If \( x \) and \( y \) are two decimal digits and \( (0.1101)_2 = (0.8xy5)_{10} \), the decimal value of \( x + y \) is \(\underline{\hspace{2cm}}\).
If \( x \) and \( y \) are two decimal digits and \( (0.1101)_2 = (0.8xy5)_{10} \), the decimal value of \( x + y \) is \(\underline{\hspace{2cm}}\).
Show Hint
Correct Answer: 3
Solution and Explanation
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: \[ \boxed{3} \]
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