What should come in place of the question mark in the following expression involving mixed fractions? 3 1/6 + 4 1/2 − 2 2/3 − 1 11/12 = ?

Introduction / Context:
This question checks your ability to work with mixed fractions, converting them to improper fractions, finding a common denominator and simplifying the result back into a mixed fraction. Careful fraction arithmetic is very important in many competitive examinations and real-life calculations where quantities are not whole numbers.

Given Data / Assumptions:

• Expression: 3 1/6 + 4 1/2 − 2 2/3 − 1 11/12.
• All numbers are positive mixed fractions and standard arithmetic rules apply.
• We must perform addition and subtraction accurately, expressing the result as a simplified mixed fraction.

Concept / Approach:
The usual strategy is to convert each mixed fraction into an improper fraction, then convert all fractions to a common denominator, perform the required additions and subtractions, and finally simplify the result. Using the least common multiple of the denominators keeps calculations manageable and reduces the amount of simplification needed at the end.

Step-by-Step Solution:
Convert each mixed fraction: 3 1/6 = (3×6 + 1)/6 = 19/6.4 1/2 = (4×2 + 1)/2 = 9/2.2 2/3 = (2×3 + 2)/3 = 8/3.1 11/12 = (1×12 + 11)/12 = 23/12.The denominators are 6, 2, 3 and 12. The least common multiple is 12.Rewrite each fraction with denominator 12: 19/6 = 38/12, 9/2 = 54/12, 8/3 = 32/12, and 23/12 stays 23/12.Now compute: (38/12 + 54/12) − (32/12 + 23/12) = 92/12 − 55/12 = 37/12.Write 37/12 as a mixed fraction: 37 ÷ 12 = 3 with remainder 1, so the result is 3 1/12.

Verification / Alternative check:
We can sum the positive fractions first: 3 1/6 + 4 1/2 ≈ 3.17 + 4.5 = 7.67 (approximately).Sum the negative fractions: 2 2/3 + 1 11/12 ≈ 2.67 + 1.92 = 4.59 (approximately).Difference is about 7.67 − 4.59 ≈ 3.08, which is close to 3 1/12 (about 3.0833). This rough check supports our exact calculation.

Why Other Options Are Wrong:
An answer around 1 7/12 or 1 5/12 is too small because the positive terms clearly outweigh the negative terms by more than 3 units.3 7/12 is larger than 3 1/12 by half a unit, and would require a numerator of 43 over 12, which does not match the correctly obtained 37 over 12.

Common Pitfalls:
Students often make mistakes when converting mixed fractions to improper fractions by miscalculating the numerator.Another pitfall is forgetting to convert all fractions to a common denominator before adding or subtracting, which produces inconsistent results.

Final Answer:
The correct simplified value of the expression is 3 1/12.