What is the product of the binary numbers 1001.01 and 11.1?

Options

A. `101110.011`
B. `100000.011`
C. `101110.101`
D. `100000.101`

Explanation

To find the product of the binary numbers 1001.01 and 11.1, we will first convert these binary numbers into decimal form, perform the multiplication, and then convert the result back into binary.

### Step 1: Convert 1001.01 from binary to decimal
The binary number 1001.01 can be converted to decimal as follows:

- The integer part (1001):
- \(1 \times 2^3 = 8\)
- \(0 \times 2^2 = 0\)
- \(0 \times 2^1 = 0\)
- \(1 \times 2^0 = 1\)

Adding these values together gives:
\[ 8 + 0 + 0 + 1 = 9 \]

- The fractional part (.01):
- \(0 \times 2^{-1} = 0\)
- \(1 \times 2^{-2} = \frac{1}{4} = 0.25\)

Adding these values gives:
\[ 0 + 0.25 = 0.25 \]

Combining both parts, we get:
\[ 1001.01_{(2)} = 9 + 0.25 = 9.25_{(10)} \]

### Step 2: Convert 11.1 from binary to decimal
The binary number 11.1 can be converted to decimal as follows:

- The integer part (11):
- \(1 \times 2^1 = 2\)
- \(1 \times 2^0 = 1\)

Adding these values together gives:
\[ 2 + 1 = 3 \]

- The fractional part (.1):
- \(1 \times 2^{-1} = \frac{1}{2} = 0.5\)

Combining both parts, we get:
\[ 11.1_{(2)} = 3 + 0.5 = 3.5_{(10)} \]

### Step 3: Multiply the decimal values
Now we will multiply the two decimal values obtained:
\[ 9.25 \times 3.5 \]

Calculating this gives:
\[ 9.25 \times 3.5 = 32.375 \]

### Step 4: Convert 32.375 from decimal to binary
To convert the decimal number 32.375 back to binary, we will separate the integer part and the fractional part.

- The integer part (32):
- \(32_{(10)}\) in binary is \(100000_{(2)}\).

- The fractional part (0.375):
- Multiply by 2: \(0.375 \times 2 = 0.75\) (integer part: 0)
- Multiply by 2: \(0.75 \times 2 = 1.5\) (integer part: 1)
- Multiply by 2: \(0.5 \times 2 = 1.0\) (integer part: 1)

So, the binary representation of 0.375 is:
\[ 0.011_{(2)} \]

Combining both parts, we get:
\[ 32.375_{(10)} = 100000.011_{(2)} \]

### Final Answer
The product of the binary numbers 1001.01 and 11.1 is:
\[ 100000.011_{(2)} \]