Question

Following are given conversion of numbers from one base system to another base system, which conversion is correct? 
I. \( (4523)_{6=(1059)_{10}} \) 
II. \( (4523)_{6=(1069)_{10}} \) 
III. \( (0.203)_{5=(0.424)_{10}} \)
IV. \( (0.203)_{5=(0.406)_{10} }\)

• A. I and III
• B. II and III
• C. I and IV
• D. II and IV

Hint:

For fractional base conversion \( (0.xyz)_b \), expand as \( x/b + y/b^2 + z/b^3 \).

Answer 1

The Correct Option is A

Step 1: Check Statement I & II (Base 6 to 10):
Convert \( (4523)_6 \) to base 10: \[ 4 \times 6^3 + 5 \times 6^2 + 2 \times 6^1 + 3 \times 6^0 \] \[ = 4(216) + 5(36) + 2(6) + 3 \] \[ = 864 + 180 + 12 + 3 \] \[ = 1059 \] Thus, Statement I is Correct and II is Incorrect. Step 2: Check Statement III & IV (Base 5 to 10):
Convert \( (0.203)_5 \) to base 10: \[ 2 \times 5^{-1} + 0 \times 5^{-2} + 3 \times 5^{-3} \] \[ = \frac{2}{5} + 0 + \frac{3}{125} \] \[ = 0.4 + 0.024 \] \[ = 0.424 \] Thus, Statement III is Correct and IV is Incorrect. Step 3: Final Answer:
Statements I and III are correct.