Question

If \( x \) and \( y \) are two decimal digits and \( (0.1101)_2 = (0.8xy5)_{10} \), the decimal value of \( x + y \) is \(\underline{\hspace{2cm}}\).

Hint:

Binary fractions can be converted to decimal by summing powers of \( \frac{1}{2} \).

Answer 1

Correct Answer: 3

Step 1: Convert the binary fraction to decimal.
\[ (0.1101)_2 = \frac{1}{2} + \frac{1}{4} + \frac{1}{16} \] \[ = 0.5 + 0.25 + 0.0625 = 0.8125 \]

Step 2: Equate with the given decimal representation.
\[ (0.8xy5)_{10} = 0.8 + \frac{x}{100} + \frac{y}{1000} + \frac{5}{10000} \] \[ = 0.8 + \frac{x}{100} + \frac{y}{1000} + 0.0005 \]

Step 3: Match decimal values.
\[ 0.8125 = 0.8005 + \frac{x}{100} + \frac{y}{1000} \] \[ \frac{x}{100} + \frac{y}{1000} = 0.012 \]

Step 4: Solve for digits.
Trying decimal digits, \( x = 1 \) and \( y = 2 \) satisfy the equation.
\[ x + y = 3 \] % Final Answer

Final Answer: \[ \boxed{3} \]