Number Base Conversion to Decimal

The question requires converting numbers from different bases (base 4, 2, 8, and 16) to the decimal system (base 10) and then arranging them in descending order.

Conversion of Number A

Convert $(30002.2)_4$ to decimal:

$ (30002.2)_4 = 3 \times 4^4 + 0 \times 4^3 + 0 \times 4^2 + 0 \times 4^1 + 2 \times 4^0 + 2 \times 4^{-1} $

$ = 3 \times 256 + 0 + 0 + 0 + 2 \times 1 + 2 \times \frac{1}{4} $

$ = 768 + 2 + 0.5 = 770.5 $

So, $\textbf{A} = 770.5_{10}$.

Conversion of Number B

Convert $(100000.11)_2$ to decimal:

$ (100000.11)_2 = 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 + 1 \times 2^{-1} + 1 \times 2^{-2} $

$ = 1 \times 32 + 0 + 0 + 0 + 0 + 0 + 1 \times \frac{1}{2} + 1 \times \frac{1}{4} $

$ = 32 + 0.5 + 0.25 = 32.75 $

So, $\textbf{B} = 32.75_{10}$.

Conversion of Number C

Convert $(666.2)_8$ to decimal:

$ (666.2)_8 = 6 \times 8^2 + 6 \times 8^1 + 6 \times 8^0 + 2 \times 8^{-1} $

$ = 6 \times 64 + 6 \times 8 + 6 \times 1 + 2 \times \frac{1}{8} $

$ = 384 + 48 + 6 + 0.25 = 438.25 $

So, $\textbf{C} = 438.25_{10}$.

Conversion of Number D

Convert $(D5.6)_{16}$ to decimal. Note that 'D' in hexadecimal is 13 in decimal.

$ (D5.6)_{16} = (13 \times 16^1 + 5 \times 16^0 + 6 \times 16^{-1}) $

$ = 13 \times 16 + 5 \times 1 + 6 \times \frac{1}{16} $

$ = 208 + 5 + 0.375 = 213.375 $

So, $\textbf{D} = 213.375_{10}$.

Decimal Values Summary

The decimal equivalents are:

A = 770.5

B = 32.75

C = 438.25

D = 213.375

Determining Descending Order

Arrange the decimal values in descending order:

$ 770.5 > 438.25 > 213.375 > 32.75 $

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

Therefore, the correct option is the one listing the order A, C, D, B.