Question

Caterer A charges $15 per person and $100 to set up tables. Caterer B charges $20 per person and $50 to set up tables. For how many guests will the cost of Caterer A be the same as the cost of Caterer B? What is that cost?

Two intersecting lines graphed on a coordinate plane showing number of guests on the X-axis and cost in dollars on the y-axis. The lines intersect at the point, 10, 250.

Number or guests:


Cost: $

Answer 1

To find the number of guests for which the costs of Caterer A and Caterer B are the same, we can set up equations for the total cost for each caterer based on the number of guests, denoted as \( x \).

**Caterer A** charges:
\[
\text{Cost}_A = 15x + 100
\]

**Caterer B** charges:
\[
\text{Cost}_B = 20x + 50
\]

To find the number of guests where the costs are equal, we set the two equations equal to each other:
\[
15x + 100 = 20x + 50
\]

Next, we solve for \( x \):
1. Subtract \( 15x \) from both sides:
\[
100 = 5x + 50
\]

2. Subtract 50 from both sides:
\[
50 = 5x
\]

3. Divide both sides by 5:
\[
x = 10
\]

Now that we have the number of guests (10), we can find the cost for either caterer at this number of guests.

Using Caterer A's cost:
\[
\text{Cost}_A = 15(10) + 100 = 150 + 100 = 250
\]

Thus, the number of guests where the costs are the same is \( 10 \), and the cost at that point is \( 250 \).

**Summary:**
- Number of guests: 10
- Cost: $250