If (L)M represents a number L in base-M number system, then identify the…
2025
If (L)M represents a number L in base-M number system, then identify the correct descending order of the following numbers A-D when converted to decimal number system.
A. (30022.2)4
B. (10000110.11)2
C. (777.2)8
D. (197.6)16
Choose the correct answer from the options given below :
Answer: B. A, C, D, B — ConceptIn a positional number system with base b, each digit is multiplied by a power of b determined by its position relative to the radix point. Positions…
- A.
B, D, C, A
- B.
A, C, D, B
- C.
C, A, B, D
- D.
D, B, C, A
Attempted by 13 students.
Show answer & explanation
Correct answer: B
Concept
In a positional number system with base b, each digit is multiplied by a power of b determined by its position relative to the radix point.
Positions to the left use b0, b1, b2, ... from right to left, while positions to the right use b-1, b-2, ... . The resulting products are added to obtain the decimal value.
Application
A = (30022.2)4 = 3×44 + 0×43 + 0×42 + 2×41 + 2×40 + 2×4-1 = 768 + 8 + 2 + 0.5 = 778.5.
B = (10000110.11)2 = 1×27 + 1×22 + 1×21 + 1×2-1 + 1×2-2 = 128 + 4 + 2 + 0.5 + 0.25 = 134.75.
C = (777.2)8 = 7×82 + 7×81 + 7×80 + 2×8-1 = 448 + 56 + 7 + 0.25 = 511.25.
D = (197.6)16 = 1×162 + 9×161 + 7×160 + 6×16-1 = 256 + 144 + 7 + 0.375 = 407.375.
Cross-check
Reverse conversion checks each result: 778.5 in base 4 gives (30022.2)4; 134.75 in base 2 gives (10000110.11)2; 511.25 in base 8 gives (777.2)8; and 407.375 in base 16 gives (197.6)16. Thus each decimal conversion reproduces its original numeral.
Thus, 778.5 > 511.25 > 407.375 > 134.75, so the descending order is A, C, D, B.