Question

Convert \( (153.513)_{10 \) in octal number system is}

• A. 231.408517
• B. 231.407517
• C. 231.406517
• D. 231.406617

Hint:

Notice that Option A contains the digit '8', which is strictly invalid in the octal number system (base 8 only uses digits 0-7), allowing you to eliminate it immediately.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This problem involves the conversion of a real number (containing both an integer part and a fractional part) from the Decimal system, base 10, to the Octal system, base 8,. This is a two-step process because the integer and fractional components are handled using different mathematical operations. The topic is fundamental to digital logic and data representation.
Step 2: Key Formulas and approach:
The approach involves two distinct methods:
1. For the Integer Part: Use the "Successive Division" method. Divide the number by the target base (8) repeatedly and track the remainders.
2. For the Fractional Part: Use the "Successive Multiplication" method. Multiply the fraction by the target base (8) and track the resulting integer parts.
The integer remainders are read from bottom-to-top (Reverse), while the fractional integer results are read from top-to-bottom (Forward).
Step 3: Detailed Explanation:}

Step A: Integer Conversion (153):

$153 \div 8 = 19$ with a remainder of 1.

$19 \div 8 = 2$ with a remainder of 3.

$2 \div 8 = 0$ with a remainder of 2.

Reading the remainders upwards, we get $231_8$.

Step B: Fractional Conversion (0.513):

$0.513 \times 8 = 4.104$ (Integer is 4).

$0.104 \times 8 = 0.832$ (Integer is 0).

$0.832 \times 8 = 6.656$ (Integer is 6).

$0.656 \times 8 = 5.248$ (Integer is 5).

$0.248 \times 8 = 1.984$ (Integer is 1).

$0.984 \times 8 = 7.872$ (Integer is 7).

Combining the fractional results top-to-bottom, we get $.406517_8$.

Finally, merging the two parts: $231 + .406517 = 231.406517_8$.

Step 4: Final Answer:
The decimal number $(153.513)_{10}$ is equal to $231.406517$ in the octal system.

Answer 2

All four options for the octal equivalent of \( 153.513 \) begin with \( 231.40 \), so the integer part and the first two fractional digits are not in dispute. The difference lies entirely in the remaining digits, so recomputing just those later multiplication steps precisely settles which option is correct.

The fractional conversion up to this point has already produced digits 4 and 0 from these steps:

\[ 0.513 \times 8 = 4.104 \Rightarrow 4 \]\[ 0.104 \times 8 = 0.832 \Rightarrow 0 \]

Continuing carefully from \( 0.832 \):

\[ 0.832 \times 8 = 6.656 \Rightarrow 6 \]\[ 0.656 \times 8 = 5.248 \Rightarrow 5 \]\[ 0.248 \times 8 = 1.984 \Rightarrow 1 \]\[ 0.984 \times 8 = 7.872 \Rightarrow 7 \]

This precise recomputation yields the digit sequence \( 6, 5, 1, 7 \), confirming the fractional part as \( .406517 \). Now checking each option against this sequence:

1. 231.408517: This has an 8 where the recomputation gives a 6 as the third fractional digit, so it does not match.
2. 231.407517: This has a 7 where the recomputation gives a 6 as the third fractional digit, so it does not match either.
3. 231.406517: Every digit here, \( 4, 0, 6, 5, 1, 7 \), matches the recomputed sequence exactly.
4. 231.406617: This has a 6 where the recomputation gives a 1 as the fifth fractional digit, so it does not match.

Only one option reproduces the recomputed fractional digits exactly, with no deviation at any step.

Therefore, the correct answer is 231.406517.