To find the decimal equivalent of the binary number 1101, we will break down the process step-by-step. ### Step 1: Understanding Binary Numbers Binary numbers are base-2 numbers, which means they only use two digits: 0 and 1. Each digit in a binary number represents a power of 2, starting from the rightmost digit, which represents \(2^0\). ### Step 2: Assigning Powers of 2 For the binary number 1101, we can assign powers of 2 to each digit as follows: - The rightmost digit (1) is in the \(2^0\) place. - The next digit to the left (0) is in the \(2^1\) place. - The next digit (1) is in the \(2^2\) place. - The leftmost digit (1) is in the \(2^3\) place. So, we can write it out like this: \[ \begin{align*} 1 & \times 2^3 \quad (1 \text{ in the } 2^3 \text{ place}) \\ 0 & \times 2^2 \quad (0 \text{ in the } 2^2 \text{ place}) \\ 1 & \times 2^1 \quad (1 \text{ in the } 2^1 \text{ place}) \\ 1 & \times 2^0 \quad (1 \text{ in the } 2^0 \text{ place}) \\ \end{align*} \] ### Step 3: Calculating Each Term Now, we calculate the value of each term: - \(1 \times 2^3 = 1 \times 8 = 8\) - \(0 \times 2^2 = 0 \times 4 = 0\) - \(1 \times 2^1 = 1 \times 2 = 2\) - \(1 \times 2^0 = 1 \times 1 = 1\) ### Step 4: Summing the Values Next, we sum all these values together: \[ 8 + 0 + 2 + 1 = 11 \] ### Step 5: Final Calculation Thus, the decimal equivalent of the binary number 1101 is: \[ 11 \] ### Step 6: Evaluating the Options Now, let's look at the options provided: - A. 10 - B. 12 - C. 13 - D. 14 None of these options match our calculated result of 11. Therefore, it seems there was an error in the recorded correct option. The correct answer should be **11**, which is not listed among the options. ### Explanation of Incorrect Options - **Option A (10)**: This is incorrect because it does not account for the values of the binary digits correctly. - **Option B (12)**: This is also incorrect as it miscalculates the sum of the binary values. - **Option C (13)**: This is incorrect; it is one more than the correct answer. - **Option D (14)**: This is incorrect; it is three more than the correct answer. ### Revision Summary - Binary numbers are base-2 and represent powers of 2. - Each digit's position corresponds to a power of 2, starting from \(2^0\) on the right. - The binary number 1101 converts to decimal by calculating \(1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0\). - The correct decimal equivalent of 1101 is 11, which is not listed in the provided options.