(929.19)10 ≈ ?

To convert a decimal number into another base b, the integer part and the fractional part are converted separately. The integer part is converted by dividing it repeatedly by b and reading the remainders from bottom to top. The fractional part is converted by multiplying it repeatedly by b and reading the integer digit produced at each step from top to bottom, continuing until the required number of digits is reached (a fractional part need not terminate, so it is usually rounded or truncated to a fixed number of digits).

Integer part: converting 929 to base 8 by repeated division.

1. 929 ÷ 8 = 116, remainder 1

2. 116 ÷ 8 = 14, remainder 4

3. 14 ÷ 8 = 1, remainder 6

4. 1 ÷ 8 = 0, remainder 1

Reading the remainders from bottom to top gives the integer part 1641.

Fractional part: converting 0.19 to base 8 by repeated multiplication.

1. 0.19 × 8 = 1.52 → digit 1, carry forward 0.52

2. 0.52 × 8 = 4.16 → digit 4, carry forward 0.16

3. 0.16 × 8 = 1.28 → digit 1, carry forward 0.28

This multiplication does not terminate (the next step would continue 0.28 × 8 = 2.24 → digit 2, and so on), so the fractional part is truncated to the three-digit precision used in the options: reading the digits from top to bottom gives .141.

Combining both parts, (929.19)10 ≈ (1641.141)8 — an approximation, since the octal fraction of 0.19 does not terminate exactly and must be truncated.

Cross-check both parts by converting 1641.1418 back to decimal using place values. Integer: 1×83 + 6×82 + 4×81 + 1×80 = 512 + 384 + 32 + 1 = 929, an exact match with the given integer part. Fraction: 1/8 + 4/64 + 1/512 = 0.125 + 0.0625 + 0.001953 ≈ 0.1895, which is what (1641.141)8 actually represents — a close approximation to the given 0.19 at the three-digit truncation used throughout the options.