Convert ($101.011)_2$ to base 10

Analyze the question: We need to convert the binary number 101.011 (base 2) to base 10. The part before the point and the part after the point are handled differently.

Concept: In binary, each digit is multiplied by a power of 2. Digits to the left of the point use 2⁰, 2¹, 2², ... Digits to the right use 2⁻¹, 2⁻², 2⁻³, ...

Solution:

Integer part (101):

• 1 × 2² = 4
• 0 × 2¹ = 0
• 1 × 2⁰ = 1

Sum = 5

Fractional part (.011):

• 0 × 2⁻¹ = 0
• 1 × 2⁻² = 1/4
• 1 × 2⁻³ = 1/8

Sum = 2/8 + 1/8 = 3/8

Total = 5 + 3/8 = 40/8 + 3/8 = 43/8

Check against the options: 43/8 = 5.375, which matches option E. (Option D, 77/8, would be 9.625, which doesn't match.)

Tip: For binary fractions, write the whole number as an improper fraction by putting everything over the last place value. Here 101011₂ = 32 + 8 + 2 + 1 = 43, and since there are 3 places after the point, divide by 2³ = 8, which gives 43/8 directly.