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:
We need to convert a fractional decimal number (base 10) into an octal number (base 8).


Step 2: Key Formula or Approach:
To convert a decimal number to octal, we split the number into its integer and fractional parts.
For the integer part, we repeatedly divide by 8 and record the remainders.
For the fractional part, we repeatedly multiply by 8 and record the integer parts of the successive products.


Step 3: Detailed Explanation:
First, let's convert the integer part, which is 153.
\[ 153 \div 8 = 19 \text{ with a remainder of } 1 \] \[ 19 \div 8 = 2 \text{ with a remainder of } 3 \] \[ 2 \div 8 = 0 \text{ with a remainder of } 2 \] Reading the remainders from bottom to top, we get \( 231 \).
So, \( (153)_{10} = (231)_8 \).
Next, let's convert the fractional part, which is 0.513.
\[ 0.513 \times 8 = 4.104 \implies \text{Integer part is } 4 \] \[ 0.104 \times 8 = 0.832 \implies \text{Integer part is } 0 \] \[ 0.832 \times 8 = 6.656 \implies \text{Integer part is } 6 \] \[ 0.656 \times 8 = 5.248 \implies \text{Integer part is } 5 \] \[ 0.248 \times 8 = 1.984 \implies \text{Integer part is } 1 \] \[ 0.984 \times 8 = 7.872 \implies \text{Integer part is } 7 \] Reading the extracted integer parts from top to bottom, we get \( .406517 \).
So, \( (0.513)_{10} \approx (0.406517)_8 \).
Combining the integer and fractional parts, we get \( (231.406517)_8 \).


Step 4: Final Answer:
The complete octal representation is 231.406517.