Number Base Conversion and Ordering

The question requires converting numbers from different bases (base-4, base-2, base-8, base-16) to the decimal system and then arranging them in descending order.

Convert Numbers to Decimal System

We convert each number individually:

• A. $(20003.2)_4$
  Decimal value $= (2 \times 4^4) + (0 \times 4^3) + (0 \times 4^2) + (0 \times 4^1) + (3 \times 4^0) + (2 \times 4^{-1})$
  $= (2 \times 256) + 0 + 0 + 0 + (3 \times 1) + (2 \times \frac{1}{4})$
  $= 512 + 3 + 0.5 = 515.5
• B. $(111111.11)_2$
  Decimal value $= (1 \times 2^5) + (1 \times 2^4) + (1 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) + (1 \times 2^{-1}) + (1 \times 2^{-2})$
  $= 32 + 16 + 8 + 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4}$
  $= 63 + 0.5 + 0.25 = 63.75
• C. $(732.2)_8$
  Decimal value $= (7 \times 8^2) + (3 \times 8^1) + (2 \times 8^0) + (2 \times 8^{-1})$
  $= (7 \times 64) + (3 \times 8) + (2 \times 1) + (2 \times \frac{1}{8})$
  $= 448 + 24 + 2 + 0.25 = 474.25
• D. $(C2.6)_{16}$
  (Note: C in hexadecimal is 12 in decimal)
  Decimal value $= (12 \times 16^1) + (2 \times 16^0) + (6 \times 16^{-1})$
  $= (12 \times 16) + (2 \times 1) + (6 \times \frac{1}{16})$
  $= 192 + 2 + 0.375 = 194.375

Determine Descending Order

The decimal values are:

• A: $515.5$
• C: $474.25$
• D: $194.375$
• B: $63.75$

Arranging these values in descending order (from largest to smallest):

This corresponds to the order: A, C, D, B.