Natalie is plannin a school celebration and wants to have live music and food for everyone who attends. She has found a band that will charge her $750 and a caterer who will provide snacks and drinks for $2.25 per person. If her goal is to keep the average cost per person between $2.75 and $3.25, how many people, p, must attend?

Options

A. `225ltplt325`
B. `325ltplt750`
C. `500ltplt1,000`
D. `750ltplt1,500`

Answer

To solve the problem step by step, we will first define the costs involved and then set up inequalities based on the average cost per person that Natalie wants to maintain.

### Step 1: Define the Costs
- The band charges a fixed amount of $750.
- The caterer charges $2.25 per person.

### Step 2: Calculate Total Cost Per Person
The total cost per person (C) can be expressed as:
\[ C = \frac{750}{p} + 2.25 \]
where \( p \) is the number of people attending.

### Step 3: Set Up the Inequalities
Natalie wants the average cost per person to be between $2.75 and $3.25. Therefore, we can set up the following inequalities:
\[ 2.75 < \frac{750}{p} + 2.25 < 3.25 \]

### Step 4: Solve the Left Inequality
First, we solve the left part of the inequality:
\[ 2.75 < \frac{750}{p} + 2.25 \]
Subtracting 2.25 from both sides gives:
\[ 2.75 - 2.25 < \frac{750}{p} \]
\[ 0.5 < \frac{750}{p} \]

### Step 5: Solve the Right Inequality
Now we solve the right part of the inequality:
\[ \frac{750}{p} + 2.25 < 3.25 \]
Subtracting 2.25 from both sides gives:
\[ \frac{750}{p} < 3.25 - 2.25 \]
\[ \frac{750}{p} < 1 \]

### Step 6: Combine the Inequalities
Now we have two inequalities:
1. \( 0.5 < \frac{750}{p} \)
2. \( \frac{750}{p} < 1 \)

### Step 7: Take Reciprocals
Taking the reciprocal of both sides of the inequalities (and reversing the inequality signs):
1. \( \frac{1}{0.5} > \frac{p}{750} \) which simplifies to \( 2 > \frac{p}{750} \) or \( p < 1500 \)
2. \( \frac{1}{1} > \frac{p}{750} \) which simplifies to \( 1 > \frac{p}{750} \) or \( p < 750 \)

### Step 8: Multiply by 750
Multiplying through by 750 gives:
1. \( 750 < p < 1500 \)

### Conclusion
Thus, the number of people \( p \) that must attend the celebration is between 750 and 1500.