$(929.19)_{10}=?$

Question

$(929.19)_{10}=?$

Accepted Answer

The correct option is (1):
To accurately convert a decimal number with a fractional part, such as $(929.19)*{10}$, to the octal system (base 8), you must treat the integer and fractional parts separately. The integer part, 929, is converted by the method of successive division by 8, collecting the remainders from bottom to top. This procedure results in $929*{10} = 1641\_8$. The fractional part, 0.19, is converted by the method of successive multiplication by 8, collecting the integer parts from top to bottom. The multiplication steps are $0.19 \times 8 = 1.52$, $0.52 \times 8 = 4.16$, and $0.16 \times 8 = 1.28$, yielding the octal fraction $0.141...*8$. Combining the two parts, we get $(929.19)*{10} \approx (1641.141)\_8$, which matches Option 1.

The incorrect option is (2):
Option (2) is $(1640.41)*8$. This result is mathematically incorrect because the integer part $1640\_8$ converts back to $928*{10}$ ($1\cdot8^3 + 6\cdot8^2 + 4\cdot8^1 + 0\cdot8^0$), which is close to but not the correct integer $929\_{10}$. Furthermore, the fractional part $0.41\_8$ represents $4/8 + 1/64 = 0.515625\_{10}$, which is a substantial error compared to the actual decimal fraction of $0.19\_{10}$. This option shows inaccuracies in both parts of the conversion, indicating arithmetic mistakes in the student’s division and multiplication steps.

The incorrect option is (3):
Option (3) is $(1632.141)*8$. While the fractional part $0.141\_8$ is correct up to three octal places, the integer part $1632\_8$ is demonstrably wrong. Converting $1632\_8$ back to decimal, we find it equals $922*{10}$ ($1\cdot8^3 + 6\cdot8^2 + 3\cdot8^1 + 2\cdot8^0$). Since the original number's integer part is $929\_{10}$, this option contains a clear error in the successive division for the integer conversion. The mistake suggests that the remainders collected during the division process were recorded incorrectly.

The incorrect option is (4):
Option (4) is $(1543.34)*8$. This option is incorrect for both the integer and fractional parts. The integer $1543\_8$ equals $867*{10}$ ($1\cdot8^3 + 5\cdot8^2 + 4\cdot8^1 + 3\cdot8^0$), representing a large deviation from the correct integer value of $929\_{10}$. Likewise, the fractional component $0.34\_8$ converts to $0.4375\_{10}$, which is far from $0.19\_{10}$. Such a comprehensive error suggests the conversion was performed using an incorrect base, a miscalculation in the initial division, or a mix-up in the sequence of remainders/integer products.