A hotelier bought 37 chairs and 29 tables for ₹95,000. He also bought 37 tables and 29 chairs for ₹1,03,000. For another hotel he proposes to buy chairs and tables worth 296,000 such that he has four chairs for every table. How many tables does he buy?

Let's solve this step by step.

### Step 1: Define the variables.
Let:
- \( x \) be the cost of one chair.
- \( y \) be the cost of one table.

### Step 2: Set up the equations.
Based on the problem statement, we can write the following two equations:

1. For 37 chairs and 29 tables:
   \[
   37x + 29y = 95,000
   \]

2. For 37 tables and 29 chairs:
   \[
   29x + 37y = 1,03,000
   \]

### Step 3: Solve the system of equations.
We can solve these two equations using the elimination method:

- Multiply the first equation by 29 and the second equation by 37 to eliminate \(x\):

\[
(37x + 29y) \times 29 = 95,000 \times 29
\]
\[
(29x + 37y) \times 37 = 1,03,000 \times 37
\]

This gives us:
\[
1073x + 841y = 27,55,000
\]
\[
1073x + 1369y = 38,11,000
\]

- Subtract the first equation from the second:
\[
(1073x + 1369y) - (1073x + 841y) = 38,11,000 - 27,55,000
\]

This simplifies to:
\[
528y = 10,56,000
\]
\[
y = \frac{10,56,000}{528} = 2000
\]

Now, substitute \( y = 2000 \) into one of the original equations to find \( x \):

Substituting into \( 37x + 29(2000) = 95,000 \):
\[
37x + 58,000 = 95,000
\]
\[
37x = 37,000
\]
\[
x = \frac{37,000}{37} = 1000
\]

### Step 4: Determine the number of tables.
For the other hotel, the total cost is ₹296,000, and he wants four chairs for every table. Let \( t \) be the number of tables.

The cost of \( t \) tables and \( 4t \) chairs is:
\[
4t \times 1000 + t \times 2000 = 296,000
\]
\[
4000t + 2000t = 296,000
\]
\[
6000t = 296,000
\]
\[
t = \frac{296,000}{6000} = 49.33
\]

Since \( t \) must be an integer, and \( t = 49 \) gives a total less than ₹296,000 while \( t = 50 \) exceeds the amount, we see that \( t = 49 \) is the correct solution.

### Answer:
He buys **49 tables**.

### Would you like more details or have any questions?
Here are 5 related questions to explore:

1. What are some other methods to solve a system of linear equations?
2. How would the solution change if the prices of chairs and tables were altered?
3. Can we solve this problem using matrix methods?
4. How does the ratio of chairs to tables impact the total cost?
5. What happens if the requirement for the number of chairs per table changes?

**Tip:** When working with linear equations, always double-check the calculations, especially when dealing with large numbers, to avoid simple mistakes.

Learn how to solve a complex problem involving the purchase of chairs and tables by a hotelier using a system of linear equations. Includes step-by-step calculations and solutions. Suitable for students in Grades 9-12.

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