Linked subgroup averages: avg(a,b,c)=11; avg(c,d,e)=17; avg(e,f)=22; avg(e,c)=17. Find the average of a, b, c, d, e, f.

Introduction / Context:
We are given several overlapping averages. Converting each to totals yields linear relations among the variables. The goal is the grand total divided by 6, which can be expressed using the provided sums and the known e + c relation.

Given Data / Assumptions:

• a + b + c = 33
• c + d + e = 51
• e + f = 44
• e + c = 34

Concept / Approach:
Express T = a + b + c + d + e + f using known sums. Replace a + b by 33 - c, d by 51 - c - e, and f by 44 - e. Substitute and simplify, then divide by 6 for the average.

Step-by-Step Solution:

Verification / Alternative check:
Plugging back: If average is 47/3, total is 94. All intermediate sums remain satisfied since we used identities only.

Why Other Options Are Wrong:

• 18 1/2, 21 1/3, 16 1/2, 17 1/6: do not equal 94/6.

Common Pitfalls:
Algebra slips when substituting or forgetting to use e + c = 34, leading to an expression still in terms of c and e.

Final Answer:
15 2/3