The decimal number corresponding to the binary number `(111000.0101)_2` is :

Options

A. `(5.6312)_(10)`
B. `(56.3125)_(10)`
C. `(563.125)_(10)`
D. `(5631.2)_(10)`

Answer

To convert the binary number \( (111000.0101)_2 \) to its decimal equivalent, we can follow these steps:

### Step 1: Separate the integer and fractional parts
The binary number \( 111000.0101 \) can be separated into two parts:
- Integer part: \( 111000 \)
- Fractional part: \( 0101 \)

### Step 2: Convert the integer part to decimal
To convert the integer part \( 111000 \) to decimal, we multiply each bit by \( 2 \) raised to the power of its position, starting from the right (0-indexed):

\[ 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 \]

Calculating each term:
- \( 1 \times 2^5 = 32 \)
- \( 1 \times 2^4 = 16 \)
- \( 1 \times 2^3 = 8 \)
- \( 0 \times 2^2 = 0 \)
- \( 0 \times 2^1 = 0 \)
- \( 0 \times 2^0 = 0 \)

Now, sum these values:
\[ 32 + 16 + 8 + 0 + 0 + 0 = 56 \]

### Step 3: Convert the fractional part to decimal
To convert the fractional part \( 0101 \) to decimal, we multiply each bit by \( 2 \) raised to the negative power of its position, starting from the left (1-indexed):

\[ 0 \times 2^{-1} + 1 \times 2^{-2} + 0 \times 2^{-3} + 1 \times 2^{-4} \]

Calculating each term:
- \( 0 \times 2^{-1} = 0 \)
- \( 1 \times 2^{-2} = 0.25 \)
- \( 0 \times 2^{-3} = 0 \)
- \( 1 \times 2^{-4} = 0.0625 \)

Now, sum these values:
\[ 0 + 0.25 + 0 + 0.0625 = 0.3125 \]

### Step 4: Combine the integer and fractional parts
Now, we combine the results from the integer and fractional parts:
\[ 56 + 0.3125 = 56.3125 \]

### Final Answer
The decimal number corresponding to the binary number \( (111000.0101)_2 \) is:
\[ \boxed{56.3125} \]

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