Question
Consider the following descriptions, each representing a relation. Description #1: The number of chairs in the restaurant is equal to three times the number of tables. Description #2: The number of tables in the restaurant is equal to one-third the number of chairs. Description #3: The number of tables in the restaurant is equal to three times the number of servers.
Answers
GPT-4o mini
To express the relationships described, we can define some variables:
- Let \( C \) denote the number of chairs.
- Let \( T \) denote the number of tables.
- Let \( S \) denote the number of servers.
Now we can translate each description into equations.
**Description #1:**
"The number of chairs in the restaurant is equal to three times the number of tables."
\[ C = 3T \]
**Description #2:**
"The number of tables in the restaurant is equal to one-third the number of chairs."
\[ T = \frac{1}{3}C \]
**Description #3:**
"The number of tables in the restaurant is equal to three times the number of servers."
\[ T = 3S \]
Now we have a system of equations:
1. \( C = 3T \)
2. \( T = \frac{1}{3}C \)
3. \( T = 3S \)
From these equations, we can analyze the relationships among chairs, tables, and servers:
1. From equations 1 and 2, you can substitute \( C = 3T \) into the second equation:
\[
T = \frac{1}{3}(3T) \Rightarrow T = T
\]
This is consistent.
2. Next, from the first equation, we can express \( T \) in terms of \( C \) as:
\[
T = \frac{C}{3}
\]
This is also consistent with equation 2.
3. Combining equations 2 and 3, we can express the relationship between tables and servers:
\[
T = 3S \rightarrow \frac{C}{3} = 3S \rightarrow C = 9S
\]
Now we have:
- \( C = 3T \)
- \( T = 3S \)
- \( C = 9S \)
Thus, you can conclude the following relationships through these equations:
- The number of chairs is nine times the number of servers (\( C = 9S \)).
- The number of tables is three times the number of servers (\( T = 3S \)).
- The number of chairs is three times the number of tables (\( C = 3T \)), consistent with other equations.
These equations help to clearly define the relationships between chairs, tables, and servers in the restaurant.
- Let \( C \) denote the number of chairs.
- Let \( T \) denote the number of tables.
- Let \( S \) denote the number of servers.
Now we can translate each description into equations.
**Description #1:**
"The number of chairs in the restaurant is equal to three times the number of tables."
\[ C = 3T \]
**Description #2:**
"The number of tables in the restaurant is equal to one-third the number of chairs."
\[ T = \frac{1}{3}C \]
**Description #3:**
"The number of tables in the restaurant is equal to three times the number of servers."
\[ T = 3S \]
Now we have a system of equations:
1. \( C = 3T \)
2. \( T = \frac{1}{3}C \)
3. \( T = 3S \)
From these equations, we can analyze the relationships among chairs, tables, and servers:
1. From equations 1 and 2, you can substitute \( C = 3T \) into the second equation:
\[
T = \frac{1}{3}(3T) \Rightarrow T = T
\]
This is consistent.
2. Next, from the first equation, we can express \( T \) in terms of \( C \) as:
\[
T = \frac{C}{3}
\]
This is also consistent with equation 2.
3. Combining equations 2 and 3, we can express the relationship between tables and servers:
\[
T = 3S \rightarrow \frac{C}{3} = 3S \rightarrow C = 9S
\]
Now we have:
- \( C = 3T \)
- \( T = 3S \)
- \( C = 9S \)
Thus, you can conclude the following relationships through these equations:
- The number of chairs is nine times the number of servers (\( C = 9S \)).
- The number of tables is three times the number of servers (\( T = 3S \)).
- The number of chairs is three times the number of tables (\( C = 3T \)), consistent with other equations.
These equations help to clearly define the relationships between chairs, tables, and servers in the restaurant.