Find the decimal equivalent of octal number $127.54_8$:

Question

Accepted Answer

The correct option is (3):
The decimal equivalent of an octal number is found by multiplying each octal digit by the corresponding power of the base (8) and summing the results. For the integer part ($127\_8$), the digits are multiplied by increasing positive powers of 8, starting from $8^0$ for the rightmost digit. Thus, the integer part is calculated as: $(1 \times 8^2) + (2 \times 8^1) + (7 \times 8^0) = 64 + 16 + 7 = 87$. For the fractional part ($0.54\_8$), the digits are multiplied by increasing negative powers of 8, starting from $8^{-1}$ for the first digit after the octal point. The fractional part is calculated as: $(5 \times 8^{-1}) + (4 \times 8^{-2}) = (5/8) + (4/64) = 0.625 + 0.0625 = 0.6875$. Combining the integer and fractional results, the decimal equivalent is $87.6875\_{10}$.

The incorrect option is (1):
The result $78.6875\_{10}$ is incorrect because, while the fractional part $0.6875$ is correct, the integer part $78$ is a result of an error in the calculation of the integer component. The correct calculation is $64 + 16 + 7 = 87$. This mistake likely stems from an arithmetic error during the summation, or an incorrect assignment of the positional weight for one of the digits in the integer part of the octal number. It is a common mistake to miscalculate the total value of the weighted terms, but the correct application of the powers of 8 is fundamental to the conversion process.

The incorrect option is (2):
The result $48.6875\_{10}$ has a significantly incorrect integer component of $48$. This suggests a major conceptual error in assigning the powers of the base, such as accidentally treating the octal number as if it were in base 4 or base 6, or simply making a significant arithmetic blunder. The power of $8^2$, which is $64$, dominates the integer sum, and obtaining $48$ indicates a fundamental miscalculation of the most significant digit's value. The base conversion formula must be rigorously applied, using $8^2, 8^1, 8^0$ for the integer digits.

The incorrect option is (4):
The result $67.6875\_{10}$ is incorrect due to a miscalculation in the integer part of the conversion. Since the correct integer part is $87$, a result of $67$ shows an error of $20$. This kind of error often occurs when students correctly identify the positional weights ($64, 8, 1$) but then make an arithmetic mistake in the final addition, perhaps miscalculating $2 \times 8$ or the sum of $64+16+7$. The base conversion method itself is based on the weighted sum of digits, and any deviation from $87$ indicates a flaw in this crucial calculation step.