Convert the binary number 1010.011 to its decimal equivalent. | Technical MCQ | Examoogle

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Convert the binary number 1010.011 to its decimal equivalent.

10.375

10.5

11.25

9.875

Correct Answer

⚙️ TE • Technical Concept & PrincipleTechnicalDigital Electronics

Option A

10.375

Quick Summary:

To convert a binary number with a fractional part to its decimal equivalent, the integer part is converted by multiplying each bit by increasing powers of $2$ starting from $2^0$ from right to left, and the fractional part is converted by multiplying each bit by decreasing negative powers of $2$ starting from $2^{-1}$ from left to right.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$N = \sum_{i=-m}^{n} b_i \times 2^i$ — general conversion formula for binary to decimal

$2^{-1} = 0.5, \; 2^{-2} = 0.25, \; 2^{-3} = 0.125$ — fractional binary weights

⚙️ Working Principle

The given binary number is $1010.011_2$ . For the integer part $1010_2$ , the positional weights are $2^3 = 8$ , $2^2 = 4$ , $2^1 = 2$ , and $2^0 = 1$ , giving $(1 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) = 8 + 0 + 2 + 0 = 10_{10}$ . For the fractional part $0.011_2$ , the positional weights are $2^{-1} = 0.5$ , $2^{-2} = 0.25$ , and $2^{-3} = 0.125$ , giving $(0 \times 0.5) + (1 \times 0.25) + (1 \times 0.125) = 0 + 0.25 + 0.125 = 0.375_{10}$ . Combining both gives $10 + 0.375 = 10.375_{10}$ .

📌 Key Points

▸
The integer part uses non-negative powers of $2$ ( $2^0, 2^1, 2^2, \dots$ ).
▸
The fractional part uses negative powers of $2$ ( $2^{-1}, 2^{-2}, 2^{-3}, \dots$ ).
▸
Binary $1010_2$ equals decimal $10_{10}$ .
▸
Binary $0.011_2$ equals decimal $0.375_{10}$ .

✅ Advantages

▸
Provides a direct and unambiguous method for radix conversion.
▸
Essential for understanding digital computer data representation.

❌ Disadvantages / Limitations

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Manual calculations can be tedious for large bit widths.
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Fractions that do not terminate in binary can lead to recurring or rounded decimal values.

🛠️ Applications / Uses

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Digital system design and microprocessor arithmetic logic units.
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Embedded systems and low-level programming where data is represented in binary format.

📄 Additional Information

▸
Option A ( $10.375$ ) is correctly calculated as $10 + 0.375$ .
▸
Option B ( $10.5$ ) corresponds to a fractional part of $0.1_2$ .
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Option C ( $11.25$ ) corresponds to an incorrect integer or fraction conversion.
▸
Option D ( $9.875$ ) represents an incorrect calculation of the integer or fractional weights.

📊 Diagram / Illustration

✅

A is correct — The binary number $1010.011_2$ converts to the decimal equivalent $10.375_{10}$ .

Core Concepts Used

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Number Systems Binary to Decimal Conversion Fractional Binary Weights

💡 EXAM TIP

Always ensure clear separation between the integer and fractional parts when performing radix conversions to avoid sign or positional errors in competitive exams.

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