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

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.