To find the decimal equivalent of the binary number 1101, 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 \(2^0\). 2. **Identifying the Place Values**: - For the binary number 1101, we can break it down 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 the binary number 1101 with its place values: \[ 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 \] 3. **Calculating Each Term**: - Now, we calculate each term: - \(1 \cdot 2^3 = 1 \cdot 8 = 8\) - \(1 \cdot 2^2 = 1 \cdot 4 = 4\) - \(0 \cdot 2^1 = 0 \cdot 2 = 0\) - \(1 \cdot 2^0 = 1 \cdot 1 = 1\) 4. **Adding the Values Together**: - Now, we add all these values together: \[ 8 + 4 + 0 + 1 = 13 \] 5. **Final Result**: - Therefore, the decimal equivalent of the binary number 1101 is **13**. ### Why Option C is Correct - The correct answer is **C. 13** because we have followed the binary to decimal conversion process accurately, and the calculations confirm that the total is indeed 13. ### Why Other Options are Incorrect - **A. 11**: This is incorrect because the sum of the place values (8 + 4 + 0 + 1) does not equal 11. - **B. 12**: This is also incorrect for the same reason; the calculated total does not match 12. - **D. 14**: This option is incorrect as well; the total of the place values is not 14. ### Common Pitfalls - **Misunderstanding Place Values**: It's crucial to remember that each digit in a binary number represents a power of 2, and the rightmost digit represents the smallest power (0). - **Forgetting to Add All Terms**: Sometimes, students may forget to include all terms in their final addition, especially if a digit is 0. - **Confusing Binary with Decimal**: Ensure you are not mixing up the binary and decimal systems; binary uses only 0s and 1s. ### Revision Summary - Binary numbers are converted to decimal by summing the products of each digit and its corresponding power of 2. - The binary number 1101 translates to \(1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0\). - The final decimal equivalent of 1101 is 13, making option C the correct answer. - Always check your calculations and ensure you understand the place values in binary.