Let's break the problem down step-by-step:

Step 1: Define the variables

• Let the number of students before the 5 students were added be $x$.
• The total marks of these $x$ students would then be $75x$ (because their average is 75).

Step 2: Calculate the new total number of students

• When 5 more students are added, the new number of students becomes $x + 5$.
• The new average is given as 78, so the total marks of all $x + 5$ students can be expressed as $78(x + 5)$.

Step 3: Calculate the total marks of the 5 students added

• The marks of the 5 students added are: 80, 85, 89, 81, and 82.
• Their total marks are: $80 + 85 + 89 + 81 + 82 = 417$

Step 4: Set up the equation for the total marks

The total marks of all $x + 5$ students can be expressed in two ways:

• From the original students: $75x$
• From the 5 new students: 417

So, the total marks of $x + 5$ students is: $75x + 417$ We also know that the total marks of all $x + 5$ students is $78(x + 5)$. So, we can set up the following equation: $75x + 417 = 78(x + 5)$

Step 5: Solve for $x$

Expand both sides of the equation: $75x + 417 = 78x + 390$ Now, move the terms involving $x$ to one side and constants to the other: $417 - 390 = 78x - 75x$ $27 = 3x$ $x = \frac{27}{3} = 9$

Step 6: Total number of students

The total number of students is $x + 5 = 9 + 5 = 14$.

Final Answer:

The total number of students in the class is 14.

Would you like further details on this solution or have any other questions? Here are some related questions you could explore:

1. How would the average change if the 5 students had different marks?
2. What if the average of the original students was different, say 80 instead of 75?
3. How would the total marks change if fewer students were added to the class?
4. How could we calculate the new average if the 5 students had a lower total sum of marks?
5. How would the solution change if there were more than 5 students added?

Tip: When dealing with averages, remember that they can be recalculated by multiplying the average by the number of items (students, in this case), then adding or subtracting from that total to account for any changes.