Question

the table below shows the amount that a catering company charges based on the number of people at an event. which of the following equations shows the relationship between c, the amount the company charges based on p, the number of people at the event?

people (p)	total charge (c)
50	$725
75	$1,050
100	$1,375

options:

• $c = 13p + 75$
• $c = 16p$
• $p = 13c + 75$

Explanation:

Step1: Identify linear form

The relationship is linear: $c = mp + b$, where $m$ is the rate per person, $b$ is fixed cost.

Step2: Calculate slope $m$

Use two points $(25, 400)$ and $(50, 725)$:
$$m = \frac{725 - 400}{50 - 25} = \frac{325}{25} = 13$$

Step3: Solve for fixed cost $b$

Substitute $m=13$, $p=25$, $c=400$ into $c=mp+b$:
$$400 = 13(25) + b$$
$$400 = 325 + b$$
$$b = 400 - 325 = 75$$

Step4: Verify with other points

Check $p=75$: $c=13(75)+75=975+75=1050$, matches the table.
Check $p=100$: $c=13(100)+75=1300+75=1375$, matches the table.

Answer:

c=13p+75