Find the value of 1101112 + 101002

Analysis

This is a problem involving binary addition (base 2). In binary, the only digits allowed are 0 and 1. The rules for addition are:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 10 (write 0, carry 1)
• 1 + 1 + 1 (including a carry) = 11 (write 1, carry 1)

Step-by-Step Solution

To solve this, we align the numbers vertically and add them column by column from right to left:

    1 1 0 1 1 1
+     1 0 1 0 0
---------------

Step 1 (Rightmost column): 1 + 0 = 1 Step 2 (Second column): 1 + 0 = 1 Step 3 (Third column): 1 + 1 = 10. Write 0, carry 1. Step 4 (Fourth column): 0 + 0 + (carry 1) = 1 Step 5 (Fifth column): 1 + 1 = 10. Write 0, carry 1. Step 6 (Sixth column): 1 + (carry 1) = 10. Write 10.

Putting it all together:

  (1) (1)   (1)
    1 1 0 1 1 1
+     1 0 1 0 0
---------------
  1 0 0 1 0 1 1

Verification

• $110111_2 = (1 \times 2^5) + (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 32 + 16 + 0 + 4 + 2 + 1 = 55_{10}$
• $10100_2 = (1 \times 2^4) + (0 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (0 \times 2^0) = 16 + 0 + 4 + 0 + 0 = 20_{10}$
• $55 + 20 = 75_{10}$

Now, convert $75_{10}$ back to binary:

• $75 \div 2 = 37$ remainder 1
• $37 \div 2 = 18$ remainder 1
• $18 \div 2 = 9$ remainder 0
• $9 \div 2 = 4$ remainder 1
• $4 \div 2 = 2$ remainder 0
• $2 \div 2 = 1$ remainder 0
• $1 \div 2 = 0$ remainder 1

Reading the remainders from bottom to top, we get 1001011.

Conclusion

The result of the addition is $1001011_2$. This matches option C.

Exam Tip: When adding binary numbers, always keep your columns straight. If you get confused, convert the binary numbers to base 10 (decimal), add them, and then convert your final answer back to base 2. It takes a little longer, but it is a foolproof way to check your work!