Solving for Tables with 3 Chairs in a Restaurant

In the context of running a restaurant, understanding the seating capacity and distribution of tables is crucial for optimal space utilization. Let's explore a problem where a restaurant has 16 tables, each with either 3, 4, or 6 chairs. The restaurant can accommodate 72 people, and we know that the tables with 3 or 4 chairs can accommodate 36 people. What is the number of tables with 3 chairs?

Defining Variables and Formulating Equations

Let's define the variables for the number of tables:

x: Number of tables with 3 chairs. y: Number of tables with 4 chairs. z: Number of tables with 6 chairs.

We have the following equations based on the given information:

The total number of tables: x y z 16 The total number of people that can be accommodated by the tables with 3 and 4 chairs: 3x 4y 36 The total accommodation capacity of the restaurant: 3x 4y 6z 72

Simplifying Equations Step-by-Step

From the second equation, (3x 4y 36), we can express (4y):

4y 36 - 3x

Next, substituting (4y) into the third equation:

36 - 3x 6z 72

This simplifies to:

6z 72 - 36 - 3x 6z 36 - 3x z 6 - frac{1}{2} x

Now substitute (z) into the first equation:

x y (6 - frac{1}{2} x) 16 x y 6 - frac{1}{2} x 16 y frac{1}{2} x 10

Now we have two equations:

1. (4y 36 - 3x) 2. (y frac{3}{2} x 10)

Final Valid Solutions

From the second equation, we can express (y):

y 10 - frac{3}{2} x

Now substitute (y) into the first equation:

4(10 - frac{3}{2} x) 36 - 3x 40 - 6x 36 - 3x

Rearranging gives:

40 - 36 6x - 3x 4 3x x frac{4}{3}

Since (x) must be a whole number, let's check our equations for integer solutions:

Testing Integer Values for (x)

If (x 0) 4y 36 rightarrow y 9 rightarrow z 16 - 9 7 text{ (not valid)} If (x 4) 34 4y 36 rightarrow 12 4y 36 rightarrow 4y 24 rightarrow y 6 rightarrow z 16 - 4 - 6 6 text{ (valid)} If (x 8) 38 4y 36 rightarrow 24 4y 36 rightarrow 4y 12 rightarrow y 3 rightarrow z 16 - 8 - 3 5 text{ (valid)} If (x 12) 312 4y 36 rightarrow 36 4y 36 rightarrow 4y 0 rightarrow y 0 rightarrow z 16 - 12 4 text{ (valid)}

The valid solutions for tables with 3 chairs are:

4 tables with 3 chairs 8 tables with 3 chairs 12 tables with 3 chairs

Conclusion

To find the possible number of tables with 3 chairs, the values can be 4, 8, or 12 based on the combinations satisfying the equations. However, for a specific answer, we conclude with one of the valid solutions:

There are 4 tables with 3 chairs.

Understanding these solutions can help in optimizing the space and managing the seating capacity in any restaurant setting.