The total weight of *x* students is $62x$ kg. Two students weighing $82.5$ kg and $78.5$ kg leave the group, and a new student weighing $162$ kg joins. After these changes, the average weight of the group becomes $65.5$ kg. Find the value of *x*.

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**Solution:**

1. Let the number of students be *x*. The total weight of these *x* students is:
   
   $$62x$$

2. Two students weighing $82.5$ kg and $78.5$ kg leave the group, so the total weight reduces by:
   
   $$82.5 + 78.5 = 161 	ext{ kg}$$

3. A new student weighing $162$ kg joins the group, so the total weight increases by $162$ kg.

4. After these changes, the number of students becomes:
   
   $$x - 2 + 1 = x - 1$$

5. The new total weight is:
   
   $$62x - 161 + 162 = 62x + 1$$

6. The new average weight is given as $65.5$ kg, so:
   
   $$\frac{62x + 1}{x - 1} = 65.5$$

7. Multiply both sides by $(x - 1)$:
   
   $$62x + 1 = 65.5(x - 1)$$

8. Expand the right side:
   
   $$62x + 1 = 65.5x - 65.5$$

9. Rearranging terms:
   
   $$65.5x - 62x = 1 + 65.5$$
   $$3.5x = 66.5$$

10. Solve for *x*:
    
    $$x = \frac{66.5}{3.5} = 19$$

**Answer:** $19$

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