Understanding the Budget Constraint

Oct 13, 2025 by miodeya.com 143 views

A High School Coach's Budget Dilemma: Finding the Maximum Cost of Each Pair of Athletic Shorts

As a high school coach, buying new athletic shorts for the basketball team can be a daunting task, especially when there's a budget constraint. In this scenario, the coach needs to purchase 15 pairs of shorts, and the total cost must not exceed $200. The question is, what is the maximum cost of each pair of shorts, denoted as $s$ , that the coach can afford? To answer this, we need to set up an inequality that represents the coach's budget constraint.

Understanding the Budget Constraint

Let's break down the coach's budget constraint. The coach needs to buy 15 pairs of shorts, and the total cost must not exceed $200. This means that the sum of the costs of all 15 pairs of shorts must be less than or equal to $200. Mathematically, this can be represented as:

15s \leq 200

where $s$ is the cost of each pair of shorts.

Solving the Inequality

To find the maximum cost of each pair of shorts, we need to solve the inequality $15s \leq 200$ . We can start by dividing both sides of the inequality by 15:

s \leq \frac{200}{15}

Simplifying the right-hand side of the inequality, we get:

s \leq 13.33

This means that the maximum cost of each pair of shorts is $13.33.

Interpretation

The inequality $s \leq 13.33$ represents the maximum cost of each pair of shorts that the coach can afford. This means that if the coach wants to stay within the budget of $200, the cost of each pair of shorts must not exceed $13.33.

Conclusion

In conclusion, the inequality that represents the maximum cost of each pair of athletic shorts is $s \leq 13.33$ . This means that the coach can afford to spend up to $13.33 on each pair of shorts, as long as the total cost of all 15 pairs does not exceed $200.

Example Use Case

Suppose the coach finds a pair of athletic shorts that costs $12.99. In this case, the coach can afford to buy 15 pairs of shorts, as the total cost would be:

15 \times 12.99 = 195.45

This is less than the budget of $200, so the coach can purchase the shorts.

Tips and Variations

If the coach wants to buy more than 15 pairs of shorts, the maximum cost of each pair would be lower.
If the coach wants to buy fewer than 15 pairs of shorts, the maximum cost of each pair would be higher.
If the coach wants to buy a different type of athletic wear, such as t-shirts or socks, the budget constraint would be different.

Mathematical Background

The inequality $s \leq 13.33$ is a linear inequality, which means that it can be represented graphically as a line on a number line. The solution to the inequality is all values of $s$ that are less than or equal to 13.33.

Real-World Applications

This problem has real-world applications in various fields, such as:

Budgeting: The coach's budget constraint is a classic example of a linear inequality in budgeting.
Economics: The coach's decision to buy athletic shorts is an example of a consumer's decision-making process.
Business: The coach's budget constraint is an example of a business's financial constraint.

Conclusion

In conclusion, the inequality $s \leq 13.33$ represents the maximum cost of each pair of athletic shorts that the coach can afford. This means that the coach can spend up to $13.33 on each pair of shorts, as long as the total cost of all 15 pairs does not exceed $200.
A High School Coach's Budget Dilemma: Q&A

In our previous article, we explored the budget dilemma of a high school coach who needs to buy new athletic shorts for the basketball team. The coach must spend less than $200 on the shorts, and we determined that the maximum cost of each pair of shorts is $13.33. In this article, we'll answer some frequently asked questions about the coach's budget dilemma.

Q: What if the coach wants to buy more than 15 pairs of shorts?

A: If the coach wants to buy more than 15 pairs of shorts, the maximum cost of each pair would be lower. For example, if the coach wants to buy 20 pairs of shorts, the maximum cost of each pair would be:

20s \leq 200

Simplifying the inequality, we get:

s \leq 10

This means that the maximum cost of each pair of shorts would be $10.

Q: What if the coach wants to buy fewer than 15 pairs of shorts?

A: If the coach wants to buy fewer than 15 pairs of shorts, the maximum cost of each pair would be higher. For example, if the coach wants to buy 10 pairs of shorts, the maximum cost of each pair would be:

10s \leq 200

Simplifying the inequality, we get:

s \leq 20

This means that the maximum cost of each pair of shorts would be $20.

Q: What if the coach wants to buy a different type of athletic wear?

A: If the coach wants to buy a different type of athletic wear, such as t-shirts or socks, the budget constraint would be different. For example, if the coach wants to buy 15 t-shirts, the maximum cost of each t-shirt would be:

15t \leq 200

Simplifying the inequality, we get:

t \leq 13.33

This means that the maximum cost of each t-shirt would be $13.33.

Q: How can the coach ensure that they stay within the budget?

A: The coach can ensure that they stay within the budget by:

Setting a budget for each pair of shorts
Researching and comparing prices of different brands and styles
Considering alternative options, such as buying last year's model or looking for discounts
Keeping track of expenses and making adjustments as needed

Q: What if the coach wants to buy athletic shorts from a different brand?

A: If the coach wants to buy athletic shorts from a different brand, the price may be different. However, the coach can still use the same budget constraint to determine the maximum cost of each pair of shorts. For example, if the coach wants to buy athletic shorts from a brand that costs $15 per pair, the coach can still buy 15 pairs of shorts as long as the total cost does not exceed $200.

Q: Can the coach buy athletic shorts from a different store?

A: Yes, the coach can buy athletic shorts from a different store. However, the coach should still consider the budget constraint and ensure that the total cost does not exceed $200.

Conclusion

In conclusion, the coach's budget dilemma is a classic example of a linear inequality in budgeting. By understanding the budget constraint and making informed decisions, the coach can ensure that they stay within the budget and make the most of their resources.

Tips and Variations

Consider alternative options, such as buying last year's model or looking for discounts
Research and compare prices of different brands and styles
Keep track of expenses and make adjustments as needed
Consider buying athletic shorts from a different store or brand

Mathematical Background

Real-World Applications

This problem has real-world applications in various fields, such as:

Budgeting: The coach's budget constraint is a classic example of a linear inequality in budgeting.
Economics: The coach's decision to buy athletic shorts is an example of a consumer's decision-making process.
Business: The coach's budget constraint is an example of a business's financial constraint.