Seats of physics, Chemistry and Mathematics in a school are in the ratio 4: 5: 6. There is a proposal to increase these seats by 75 in each department. What were the total number of seats in the school finally?

Let the original number of seats for Physics, Chemistry, and Mathematics be in the ratio 4:5:6.

Let the common factor for these seats be \( x \). Therefore, the number of seats in each department initially is:
- Physics: \( 4x \)
- Chemistry: \( 5x \)
- Mathematics: \( 6x \)

The total number of seats initially is:
\[
\text{Total initial seats} = 4x + 5x + 6x = 15x
\]

According to the proposal, each department will receive an additional 75 seats. Therefore, the new number of seats in each department is:
- Physics: \( 4x + 75 \)
- Chemistry: \( 5x + 75 \)
- Mathematics: \( 6x + 75 \)

The total number of seats after the increase is:
\[
\text{Total final seats} = (4x + 75) + (5x + 75) + (6x + 75)
\]
Simplifying the expression:
\[
\text{Total final seats} = 15x + 225
\]

Now, we are given that the number of seats in each department increases by 75, but we are not given the final total directly. Without the specific value of \( x \), this is the general form. But if you are provided with a total before or after, we can calculate \( x \).

Would you like more details on calculating \( x \), or do you have more specific information about initial or final totals?

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Here are five questions to expand this:
1. What was the initial total number of seats in the school before the increase?
2. How many seats were increased in total across all departments?
3. What is the percentage increase in the total number of seats after the change?
4. How many total seats does each department have after the increase?
5. How would the total seats change if a different number of seats were added?

**Tip:** When dealing with ratios, always convert to a common factor to understand the original proportions and then apply changes step by step.

This problem explores the ratio of seats in Physics, Chemistry, and Mathematics in a school. Learn how to calculate the total number of seats after a proposed increase of 75 seats in each department using algebraic methods. Suitable for students in Grades 6-8.

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