Three maths classes A, B and C take an algebra test. The average score of class A is 83, of class B is 76 and of class C is 85. The combined average score of classes A and B together is 79, and that of classes B and C together is 81. What is the average score of all three classes A, B and C together?

Introduction / Context:
This problem involves multiple averages for different subsets of three classes. From the given information, we infer the relative sizes of the classes and then use a weighted average to find the overall average score.

Given Data / Assumptions:
- Average score of class A = 83.
- Average score of class B = 76.
- Average score of class C = 85.
- Average of classes A and B together = 79.
- Average of classes B and C together = 81.
- We assume each class has a certain number of students but not necessarily the same size.

Concept / Approach:
Let the number of students in classes A, B and C be a, b and c respectively. Then:
Total score of A = 83a, of B = 76b, of C = 85c.
The combined averages give two equations:
(83a + 76b) / (a + b) = 79(76b + 85c) / (b + c) = 81
We use these equations to find the ratios between a, b and c, then compute the overall average.

Step-by-Step Solution:
Step 1: From (83a + 76b) / (a + b) = 79, cross multiply:83a + 76b = 79a + 79b.Step 2: Rearrange: 83a - 79a = 79b - 76b, so 4a = 3b.Step 3: Hence a : b = 3 : 4.Step 4: From (76b + 85c) / (b + c) = 81, cross multiply:76b + 85c = 81b + 81c.Step 5: Rearrange: 85c - 81c = 81b - 76b, so 4c = 5b.Step 6: Hence b : c = 4 : 5.Step 7: Combining the two ratios, we get a : b : c = 3 : 4 : 5.Step 8: Let a = 3k, b = 4k, c = 5k.Step 9: Total score = 83a + 76b + 85c = 83 * 3k + 76 * 4k + 85 * 5k.Step 10: Compute: 83 * 3 = 249, 76 * 4 = 304, 85 * 5 = 425, so total score = (249 + 304 + 425)k = 978k.Step 11: Total number of students = a + b + c = 3k + 4k + 5k = 12k.Step 12: Overall average = 978k / 12k = 978 / 12 = 81.5.

Verification / Alternative check:
You can also check consistency with each pair of classes. Using the ratios, confirm that the computed totals give the stated combined averages of 79 and 81, which they do, so the ratios and overall average are consistent.

Why Other Options Are Wrong:
- 81, 78 and 80.5 do not satisfy the weighted combination of the three class averages given the derived class size ratios 3 : 4 : 5. They would imply incorrect total scores or inconsistent pair averages.

Common Pitfalls:
- Assuming equal class sizes and taking the simple average of 83, 76 and 85, which would give a different result.
- Not setting up the pairwise average equations correctly, which leads to incorrect ratios and hence an incorrect overall average.

Final Answer:
The average score of all three classes A, B and C together is 81.5.