If (L)M denotes the number L written in base M, identify the correct ascending…
2024
If (L)M denotes the number L written in base M, identify the correct ascending order of the following numbers after converting them to decimal.
A–D denote the four given numbers:
A. (11001.11)2
B. (123.2)4
C. (31.5)8
D. (19.6)16
Answer: D. D,C,A,B — ConceptIn a positional numeral with base b, a digit at position i contributes digit × bi. Digits to the right of the radix point use negative powers. To…
- A.
D,A,C,B
- B.
B,C,D,A
- C.
C,D,A,B
- D.
D,C,A,B
Attempted by 10 students.
Show answer & explanation
Correct answer: D
Concept
In a positional numeral with base b, a digit at position i contributes digit × bi. Digits to the right of the radix point use negative powers.
To compare numerals written in different bases, expand each numeral in powers of its own base, convert every value to decimal, and then compare the decimal values.
Application
For A: (11001.11)2 = 1×24 + 1×23 + 0×22 + 0×21 + 1×20 + 1×2−1 + 1×2−2 = 16 + 8 + 0 + 0 + 1 + 0.5 + 0.25 = 25.75.
For B: (123.2)4 = 1×42 + 2×41 + 3×40 + 2×4−1 = 16 + 8 + 3 + 0.5 = 27.5.
For C: (31.5)8 = 3×81 + 1×80 + 5×8−1 = 24 + 1 + 0.625 = 25.625.
For D: (19.6)16 = 1×161 + 9×160 + 6×16−1 = 16 + 9 + 0.375 = 25.375.
Cross-check
The decimal values satisfy 25.375 < 25.625 < 25.75 < 27.5. The fractional contributions also use the reciprocal of each stated base.
Therefore, the ascending order is D,C,A,B.