To find the decimal equivalent of the binary number 101101, we will follow a step-by-step process to convert it from binary (base 2) to decimal (base 10). ### Step-by-Step Explanation 1. **Understanding Binary Numbers**: - Binary numbers are composed of only two digits: 0 and 1. Each digit represents a power of 2, starting from the rightmost digit, which represents \(2^0\), the next one to the left represents \(2^1\), and so on. 2. **Identifying the Binary Number**: - The binary number given is 101101. We will break it down from right to left: - The rightmost digit (1) is in the \(2^0\) place. - The next digit (0) is in the \(2^1\) place. - The next digit (1) is in the \(2^2\) place. - The next digit (1) is in the \(2^3\) place. - The next digit (0) is in the \(2^4\) place. - The leftmost digit (1) is in the \(2^5\) place. 3. **Calculating the Decimal Value**: - Now, we will calculate the decimal value by multiplying each binary digit by its corresponding power of 2 and then summing the results: \[ \text{Decimal Value} = (1 \times 2^5) + (0 \times 2^4) + (1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) \] - Breaking it down: - \(1 \times 2^5 = 1 \times 32 = 32\) - \(0 \times 2^4 = 0 \times 16 = 0\) - \(1 \times 2^3 = 1 \times 8 = 8\) - \(1 \times 2^2 = 1 \times 4 = 4\) - \(0 \times 2^1 = 0 \times 2 = 0\) - \(1 \times 2^0 = 1 \times 1 = 1\) - Now, we sum these values: \[ 32 + 0 + 8 + 4 + 0 + 1 = 45 \] 4. **Final Answer**: - Therefore, the decimal equivalent of the binary number 101101 is **45**. ### Explanation of Other Options - **Option B (22)**: - This is incorrect because the calculations show that the total is 45, not 22. - **Option C (27)**: - This is also incorrect. The breakdown of the binary number does not yield 27 when calculated correctly. - **Option D (23)**: - This option is incorrect as well. The correct calculation leads to 45, not 23. ### Common Pitfalls - **Miscounting Powers**: Ensure that you start counting powers of 2 from the rightmost digit (starting at \(2^0\)). - **Forgetting to Multiply**: Each binary digit must be multiplied by its corresponding power of 2 before summing. - **Adding Incorrectly**: Double-check your addition to avoid simple arithmetic errors. ### Revision Summary - Binary numbers are converted to decimal by multiplying each digit by its corresponding power of 2. - The binary number 101101 converts to decimal as follows: \(1 \times 32 + 0 \times 16 + 1 \times 8 + 1 \times 4 + 0 \times 2 + 1 \times 1 = 45\). - The correct answer is **A. 45**. - Always verify each step in the conversion process to avoid mistakes.