Question

Add the binary numbers \(10011\) and \(1001\) in both decimal and binary forms. The values are:

• A. \(11100\) in binary, \(28\) in decimal
• B. \(00011\) in binary, \(20\) in decimal
• C. \(10101\) in binary, \(18\) in decimal
• D. \(11101\) in binary, \(19\) in decimal

Hint:

Quick Method: \[ 10011_2=19 \] \[ 1001_2=9 \] \[ 19+9=28 \] \[ 28=(11100)_2 \] Always verify binary addition by converting the result back to decimal.

Answer 1

The Correct Option is D

Concept: Binary addition can be performed directly using binary rules or by first converting binary numbers into decimal form. Binary addition rules: \[ 0+0=0 \] \[ 0+1=1 \] \[ 1+0=1 \] \[ 1+1=10 \]

Step 1: Convert \(10011_2\) into decimal form.
\[ 10011_2 = 1(2^4)+0(2^3)+0(2^2)+1(2^1)+1(2^0) \] \[ =16+0+0+2+1 \] \[ =19 \]

Step 2: Convert \(1001_2\) into decimal form.
\[ 1001_2 = 1(2^3)+0(2^2)+0(2^1)+1(2^0) \] \[ =8+1 \] \[ =9 \]

Step 3: Add the decimal values.
\[ 19+9=28 \]

Step 4: Perform binary addition directly.
\[ 10011 \] \[ +01001 \] \[ _ _ _ _ \] \[ 11100 \] Thus, \[ 10011_2+1001_2=11100_2 \]

Step 5: Verify the result.
\[ 11100_2 = 1(16)+1(8)+1(4)+0(2)+0(1) \] \[ =16+8+4 \] \[ =28 \] Hence, \[ 11100_2 = 28_{10} \] \[ \boxed{\text{Correct Answer = Option (A)}} \] Note: The option key printed as “11101 in binary, 19 in decimal” is mathematically incorrect. The correct sum is \(11100_2 = 28_{10}\).