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.

Answer 2

This question asks for the octal equivalent of the decimal number \( 153.513 \). Since a mixed number has both an integer and a fractional part, each part is converted separately, but here the fractional part is explained through place value alongside a check that verifies the final answer against the original number.

First, the integer part \( 153 \) is converted using successive division by 8, since this step is unavoidable regardless of the method used for the fractional part:

\[ 153 \div 8 = 19 \text{ remainder } 1 \]\[ 19 \div 8 = 2 \text{ remainder } 3 \]\[ 2 \div 8 = 0 \text{ remainder } 2 \]

Reading the remainders bottom to top gives \( 231_8 \) for the integer part.

For the fractional part, each digit obtained by multiplying by 8 represents a place value of \( 8^{-1}, 8^{-2}, 8^{-3} \), and so on. Multiplying the fractional part by 8 repeatedly and reading off the integer portion at each stage builds up these place values one at a time:

\[ 0.513 \times 8 = 4.104 \Rightarrow \text{digit at } 8^{-1} \text{ is } 4 \]\[ 0.104 \times 8 = 0.832 \Rightarrow \text{digit at } 8^{-2} \text{ is } 0 \]\[ 0.832 \times 8 = 6.656 \Rightarrow \text{digit at } 8^{-3} \text{ is } 6 \]\[ 0.656 \times 8 = 5.248 \Rightarrow \text{digit at } 8^{-4} \text{ is } 5 \]\[ 0.248 \times 8 = 1.984 \Rightarrow \text{digit at } 8^{-5} \text{ is } 1 \]\[ 0.984 \times 8 = 7.872 \Rightarrow \text{digit at } 8^{-6} \text{ is } 7 \]

This gives the fractional octal digits \( .406517 \), so the full result is \( 231.406517_8 \).

To check this is right, convert it back toward decimal: the integer part \( 231_8 = (2 \times 64) + (3 \times 8) + 1 = 153 \), which matches exactly. The fractional part \( .406517_8 \) works out to approximately \( 0.513 \) when summed using the same place values, which also matches the original number closely.

Therefore, the correct answer is 231.406517.