Question

What is the decimal equivalent of binary number \(101101\)?

• A. \(43\)
• B. \(45\)
• C. \(47\)
• D. \(49\)

Hint:

For binary-to-decimal conversion, remember: \[ 2^0=1,\quad2^1=2,\quad2^2=4,\quad2^3=8,\quad2^4=16,\quad2^5=32. \] These powers appear frequently in competitive examinations.

Answer 1

The Correct Option is B

Concept: A binary number consists only of digits \(0\) and \(1\). Each digit represents a power of \(2\) depending upon its position from right to left. For a binary number \[ b_nb_{n-1}\cdots b_1b_0, \] its decimal value is calculated as \[ b_n2^n+b_{n-1}2^{n-1}+\cdots+b_12^1+b_02^0. \] Thus, to convert a binary number into decimal form, we multiply each digit by its corresponding power of \(2\) and add the results.

Step 1: Write the positional values of the binary digits.
Given binary number: \[ 101101 \] Assign powers of \(2\) from right to left: \[ \begin{array}{cccccc} 1 & 0 & 1 & 1 & 0 & 1\\ 2^5 & 2^4 & 2^3 & 2^2 & 2^1 & 2^0 \end{array} \]

Step 2: Multiply each digit by its positional value.
\[ 1\times2^5 + 0\times2^4 + 1\times2^3 + 1\times2^2 + 0\times2^1 + 1\times2^0 \] \[ =32+0+8+4+0+1 \]

Step 3: Add all the obtained values.
\[ 32+8+4+1=45 \] Therefore, \[ (101101)_2=(45)_{10} \] \[ \boxed{45} \]