In a restaurant 35 visitors can have lunch at a time. If the number of visitors increases by 7 , then the expense of the restaurant on food increases by ₹ while the average expenditure per head decreases by ₹. Find the original expenditure of the restaurant.

Let's break down the problem and find the original expenditure of the restaurant.

### Given:
- The restaurant can accommodate 35 visitors at a time.
- If the number of visitors increases by 7, the total expense increases by ₹ (we need to determine this value) and the average expenditure per head decreases by ₹ (we also need to determine this value).

### Let:
- \( x \) be the original total expenditure when there are 35 visitors.
- \( y \) be the original average expenditure per visitor.

So, the total expenditure is \( x = 35y \).

When the number of visitors increases by 7, the new number of visitors is \( 35 + 7 = 42 \).

Let the new average expenditure per visitor be \( y' \).

We are given that the total expenditure increases by ₹ \( z \) (let's call this amount \( z \)), and the average expenditure per visitor decreases by ₹ \( d \) (let's call this amount \( d \)).

So:
- The new total expenditure \( x + z \) is equal to the new average expenditure per head multiplied by the new number of visitors: \( 42 \times (y - d) \).

This gives us the equation:
\[
x + z = 42 \times (y - d)
\]
But we also have:
\[
x = 35y
\]

Now, substituting \( x = 35y \) into the equation:
\[
35y + z = 42 \times (y - d)
\]

Expanding and simplifying the equation:
\[
35y + z = 42y - 42d
\]
\[
z = 42y - 42d - 35y
\]
\[
z = 7y - 42d
\]

This equation relates the increase in expenditure (\( z \)) to the original average expenditure per visitor (\( y \)) and the decrease in average expenditure per head (\( d \)).

To solve for the original expenditure \( x \), we need specific values for \( z \) and \( d \).

If you provide the values for the increase in expenditure (\( z \)) and the decrease in average expenditure (\( d \)), I can find the exact original expenditure \( x \) for you.

Would you like to provide those values, or should I proceed with an assumption to illustrate the solution?

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### Follow-Up Questions:

1. Do you have specific values for the increase in total expenditure or the decrease in average expenditure per head?
2. Would you like to explore the relationship between the number of visitors and total expenditure further?
3. How does the change in visitor numbers typically affect the restaurant's expenses?
4. Would solving for the expenditure with assumed values be helpful?
5. Are you interested in understanding how this problem relates to real-world business scenarios?
6. Do you want to discuss the role of averages and totals in calculating expenditures?
7. Would you like to explore more problems related to average and total costs?
8. Are you interested in how to derive similar formulas for other business calculations?

**Tip:** In word problems, clearly identifying what each variable represents helps in setting up equations accurately.

Explore how to determine the original expenditure of a restaurant using algebraic equations. Understand the impact of increased visitor numbers on total and average expenditures. Suitable for advanced high school students and above.

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