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

Introduction:
This question checks understanding of weighted averages and how class sizes affect an overall average. The averages of three different classes and the averages of pairs of classes are given. From these relations, we can find the relative sizes of the classes and then compute the combined average of all students in classes A, B and C together.

Given Data / Assumptions:
- Average score of class A = 83. - Average score of class B = 76. - Average score of class C = 85. - Average score of classes A and B together = 79. - Average score of classes B and C together = 81. - We are asked for the average score of all students in classes A, B and C together.

Concept / Approach:
Let the number of students in classes A, B and C be a, b and c respectively. Then the total scores of the three classes are 83a, 76b and 85c. Using the given combined averages of A and B, and B and C, we can set up equations involving a, b and c. From these equations we can find ratios between a, b and c. Once the ratios are known, the overall average is the total score divided by the total number of students.

Step-by-Step Solution:
Step 1: From the average of A and B, we have (83a + 76b) / (a + b) = 79. Step 2: Multiply both sides by (a + b): 83a + 76b = 79a + 79b. Step 3: Rearrange: 83a - 79a = 79b - 76b which gives 4a = 3b, so b = (4/3)a. Step 4: From the average of B and C, (76b + 85c) / (b + c) = 81. Step 5: Multiply both sides: 76b + 85c = 81b + 81c, so 85c - 81c = 81b - 76b which gives 4c = 5b, so c = (5/4)b. Step 6: Substitute b = (4/3)a into c = (5/4)b to get c = (5/4)*(4/3)a = (5/3)a. Step 7: Total score of all students = 83a + 76b + 85c = 83a + 76*(4/3)a + 85*(5/3)a. Step 8: Compute the coefficients: 76*(4/3) = 304/3 and 85*(5/3) = 425/3. So total score = 83a + (304/3)a + (425/3)a. Step 9: Combine terms: (304/3 + 425/3) = 729/3 = 243. Thus total score = 83a + 243a = 326a. Step 10: Total number of students = a + b + c = a + (4/3)a + (5/3)a = a + 3a = 4a. Step 11: Overall average = total score / total students = 326a / 4a = 81.5.

Verification / Alternative Check:
We can check that the derived ratios a : b : c = 3 : 4 : 5 are consistent. If a = 3, b = 4 and c = 5, total students = 12. Then total score = 83*3 + 76*4 + 85*5. This equals 249 + 304 + 425 = 978. Average = 978 / 12 = 81.5, which matches the result. The pairwise averages also work out correctly, confirming our solution.

Why Other Options Are Wrong:
Options 78, 79.5, 80 and 82 do not use the correct ratios between the class sizes and treat the averages as if each class had the same number of students or use incorrect algebra. They therefore lead to incorrect combined averages.

Common Pitfalls:
Students often assume that each class has the same number of students and simply average the three given averages. That method ignores the effect of class size on the combined average. Another common mistake is to mix up the equations when forming relations from the pairwise averages. Careful algebra and stepwise manipulation are essential.

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