Question

information about the costs of three catering companies is shown in this table.
catering company costs
acme catering company: $6 per person plus a flat $100 service charge
best foods company: $8 per person plus a flat $40 service charge
creative catering company: $10 per person with no service charge
gavin can spend no more than $300 on the catering for an event.
what is the greatest number of people that gavin can have at his event without spending more than $300 on catering?
(a) 30
(b) 32
(c) 33
(d) 37

Explanation:

Step1: Analyze each company's cost formula

Let \( n \) be the number of people.

• Acme: Cost \( C_A = 6n + 100 \)
• Best Foods: Cost \( C_B = 8n + 40 \)
• Creative: Cost \( C_C = 10n \)

Step2: Solve for \( n \) in each inequality (cost ≤ 300)

• For Acme: \( 6n + 100 \leq 300 \)

Subtract 100: \( 6n \leq 200 \)
Divide by 6: \( n \leq \frac{200}{6} \approx 33.33 \) (but check if feasible, but let's check others)

• For Best Foods: \( 8n + 40 \leq 300 \)

Subtract 40: \( 8n \leq 260 \)
Divide by 8: \( n \leq \frac{260}{8} = 32.5 \)

• For Creative: \( 10n \leq 300 \)

Divide by 10: \( n \leq 30 \)

Step3: Find the maximum \( n \) across all

Compare the maximums: 33.33 (Acme, but we need integer, so 33? Wait, wait, wait—wait, Acme's calculation: \( 6n + 100 \leq 300 \) → \( 6n \leq 200 \) → \( n \leq 33.33 \), so 33 people? But wait, let's check Best Foods: 32.5 → 32. Creative: 30. Wait, but the options have 33 as option c. Wait, but let's verify Acme with 33: \( 6*33 + 100 = 198 + 100 = 298 \leq 300 \). 34 people: \( 6*34 + 100 = 204 + 100 = 304 > 300 \). Best Foods: 32 people: \( 8*32 + 40 = 256 + 40 = 296 \leq 300 \). 33 people: \( 8*33 + 40 = 264 + 40 = 304 > 300 \). Creative: 30. So the maximum is 33 (from Acme). Wait, but the options: a)30, b)32, c)33, d)37. So the greatest number is 33.

Answer:

c) 33