Three related numbers — The first equals twice the second and thrice the third. If their average is 154, find the difference between the first and the third.

Introduction / Context:
This is a ratio-based setup translated into linear equations. By expressing each number in terms of the first (or a common variable), you can compute the total, use the average to find the exact values, and then answer the query about the difference between specific terms.

Given Data / Assumptions:

• Let the three numbers be A (first), B (second), C (third).
• A = 2B and A = 3C → B = A/2, C = A/3.
• Their average is 154 → (A + B + C)/3 = 154.

Concept / Approach:
Sum in terms of A: A + A/2 + A/3 = A * (1 + 1/2 + 1/3) = A * (11/6). The average condition gives (A * 11/6)/3 = 154, from which A is determined. Then compute C and the required difference A - C.

Step-by-Step Solution:
Average condition → (A + A/2 + A/3)/3 = 154 → A * 11/18 = 154.Solve for A: A = 154 * 18 / 11 = 14 * 18 = 252.Then C = A/3 = 252/3 = 84.Difference A - C = 252 - 84 = 168.

Verification / Alternative check:
Compute B = A/2 = 126. Average: (252 + 126 + 84)/3 = 462/3 = 154 as required.

Why Other Options Are Wrong:

• 126 / 42 / 52 / 108: These stem from mixing the relations (twice vs thrice), miscomputing the average, or subtracting the wrong pair.

Common Pitfalls:
Using 2C = A instead of A = 3C; forgetting to divide the sum by 3 for the average; arithmetic slips in fractions.

Final Answer:
168