<p>There is a school that has classes of students and each class will be having a final exam. You are given a 2D integer array <code>classes</code>, where <code>classes[i] = [pass<sub>i</sub>, total<sub>i</sub>]</code>. You know beforehand that in the <code>i<sup>th</sup></code> class, there are <code>total<sub>i</sub></code> total students, but only <code>pass<sub>i</sub></code> number of students will pass the exam.</p>

<p>You are also given an integer <code>extraStudents</code>. There are another <code>extraStudents</code> brilliant students that are <strong>guaranteed</strong> to pass the exam of any class they are assigned to. You want to assign each of the <code>extraStudents</code> students to a class in a way that <strong>maximizes</strong> the <strong>average</strong> pass ratio across <strong>all</strong> the classes.</p>

<p>The <strong>pass ratio</strong> of a class is equal to the number of students of the class that will pass the exam divided by the total number of students of the class. The <strong>average pass ratio</strong> is the sum of pass ratios of all the classes divided by the number of the classes.</p>

<p>Return <em>the <strong>maximum</strong> possible average pass ratio after assigning the </em><code>extraStudents</code><em> students. </em>Answers within <code>10<sup>-5</sup></code> of the actual answer will be accepted.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> classes = [[1,2],[3,5],[2,2]], <code>extraStudents</code> = 2
<strong>Output:</strong> 0.78333
<strong>Explanation:</strong> You can assign the two extra students to the first class. The average pass ratio will be equal to (3/4 + 3/5 + 2/2) / 3 = 0.78333.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> classes = [[2,4],[3,9],[4,5],[2,10]], <code>extraStudents</code> = 4
<strong>Output:</strong> 0.53485
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= classes.length &lt;= 10<sup>5</sup></code></li>
	<li><code>classes[i].length == 2</code></li>
	<li><code>1 &lt;= pass<sub>i</sub> &lt;= total<sub>i</sub> &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= extraStudents &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    def maxAverageRatio(self, classes: List[List[int]], extraStudents: int) -> float:
        import heapq
        from typing import List

        max_heap = []
        
        # Function to calculate potential gain from adding one student to a class
        def ratio_gain(p, t):
            # Current pass ratio
            current_ratio = p / t
            # New pass ratio if one student is added
            new_ratio = (p + 1) / (t + 1)
            # Gain in ratio
            return new_ratio - current_ratio
        
        # Initialize the heap with all classes
        for p, t in classes:
            # Push negative gain to make it a max-heap
            heapq.heappush(max_heap, (-ratio_gain(p, t), p, t))
        
        # Distribute extra students
        for _ in range(extraStudents):
            # Get the class with the maximum gain
            neg_gain, p, t = heapq.heappop(max_heap)
            # Update the class with one more student
            p += 1
            t += 1
            # Push the updated class back into the heap
            heapq.heappush(max_heap, (-ratio_gain(p, t), p, t))
        
        # Calculate the final average pass ratio
        total_ratio = 0
        while max_heap:
            _, p, t = heapq.heappop(max_heap)
            total_ratio += p / t
        
        return total_ratio / len(classes)
2135