$(929.19)_{10} = ?$

Question

$(929.19)_{10} = ?$

Accepted Answer

The correct option is (2):
The decimal number $(929.19)\_{10}$ must be converted to octal (base 8) by processing its integer and fractional parts separately. For the integer part, 929, we use the method of successive division by 8. Dividing 929 by 8 yields a quotient of 116 with a remainder of 1. Dividing 116 by 8 yields 14 with a remainder of 4. Dividing 14 by 8 yields 1 with a remainder of 6. Finally, dividing 1 by 8 yields 0 with a remainder of 1. Reading the remainders from the last to the first gives the octal integer as $1641\_8$. For the fractional part, 0.19, we use successive multiplication by 8. $0.19 \times 8 = 1.52$ (carry 1); $0.52 \times 8 = 4.16$ (carry 4); $0.16 \times 8 = 1.28$ (carry 1), and so on. Reading the carries from first to last gives the octal fraction as $0.141...\_8$. Combining the parts correctly yields the result $(1641.141)\_8$.

The incorrect option is (1):
The option $(1640.41)\_8$ is incorrect because it reflects several errors in the conversion process. The integer part, $1640\_8$, is clearly incorrect; it would be the octal equivalent of 928, not 929, which should be $1641\_8$. Moreover, the fractional part $0.41\_8$ is also incorrect. The conversion of the decimal fraction $0.19$ must begin with a multiplication result of $0.19 \times 8 = 1.52$, meaning the first digit after the radix point must be 1. Therefore, this option is numerically inaccurate for the given decimal number.

The incorrect option is (3):
The option $(1632.141)\_8$ contains the correct fractional part but the integer part is incorrect. As demonstrated in the explanation for the correct answer, the octal representation of the decimal integer 929 is $1641\_8$. The integer $1632\_8$ is the result of an arithmetic miscalculation during the repeated division process. A common mistake here would be an error in one of the remainder calculations, such as incorrectly calculating the remainder of $116 \div 8$ to be 3 instead of 4, which would lead to the third digit being 3.

The incorrect option is (4):
The option $(1543.34)\_8$ is a distant numerical approximation and contains significant errors in both parts of the conversion. The integer part $1543\_8$ is wrong, as the correct octal integer is $1641\_8$. The first digit of the correct integer is 1, not 3 as implied by $1543\_8$. Similarly, the fractional part $0.34\_8$ is incorrect. The first digit of the fractional conversion is 1, derived from the $0.19 \times 8$ operation, and not 3. This option shows a faulty application of both the division and multiplication methods for number system conversion.