The decimal number `(127.25)_(10)`, when converted to binary number, taken the form

Options

A. `(1111111.11)_(2)`
B. `(1111110.01)_(2)`
C. `(1110111.11)_(2)`
D. `(1111111.01)_(2)`

Explanation

To convert the decimal number \( (127.25)_{10} \) to binary, we need to handle the integer part (127) and the fractional part (0.25) separately.

### Step 1: Convert the Integer Part (127) to Binary

1. **Divide the integer by 2 and record the remainder:**
- \( 127 \div 2 = 63 \) remainder \( 1 \)
- \( 63 \div 2 = 31 \) remainder \( 1 \)
- \( 31 \div 2 = 15 \) remainder \( 1 \)
- \( 15 \div 2 = 7 \) remainder \( 1 \)
- \( 7 \div 2 = 3 \) remainder \( 1 \)
- \( 3 \div 2 = 1 \) remainder \( 1 \)
- \( 1 \div 2 = 0 \) remainder \( 1 \)

2. **Write the remainders in reverse order:**
- The binary representation of \( 127 \) is \( 1111111 \).

### Step 2: Convert the Fractional Part (0.25) to Binary

1. **Multiply the fractional part by 2:**
- \( 0.25 \times 2 = 0.5 \) (the integer part is \( 0 \))
- \( 0.5 \times 2 = 1.0 \) (the integer part is \( 1 \))

2. **Stop when the fractional part becomes \( 0 \):**
- The binary representation of \( 0.25 \) is \( 01 \).

### Step 3: Combine Both Parts

- Combine the integer and fractional parts:
- The binary representation of \( (127.25)_{10} \) is \( (1111111.01)_2 \).

### Final Answer

The binary representation of \( (127.25)_{10} \) is \( (1111111.01)_2 \).

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