The principal of a school is hosting a small luncheon for her staff. She plans to prepare two sandwiches for each person. Some staff members offer to bring salads and beverages.

The principal has a budget of \$225 and expects at least 16 people to attend. Sandwiches cost \$3 each.

Select all of the equations and inequalities that could represent the constraints in the situation, where $n$ is the number of people attending and $s$ is the number of sandwiches.

A$n \geq 16$
B$n \geq 32$
C$s < 2n$
D$s = 2n$
E$3n \leq 225$

F. $3s \leq 225$

Question

04-11-2026
Mathematics

Answered

The principal of a school is hosting a small luncheon for her staff. She plans to prepare two sandwiches for each person. Some staff members offer to bring salads and beverages.

The principal has a budget of \$225 and expects at least 16 people to attend. Sandwiches cost \$3 each.

Select all of the equations and inequalities that could represent the constraints in the situation, where $n$ is the number of people attending and $s$ is the number of sandwiches.

A$n \geq 16$
B$n \geq 32$
C$s < 2n$
D$s = 2n$
E$3n \leq 225$

F. $3s \leq 225$

Verified Answer :

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Identify the constraints: $n \geq 16$ (attendees), $s = 2n$ (sandwiches), and $3s \leq 225$ (budget).
Evaluate given options:
A ($n \geq 16$) is correct.
D ($s = 2n$) is correct.
F ($3s \leq 225$) is correct.
Conclude that the valid options are $\boxed{A, D, F}$.

Explanation

Analyzing the Problem Constraints

The principal is hosting a luncheon with a budget constraint of $\$225$, and each sandwich costs $\$3$. We are given that $n$ is the number of people attending and $s$ is the number of sandwiches prepared. The principal expects at least 16 people to attend, which means $n \geq 16$. The principal plans to prepare 2 sandwiches for each person, so the total number of sandwiches is $s = 2n$. Finally, the total cost of the sandwiches must not exceed the budget, so $3s \leq 225$.

Evaluating the Options

We evaluate each provided option based on the derived constraints:

A$n \geq 16$: This matches the condition that at least 16 people will attend. Thus, this option is correct.

B$n \geq 32$: This is not explicitly required as a constraint; while the number of sandwiches might exceed 32, this is not a constraint on the number of people. Thus, this option is incorrect.

C$s < 2n$: The problem states the principal plans to prepare 2 sandwiches for each person, so $s = 2n$, not $s < 2n$. Thus, this option is incorrect.

D$s = 2n$: This matches the plan to prepare 2 sandwiches for each person. Thus, this option is correct.

E$3n \leq 225$: Given $s = 2n$, the cost constraint is $3s \leq 225$, which becomes $3(2n) \leq 225$, or $6n \leq 225$. This does not match $3n \leq 225$. Thus, this option is incorrect.

F. $3s \leq 225$: This matches the requirement that the total cost (3 dollars per sandwich times the number of sandwiches) must be within the budget of 225 dollars. Thus, this option is correct.

Conclusion

Based on the analysis, the constraints that correctly represent the situation are $n \geq 16$, $s = 2n$, and $3s \leq 225$. These correspond to options A, D, and F. The final answer is $\boxed{A, D, F}$.

Examples

This type of algebraic modeling is essential for project management and event planning. For instance, when organizing a professional conference or a community event, you must balance headcount, resource requirements (like food or catering), and budget limits. By defining these constraints as mathematical equations and inequalities, you can optimize your resources, ensure your event remains within budget, and determine the capacity limits for attendees. This method provides a logical framework for making informed decisions under real-world constraints.