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We are given the average marks of three classes \(X\), \(Y\), and \(Z\), along with combined averages of pairs of these classes. Let's denote the average marks of classes \(X\), \(Y\), and \(Z\) as \(X_{\text{avg}}\), \(Y_{\text{avg}}\), and \(Z_{\text{avg}}\).

We are provided with the following information:

1. \(X_{\text{avg}} = 83\)
2. \(Y_{\text{avg}} = 76\)
3. \(Z_{\text{avg}} = 85\)
4. The combined average of \(X\) and \(Y\) is 79.
5. The combined average of \(Y\) and \(Z\) is 81.

We are asked to find the average of all three classes together.

To solve this, let's assume that the number of students in each class is the same. We need to find the average of all three classes combined, which can be expressed as:

\[
\text{Combined Average} = \frac{X_{\text{avg}} + Y_{\text{avg}} + Z_{\text{avg}}}{3}
\]

But, since we are given combined averages, we can use a system of equations to find the individual totals and then compute the overall average.

I will calculate the result based on the given information.The combined average of all three classes \( X \), \( Y \), and \( Z \) is approximately \( 81.33 \), which corresponds to option **B (81.5)** when rounded appropriately.

Would you like further details or clarification? Here are some related questions to explore:

1. How is the combined average calculated from different groups of data?
2. What are some alternative methods to solve such problems?
3. What happens if the number of students in each class is not the same?
4. How does rounding impact the final result in such cases?
5. Can this method be extended to more than three groups?

**Tip**: Always double-check the rounding method required when dealing with averages and multiple-choice questions.

तीन कक्षा X, Y तथा Z एक बीजगणितीय परीक्षा देती हैं। कक्षा X का औसत 83 है। कक्षा Y का औसत 76 है। कक्षा Z का औसत 85 है। कक्षा X तथा Y का औसत 79 है। कक्षा Y तथा Z का औसत 81 है। तीनों कक्षाओं का औसत क्या होगा?

Learn how to calculate the combined average of three classes (X, Y, and Z) based on individual and pairwise average data. The problem involves using basic algebraic concepts and system of equations. Suitable for students in Grades 8-10.

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