Convert the number 10111.11$_{two}$ to a mixed number.

To convert a number from base two (binary) to base ten (decimal), we use the positional notation method. Each digit is multiplied by 2 raised to the power of its position relative to the binary point.

1. Analyze the question

We need to convert $10111.11_2$ into a decimal mixed number. The number consists of an integer part ($10111$) and a fractional part ($0.11$).

2. Explain the concept

• Integer part: Starting from the right of the binary point and moving left, the powers of 2 are $2^0, 2^1, 2^2, 2^3, 2^4, \dots$
• Fractional part: Starting from the left of the binary point and moving right, the powers of 2 are $2^{-1}, 2^{-2}, 2^{-3}, \dots$

3. Show the solution

Step 1: Convert the integer part ($10111_2$)

• $1 \times 2^4 = 1 \times 16 = 16$
• $0 \times 2^3 = 0 \times 8 = 0$
• $1 \times 2^2 = 1 \times 4 = 4$
• $1 \times 2^1 = 1 \times 2 = 2$
• $1 \times 2^0 = 1 \times 1 = 1$
• Sum: $16 + 0 + 4 + 2 + 1 = 23$

Step 2: Convert the fractional part ($0.11_2$)

• $1 \times 2^{-1} = 1 \times \frac{1}{2} = 0.5$
• $1 \times 2^{-2} = 1 \times \frac{1}{4} = 0.25$
• Sum: $0.5 + 0.25 = 0.75$

Step 3: Combine the parts

• $23 + 0.75 = 23.75$

Step 4: Convert to a mixed number

• $0.75$ is the same as $\frac{75}{100}$, which simplifies to $\frac{3}{4}$.
• Therefore, $23.75 = 23\frac{3}{4}$.

4. Identify the answer

Comparing our result to the given options, $23\frac{3}{4}$ corresponds to option B.

5. Tip

When dealing with binary fractions, remember that the first position after the point is always "halves" ($\frac{1}{2}$), the second is "quarters" ($\frac{1}{4}$), and the third is "eighths" ($\frac{1}{8}$). This makes quick mental conversion much easier!