 in the quotient.

15. **Final Subtraction**:
- Subtract \( 7500 \) from \( 7500 \):
\[ 7500 - 7500 = 0 \]
- Since the remainder is now zero, we can stop here.

### Result:
The decimal expansion of \( \frac{14587}{1250} \) is \( 11.6648 \).