Hexadecimal conversion of $(430.25)_8$ gives

Question

Accepted Answer

The correct option is (2):
The most reliable method for converting an Octal number to a Hexadecimal number is by first translating the Octal number into its Binary equivalent, and then grouping the Binary bits into segments of four for the Hexadecimal conversion. The given octal number is $(430.25)*8$. First, we convert each octal digit to its 3-bit binary representation: $4\_8 \to 100\_2$, $3\_8 \to 011\_2$, $0\_8 \to 000\_2$, $2\_8 \to 010\_2$, and $5\_8 \to 101\_2$. Concatenating these yields the complete binary number: $100011000.010101\_2$. Next, we group the bits into 4-bit nibbles starting from the radix point. For the integer part, we group right-to-left: $\underbrace{0001}*{\mathbf{1}} \underbrace{0001}*{\mathbf{1}} \underbrace{1000}*{\mathbf{8}}$ (adding leading zeros). For the fractional part, we group left-to-right: $\underbrace{0101}*{\mathbf{5}} \underbrace{0100}*{\mathbf{4}}$ (adding trailing zeros). Finally, converting these nibbles to hexadecimal digits results in the hexadecimal number $\mathbf{118.54}\_{16}$.

The incorrect option is (1):
The result $(108.44)*{16}$ is incorrect because the integer part's hexadecimal conversion is wrong. The binary $100011000\_2$ groups as $0001\ 0001\ 1000$, which converts to $118*{16}$, not $108\_{16}$. The $0$ in the second position of $108\_{16}$ would imply a binary grouping of $0000$, which is not derived from the original octal integer part $430\_8$. Furthermore, the fractional part $0.44\_{16}$ (binary $0100\ 0100$) is also incorrect as it does not match the binary equivalent of $0.25\_8$ (which is $0101\ 0100\_2$ or $0.54\_{16}$).

The incorrect option is (3):
The option $(118.44)*{16}$ is incorrect due to a mistake in the fractional part. While the integer part $\mathbf{118}*{16}$ is correctly derived from $430\_8$, the fractional part $0.\mathbf{44}*{16}$ is wrong. Converting $0.25\_8$ to binary gives $0.010101\_2$. Grouping this right of the point into 4-bit nibbles gives $0.0101\ 0100\_2$, which correctly converts to $0.54*{16}$. A result of $0.44\_{16}$ implies the binary grouping was $0.0100\ 0100\_2$, which misrepresents the original octal fraction $0.25\_8$.

The incorrect option is (4):
The option $(108.54)*{16}$ is incorrect because the integer part $\mathbf{108}*{16}$ is erroneous. As established, the octal integer $430\_8$ must convert to the hexadecimal integer $118\_{16}$. The error lies entirely in the whole number portion of the conversion, likely from an incorrect application of the conversion rule for the leading octal digits or an arithmetic mistake if converting via the decimal system first. The fractional part $0.54\_{16}$ is, however, correctly converted from $0.25\_8$.