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 Now, we can see that the total is 11. However, we need to check our calculations again because the options provided are 10, 12, 13, and 15. Upon reviewing the binary number 1101, we realize that we made a mistake in our calculations. Let's re-evaluate the binary number: 1. **Binary Number**: 1101 2. **Powers of 2**: - \(1 \times 2^3 = 8\) - \(1 \times 2^2 = 4\) - \(0 \times 2^1 = 0\) - \(1 \times 2^0 = 1\) Now, let's sum these values correctly: \[ 8 + 4 + 0 + 1 = 13 \] ### Conclusion Thus, the decimal equivalent of the binary number 1101 is **13**. ### Answer The correct option is **C. 13**. ### Explanation of Other Options - **A. 10**: This is incorrect because the sum of the binary digits does not equal 10. - **B. 12**: This is incorrect as the calculated total is 13, not 12. - **D. 15**: This is incorrect because the binary number does not represent 15; it is less than that. ### 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. - To convert binary to decimal, multiply each binary digit by its corresponding power of 2 and sum the results. - The binary number 1101 equals 13 in decimal.