The owner of a new restaurant is ordering tables and chairs. She wants to have only tables for 2 and tables for 4. The total number of people that can be seated in the restaurant i... The owner of a new restaurant is ordering tables and chairs. She wants to have only tables for 2 and tables for 4. The total number of people that can be seated in the restaurant is 120. Complete the table to show some possible combinations of 2-seat tables and 4-seat tables that will seat 120 customers.
Understand the Problem
The question is asking for combinations of 2-seat and 4-seat tables that can seat a total of 120 people in a restaurant. We need to determine different possible values that can be used to complete the given table.
Answer
Possible combinations of 2-seat and 4-seat tables are generated where $2x + 4y = 120$.
Answer for screen readers
The possible combinations of 2-seat and 4-seat tables that can seat a total of 120 customers are:
| Number of 2-Seat Tables | Number of 4-Seat Tables |
|---|---|
| 0 | 30 |
| 2 | 29 |
| 4 | 28 |
| 6 | 27 |
| 8 | 26 |
| 10 | 25 |
| 12 | 24 |
| 14 | 23 |
| 16 | 22 |
| 18 | 21 |
| 20 | 20 |
| 22 | 19 |
| 24 | 18 |
| 26 | 17 |
| 28 | 16 |
| 30 | 15 |
| 32 | 14 |
| 34 | 13 |
| 36 | 12 |
| 38 | 11 |
| 40 | 10 |
| 42 | 9 |
| 44 | 8 |
| 46 | 7 |
| 48 | 6 |
| 50 | 5 |
| 52 | 4 |
| 54 | 3 |
| 56 | 2 |
| 58 | 1 |
| 60 | 0 |
Steps to Solve
- Define the Variables
Let $x$ be the number of 2-seat tables and $y$ be the number of 4-seat tables.
- Set Up the Equation
The relationship between the tables and the total number of customers can be expressed with the equation: $$ 2x + 4y = 120 $$
- Simplify the Equation
We can simplify this equation by dividing everything by 2: $$ x + 2y = 60 $$
- Express One Variable in Terms of the Other
To find possible combinations, solve for $y$: $$ y = \frac{60 - x}{2} $$
- Determine Possible Values for $x$
Since $y$ must be a non-negative integer:
- $60 - x \geq 0 \Rightarrow x \leq 60$
- The value of $x$ must be even (because $60 - x$ needs to be even) for $y$ to remain an integer.
- List Combinations of Tables
Now, we can plug in even values for $x$ and calculate $y$:
- For $x = 0$: $y = 30$
- For $x = 2$: $y = 29$
- For $x = 4$: $y = 28$
- Continue this until $x = 60$.
The possible combinations of 2-seat and 4-seat tables that can seat a total of 120 customers are:
| Number of 2-Seat Tables | Number of 4-Seat Tables |
|---|---|
| 0 | 30 |
| 2 | 29 |
| 4 | 28 |
| 6 | 27 |
| 8 | 26 |
| 10 | 25 |
| 12 | 24 |
| 14 | 23 |
| 16 | 22 |
| 18 | 21 |
| 20 | 20 |
| 22 | 19 |
| 24 | 18 |
| 26 | 17 |
| 28 | 16 |
| 30 | 15 |
| 32 | 14 |
| 34 | 13 |
| 36 | 12 |
| 38 | 11 |
| 40 | 10 |
| 42 | 9 |
| 44 | 8 |
| 46 | 7 |
| 48 | 6 |
| 50 | 5 |
| 52 | 4 |
| 54 | 3 |
| 56 | 2 |
| 58 | 1 |
| 60 | 0 |
More Information
This represents a linear combination problem where different configurations of tables meet a set capacity. For each even value of 2-seat tables, you can find the corresponding number of 4-seat tables to achieve the total of 120 seats.
Tips
- Forgetting that $x$ must be even, which leads to incorrect values for $y$.
- Miscalculating or misinterpreting the formula after simplifying.
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