To determine how many tables are needed for 16 people based on the given seating arrangement, we can analyze the scenario quantitatively. The seating arrangement describes how many people can be seated based on the number of tables used:
- 1 table seats 4 people
- 2 tables seat 6 people
- 3 tables seat 8 people
From this, we can observe a pattern in the number of people seated based on the number of tables:
- \( n = 1 \) table: \( p = 4 \)
- \( n = 2 \) tables: \( p = 6 \)
- \( n = 3 \) tables: \( p = 8 \)
We can see that with each additional table, the number of people that can be seated increases by 2 from the previous table count:
- From 1 to 2 tables: \( 6 - 4 = 2 \)
- From 2 to 3 tables: \( 8 - 6 = 2 \)
From the pattern, we have:
- The number of people \( p \) that can be seated with \( n \) tables appears to form a linear sequence, where \( p(n) = 2n + 2 \).
We can derive the equation to represent this situation:
\[ p(n) = 2n + 2 \]
Now we need to calculate how many tables are needed to seat 16 people. We will set up the equation:
\[ 2n + 2 = 16 \]
Subtracting 2 from both sides:
\[ 2n = 14 \]
Dividing both sides by 2:
\[ n = 7 \]
Thus, \(n = 7\) tables are required to seat 16 people.
To confirm, we can check the total seating for 7 tables using the equation:
\[ p(7) = 2(7) + 2 = 14 + 2 = 16 \]
The calculation verifies that 7 tables can indeed seat 16 people. Therefore, 7 tables are needed for 16 people.