Convert the octal number 110 (base 8) to its decimal (base 10) equivalent.
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A70 (base 10)
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B72 (base 10)
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C73 (base 10)
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D110 (base 10)
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ENone of the above
Answer
Correct Answer: 72 (base 10)
Explanation
Introduction / Context:Number base conversion is a foundational skill in digital systems. Octal (base 8) is convenient for grouping binary digits (3 bits per octal digit). Converting octal to decimal reinforces positional notation and prepares you to translate addresses, permissions, and encoded values across representations.
Given Data / Assumptions:
- Octal number: 110₈.
- We need its base-10 (decimal) value.
- Octal digits range from 0 to 7.
Concept / Approach:Use positional weights of powers of 8. For a three-digit octal number abc₈, value = a*8^2 + b*8^1 + c*8^0. Substitute digits from 110₈ to obtain the decimal sum.
Step-by-Step Solution:
Write weights: 8^2 = 64, 8^1 = 8, 8^0 = 1. Map digits: a=1, b=1, c=0. Compute: value = 1*64 + 1*8 + 0*1 = 64 + 8 + 0 = 72. Therefore, 110₈ equals 72 in decimal.Verification / Alternative check:Convert to binary first: each octal digit → 3 binary bits. 1→001, 1→001, 0→000. Thus 110₈ = 001 001 000₂ = 1001000₂. Convert binary to decimal: 64 + 8 = 72. The result agrees with the direct calculation.
Why Other Options Are Wrong:
70 or 73 are close but result from arithmetic slips (e.g., mis-weighting digits). 110 (base 10) is a misinterpretation (reading symbols without considering base). “None” is invalid because 72 is correct.Common Pitfalls:Interpreting the same digits as decimal; forgetting that octal uses powers of 8; or mis-grouping when using the binary intermediary method (ensure exactly 3 bits per octal digit).
Final Answer:72 (base 10)