What is the decimal equivalent of `(101.101)_(2)`?

Options

A. `(5.225)_(10)`
B. `(5.525)_(10)`
C. `(5.625)_(10)`
D. `(5.65)_(10)`

Answer

To find the decimal equivalent of the binary number \( (101.101)_2 \), we will convert both the integer part (left of the binary point) and the fractional part (right of the binary point) separately, and then combine the results.

### Step-by-Step Solution:

1. **Identify the binary number**:
The binary number given is \( 101.101_2 \).

2. **Convert the integer part \( 101_2 \)**:
- The integer part is \( 101 \).
- We will convert it to decimal using the formula:
\[ \text{Decimal} = b_n \times 2^n + b_{n-1} \times 2^{n-1} + \ldots + b_1 \times 2^1 + b_0 \times 2^0 \]
- For \( 101 \):
- \( 1 \times 2^2 = 1 \times 4 = 4 \)
- \( 0 \times 2^1 = 0 \times 2 = 0 \)
- \( 1 \times 2^0 = 1 \times 1 = 1 \)
- Adding these values together:
\[ 4 + 0 + 1 = 5 \]

3. **Convert the fractional part \( .101_2 \)**:
- The fractional part is \( .101 \).
- We will convert it using the formula:
\[ \text{Decimal} = b_{-1} \times 2^{-1} + b_{-2} \times 2^{-2} + b_{-3} \times 2^{-3} \]
- For \( .101 \):
- \( 1 \times 2^{-1} = 1 \times 0.5 = 0.5 \)
- \( 0 \times 2^{-2} = 0 \times 0.25 = 0 \)
- \( 1 \times 2^{-3} = 1 \times 0.125 = 0.125 \)
- Adding these values together:
\[ 0.5 + 0 + 0.125 = 0.625 \]

4. **Combine the integer and fractional parts**:
- Now, we combine the results from the integer and fractional parts:
\[ 5 + 0.625 = 5.625 \]

5. **Final Result**:
- The decimal equivalent of \( (101.101)_2 \) is \( (5.625)_{10} \).