Question

You are in charge of buying prizes for a school contest. A one-strand rosette ribbon costs 2.00 dollars and a three-strand rosette ribbon costs 3.00 dollars . You have 10 dollars to spend. What is the least number of students that can receive a ribbon?

Short answer

The least number of students that can receive a ribbon is 5.

Step-by-step solution

Determine the cost of each type of ribbon

The cost of a one-strand ribbon is \$2.00 and a three-strand ribbon is \$3.00.

Determine the cost-effectiveness of each type of ribbon

In this case, the one-strand ribbon is more cost-effective than the three-strand ribbon. A one-strand ribbon costs \$2 per student while a three-strand ribbon costs \$1 per strand but \$3 per student.

Maximize the number of ribbons purchased with a \$10 budget

With a budget of \$10, as many one-strand ribbons as possible should be purchased for cost efficiency. Since each one-strand ribbon costs \$2.00, \(\frac{10}{2} = 5\) one-strand ribbons can be purchased.

Determine the least number of students that can receive a ribbon

As each purchased ribbon can be given to one student, the least number of students that can receive a ribbon is equal to the number of ribbons purchased, which is 5.

Key concepts

Cost Analysis in Algebra

Cost analysis is an important part of algebra, especially when it comes to making decisions about purchases within a budget. In the exercise, we delve into a practical scenario where cost analysis is used to determine how to maximize the number of prizes within a given budget. The crux of cost analysis here lies in determining the unit cost and total cost of items to make the most of the available funds.

When faced with options like the one-strand and three-strand ribbons, calculating the cost per student, or per recipient, allows for effective comparison. It illuminates which option gives you more value for money, a fundamental aspect of cost analysis. By understanding that the cost per student for a one-strand ribbon at \( \(2.00 \) is less than the \( \)3.00 \) for a three-strand ribbon, we concluded the one-strand ribbon is more cost-efficient for maximizing the number of recipients. In algebraic terms, cost analysis involves comparing ratios and per-unit costs to aid in the decision-making process.

Budget Constraints

Budget constraints are a real-world issue that often require algebraic thinking to navigate. They represent the limit on the spending capacity for any given purpose. In our exercise, the budget is constrained to \( $10.00 \), serving as the upper limit for purchasing rosette ribbons.

The concept is vital in making informed choices under financial limitations. It involves the conflict between desire—wanting to give ribbons to as many students as possible—and the reality of a finite amount of money available. Algebraically, budget constraints can be expressed as inequalities or equations that have to be respected when planning purchases. Through analysis and calculation, we established that adhering to the budget means opting to buy five one-strand ribbons, effectively solving the problem under the presented constraints.

System of Equations

A system of equations is a set of equations with multiple variables that are solved together to find a common solution for those variables. While the given exercise does not involve a complex system of equations, understanding how they work can enhance our approach to solve similar problems.

In a more elaborate scenario, one might have to consider multiple types of items with different prices and budget constraints that would require setting up a system of equations. Each equation would represent a different constraint or relation, like the price of an item and the total number of items. Algebraically solving these systems can help find the optimal number of each type of ribbon while staying within budget. The concept underscores the intertwined relationship between various factors and the necessity of finding a balance that satisfies all conditions simultaneously.