Question

What is the decimal equivalent of \( (B65F)_{16} \)?

• A. \((46876)_{10}\)
• B. \((46687)_{10}\)
• C. \((48676)_{10}\)
• D. \((47686)_{10}\)

Hint:

For hexadecimal to decimal conversion: \[ (B65F)_{16} = 11(16^3)+6(16^2)+5(16)+15 \] Always remember: \[ A=10,\;B=11,\;C=12,\;D=13,\;E=14,\;F=15 \]

Answer 1

The Correct Option is B

Concept: To convert a hexadecimal number into its decimal equivalent, each digit is multiplied by the corresponding power of 16. The hexadecimal digits are: \[ A=10,\quad B=11,\quad C=12,\quad D=13,\quad E=14,\quad F=15 \] General form: \[ (a_n a_{n-1}\ldots a_1 a_0)_{16} = a_n16^n+a_{n-1}16^{n-1}+\cdots+a_116+a_0 \]

Step 1: Write the hexadecimal number in expanded form.
Given: \[ (B65F)_{16} \] Replacing \(B=11\) and \(F=15\), \[ (B65F)_{16} = 11\times16^3+6\times16^2+5\times16^1+15\times16^0 \]

Step 2: Calculate the powers of 16.
\[ 16^0=1 \] \[ 16^1=16 \] \[ 16^2=256 \] \[ 16^3=4096 \]

Step 3: Multiply each digit by its positional weight.
\[ 11\times4096=45056 \] \[ 6\times256=1536 \] \[ 5\times16=80 \] \[ 15\times1=15 \]

Step 4: Add all values.
\[ 45056+1536+80+15 \] \[ =46592+80+15 \] \[ =46672+15 \] \[ =46687 \] Therefore, \[ (B65F)_{16}=(46687)_{10} \] \[ \boxed{\text{Correct Answer = Option (B)}} \]