The expenses of a lunch are partly constant and partly proportional to the number of guests. The expenses amount to Rs. 65 for 7 guests and Rs. 97 for 11 guests. How much the expenses for 18 guests will amount to?

Let the total expense of the lunch be modeled as:

\[
E = C + k \cdot g
\]

where:
- \(E\) is the total expense,
- \(C\) is the constant part of the expense (which doesn't depend on the number of guests),
- \(k\) is the proportional part (expense per guest),
- \(g\) is the number of guests.

We are given two conditions:
1. For 7 guests, the total expense is Rs. 65:
   \[
   65 = C + 7k
   \]
2. For 11 guests, the total expense is Rs. 97:
   \[
   97 = C + 11k
   \]

We can solve this system of linear equations to find \(C\) and \(k\).

### Step 1: Subtract the two equations to eliminate \(C\):

\[
(97 - 65) = (C + 11k) - (C + 7k)
\]
\[
32 = 4k \quad \Rightarrow \quad k = \frac{32}{4} = 8
\]

### Step 2: Substitute \(k = 8\) into one of the original equations (e.g., \(65 = C + 7k\)):

\[
65 = C + 7 \times 8
\]
\[
65 = C + 56 \quad \Rightarrow \quad C = 65 - 56 = 9
\]

### Step 3: Find the expenses for 18 guests:

Now that we know \(C = 9\) and \(k = 8\), we can use the equation \(E = C + k \cdot g\) to find the total expenses for 18 guests:

\[
E = 9 + 8 \times 18
\]
\[
E = 9 + 144 = 153
\]

Thus, the total expenses for 18 guests will amount to **Rs. 153**.

Would you like further details or have any questions?

### Related Questions:
1. What would the expenses be for 25 guests using the same formula?
2. How would the expenses change if the constant part increased by Rs. 5?
3. Can we express the relationship between expenses and guests in terms of a linear equation?
4. What would happen if the cost per guest doubled while the constant part remained unchanged?
5. How could we solve this problem using matrix methods?

### Tip:
In problems involving proportional relationships, it's helpful to express the problem in terms of equations, then eliminate variables by subtracting equations.

Learn how to calculate the expenses of a lunch which is partly constant and partly proportional to the number of guests. Given the costs for 7 and 11 guests, find the expenses for 18 guests using linear equations and proportionality. Suitable for students in Grades 8-10.

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