We are given the average scores of three classes A, B, and C and some combined averages between them. Let's break down the information step by step:

• Average score of class A = 83
• Average score of class B = 76
• Average score of class C = 85
• Average score of class A and B combined = 79
• Average score of class B and C combined = 81

Step 1: Let the number of students in class A be $a$, in class B be $b$, and in class C be $c$.

Using the definition of average, we can write the following equations based on the information given:

For class A: $\frac{\text{Total score of A}}{a} = 83 \quad \Rightarrow \quad \text{Total score of A} = 83a$

For class B: $\frac{\text{Total score of B}}{b} = 76 \quad \Rightarrow \quad \text{Total score of B} = 76b$

For class C: $\frac{\text{Total score of C}}{c} = 85 \quad \Rightarrow \quad \text{Total score of C} = 85c$

For the combined average of classes A and B: $\frac{83a + 76b}{a + b} = 79 \quad \Rightarrow \quad 83a + 76b = 79(a + b)$ Expanding the equation: $83a + 76b = 79a + 79b \quad \Rightarrow \quad 4a = 3b \quad \Rightarrow \quad a = \frac{3}{4}b$

For the combined average of classes B and C: $\frac{76b + 85c}{b + c} = 81 \quad \Rightarrow \quad 76b + 85c = 81(b + c)$ Expanding this equation: $76b + 85c = 81b + 81c \quad \Rightarrow \quad 4c = 5b \quad \Rightarrow \quad c = \frac{5}{4}b$

Step 2: Calculate the total score for classes A, B, and C

We know the relationships between the sizes of the classes: $a = \frac{3}{4}b, \quad c = \frac{5}{4}b$ The total score of all students in A, B, and C combined is: $\text{Total score} = 83a + 76b + 85c$

Substitute $a = \frac{3}{4}b$ and $c = \frac{5}{4}b$ into the total score expression: $\text{Total score} = 83\left(\frac{3}{4}b\right) + 76b + 85\left(\frac{5}{4}b\right)$ $\text{Total score} = \frac{249}{4}b + 76b + \frac{425}{4}b$ $\text{Total score} = \frac{249b + 304b + 425b}{4} = \frac{978b}{4}$

Step 3: Calculate the total number of students

The total number of students in A, B, and C combined is: $\text{Total number of students} = a + b + c = \frac{3}{4}b + b + \frac{5}{4}b = \frac{3b + 4b + 5b}{4} = \frac{12b}{4} = 3b$

Step 4: Calculate the average score of all students

The average score of all students in A, B, and C is: $\text{Average score} = \frac{\text{Total score}}{\text{Total number of students}} = \frac{\frac{978b}{4}}{3b} = \frac{978}{12} = 81.5$

Thus, the average score of the students in classes A, B, and C is 81.5.

The correct answer is 81.5.

Would you like further explanation or details on any step?

Relative Questions:

1. How do you calculate the combined average of two groups?
2. How does the number of students in each class affect the overall average?
3. What would happen if one class had a significantly higher number of students?
4. Can you explain how averages change when combining two or more groups with different averages?
5. How would you calculate the median score instead of the average?

Tip: When combining averages, it's important to account for the size of each group to get an accurate overall average.