Arrange the following in ascending order of magnitude $110_{two}, 31_{eight}, 42_{five}$

Analyze the question: The three numbers are in different bases. To compare them, convert each to base 10, then arrange from smallest to largest.

Concept: In base b, each digit is multiplied by a power of b, starting from b⁰ on the right.

Solution:

1. 110₂ = 1×2² + 1×2¹ + 0×2⁰ = 4 + 2 + 0 = 6
2. 31₈ = 3×8¹ + 1×8⁰ = 24 + 1 = 25
3. 42₅ = 4×5¹ + 2×5⁰ = 20 + 2 = 22

Ascending order: 6 < 22 < 25, which gives

110₂ < 42₅ < 31₈

Checking the options:

• A. Uses 31₅ and 42₈, which are not the given numbers, so it is wrong.
• B. 42₅ (22), 110₂ (6), 31₈ (25) is not in order, so it is wrong.
• C. 42₅ (22), 31₈ (25), 110₂ (6) is not in order, so it is wrong.
• D. 110₂ (6), 42₅ (22), 31₈ (25) is in ascending order, so it is correct.

Tip: In base-comparison questions, always convert everything to base 10 first. Also check that the digits and bases in an option match the original question, as option A tries to trick you by swapping bases.