Linked averages for overlapping triplets: Of four numbers a, b, c, d, the average of the first three is 15, the average of the last three is 16, and the last number d is 19. Find the first number a.

Introduction / Context:When averages of overlapping groups are known, sums can be equated to determine individual elements.Given Data / Assumptions:

• (a + b + c)/3 = 15
• (b + c + d)/3 = 16
• d = 19

Concept / Approach:Compute sums from averages, eliminate common parts, and solve for the unknown a.Step-by-Step Solution:

Verification / Alternative check:Check last three: b + c + d = 29 + 19 = 48, average = 16. Works.Why Other Options Are Wrong:

• 15, 18, 19: They do not satisfy both average relations simultaneously.

Common Pitfalls:Incorrectly averaging the two averages instead of using sums.Final Answer: