Number system conversion — decimal to octal (with fraction): Find the octal equivalent of the decimal number 324.987 (rounded to at least three fractional octal digits).

Introduction / Context:
Converting decimal numbers to octal involves separate handling of the integer and fractional parts. The integer part is repeatedly divided by 8, while the fractional part is repeatedly multiplied by 8. Octal is base-8, so each step cleanly extracts digits without resorting to binary intermediates.

Given Data / Assumptions:

• Decimal value: 324.987.
• We need a correct octal equivalent to at least three fractional digits.
• Rounding or truncation is acceptable after three fractional octal digits, as typical in MCQs.

Concept / Approach:
For the integer part: use repeated division by 8 to obtain digits from least significant to most. For the fractional part: multiply by 8, record the integer part as the next octal digit, and iterate on the fractional remainder.

Step-by-Step Solution:
Integer part 324: 324 / 8 = 40 remainder 4 → LSB digit 4.40 / 8 = 5 remainder 0 → next digit 0.5 / 8 = 0 remainder 5 → MSB digit 5. Integer octal = 504.Fractional part 0.987: 0.987 * 8 = 7.896 → digit 7; remainder 0.896.0.896 * 8 = 7.168 → digit 7; remainder 0.168.0.168 * 8 = 1.344 → digit 1; remainder 0.344 (further digits would be 2, 6, 0, …).Combine: 324.987 ≈ 504.771 in octal (to three fractional octal digits).

Verification / Alternative check:
Back-convert: 504.771₈ ≈ 58^2 + 08 + 4 + 7/8 + 7/8^2 + 1/8^3 = 320 + 0 + 4 + 0.875 + 0.109375 + 0.001953125 ≈ 324.9863, closely matching 324.987 with small truncation error.

Why Other Options Are Wrong:
640.781₈ and 815.234₈ decode to very different decimal values; 90.987 mixes bases incorrectly; they do not match 324.987.

Common Pitfalls:
Reversing division/multiplication steps, or rounding too early and losing accuracy. Keep integer and fractional conversions separate.

Final Answer:
504.771.