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 \(\underline{\hspace{2cm}}\).
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Correct Answer: 3
Solution and Explanation
To solve for \( x+y \), first convert the binary number \( (0.1101)_2 \) to its decimal equivalent.
Step 1: Binary to Decimal Conversion
The binary number \( 0.1101_2 \) is read as \( 1 \times 2^{-1} + 1 \times 2^{-2} + 0 \times 2^{-3} + 1 \times 2^{-4} \).
Calculate each term:
- \(1 \times 2^{-1} = 0.5\)
- \(1 \times 2^{-2} = 0.25\)
- \(0 \times 2^{-3} = 0\)
- \(1 \times 2^{-4} = 0.0625\)
Sum these to get the decimal equivalent: \(0.5 + 0.25 + 0 + 0.0625 = 0.8125_{10}\).
Step 2: Matching Decimal Places
The problem states \( (0.1101)_2 = (0.8xy5)_{10} \). Therefore, \(0.8125_{10} = 0.8xy5_{10}\).
Convert \(0.8xy5\) to a form we can solve: \(0.8 + 0.x + 0.0y + 0.0005\).
Combine known terms: \(0.8 + 0.0005 = 0.8005\).
Thus, we have \(0.8005 + 0.1x + 0.01y = 0.8125\).
Step 3: Solving for \(x\) and \(y\)
Rearrange the equation: \(0.1x + 0.01y = 0.012\).
Multiply through by 100 to clear decimals: \(10x + y = 1.2\).
Considering \(x\) and \(y\) must be digits (0-9), test integer values:
If \(x=1\), \(10(1)+y=1.2\) becomes \(10+y=12\Rightarrow y=2\).
Check the values: \(x=1, y=2\) satisfies the conditions.
Step 4: Compute \(x+y\) and Verify
The sum \(x+y=1+2=3\).
Since the range specified is 3,3, the solution fits the range.
Therefore, the decimal value of \( x+y \) is 3.
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