Inequality For Dan's Potato And Grape Purchase

This article breaks down a mathematical problem step-by-step, focusing on translating a real-world scenario into an algebraic inequality. We'll explore how to represent the cost of Dan's potato and grape purchase using variables and then construct an inequality that captures the given constraint. This approach is fundamental in algebra, allowing us to model and solve a wide array of practical problems.

Problem Statement

Dan went to the market and bought two types of produce potatoes and grapes. He purchased $x$ pounds of potatoes at a price of $0.85 per pound*. Additionally, he bought *$y$ pounds of grapes priced at $1.29 per pound. The critical piece of information is that the total cost of his purchase was less than $5. Our task is to identify which inequality accurately represents this scenario.

Understanding the Components of the Problem

To construct the correct inequality, we need to break down the problem into its fundamental components:

1. Cost of Potatoes: The cost of the potatoes is determined by multiplying the quantity purchased (x pounds) by the price per pound ($0.85). This can be expressed as 0.85x.
2. Cost of Grapes: Similarly, the cost of the grapes is calculated by multiplying the quantity (y pounds) by the price per pound ($1.29). This gives us 1.29y.
3. Total Cost: The total cost of Dan's purchase is the sum of the cost of potatoes and the cost of grapes, which is 0.85x + 1.29y.
4. The Constraint: The problem states that the total cost was less than $5. This is a crucial piece of information that dictates the inequality sign we will use.

Constructing the Inequality

Now that we've identified the components, we can construct the inequality. We know that the total cost (0.85x + 1.29y) must be less than $5. In mathematical notation, "less than" is represented by the "<" symbol. Therefore, the inequality that represents Dan's purchase is:

0.85x + 1.29y < 5

This inequality states that the sum of the cost of the potatoes (0.85x) and the cost of the grapes (1.29y) is strictly less than $5.

Analyzing the Answer Choices

Let's examine the answer choices provided in the original problem:

A. $1.29x + 0.85y < 5$ B. $1.29x + 0.85y > 5$ C. $0.85x + 1.29y > 5$ D. $0.85x + 1.29y < 5$

By comparing our derived inequality (0.85x + 1.29y < 5) with the answer choices, we can see that option D is the correct representation of Dan's purchase. Option A has the coefficients reversed, while options B and C use the incorrect inequality sign (>), indicating a total cost greater than $5.

Why Other Options are Incorrect

It's important to understand why the other options are incorrect to solidify our understanding of the problem:

• Option A ($1.29x + 0.85y < 5$): This option incorrectly assigns the price per pound. It suggests that potatoes cost $1.29 per pound and grapes cost $0.85 per pound, which contradicts the problem statement.
• Option B ($1.29x + 0.85y > 5$): This option not only reverses the prices but also uses the "greater than" (>) symbol. This would represent a scenario where the total cost is more than $5, which is the opposite of what the problem states.
• Option C ($0.85x + 1.29y > 5$): This option correctly assigns the prices but uses the "greater than" (>) symbol. Again, this represents a total cost exceeding $5, which is not the scenario described in the problem.

Real-World Applications and Importance

Translating real-world scenarios into mathematical inequalities is a fundamental skill in various fields, including:

• Budgeting: Inequalities are used to represent budget constraints, such as limiting spending on different categories.
• Resource Allocation: Companies use inequalities to optimize the allocation of resources, ensuring they don't exceed available supplies.
• Optimization Problems: Many real-world problems involve finding the best solution within certain constraints, which are often expressed as inequalities.
• Science and Engineering: Inequalities are used in various scientific and engineering applications, such as determining the range of acceptable values for parameters in a system.

Key Takeaways

This problem demonstrates the importance of carefully translating word problems into mathematical expressions. Here are some key takeaways:

• Identify the Variables: Determine what quantities are represented by variables (in this case, x and y).
• Translate Words into Math: Understand how keywords like "per," "total," and "less than" translate into mathematical operations and symbols.
• Pay Attention to Units: Ensure that the units are consistent throughout the equation or inequality.
• Check Your Answer: Compare your derived inequality with the answer choices and make sure it logically represents the problem statement.

By mastering these skills, you can confidently tackle a wide range of mathematical problems that arise in everyday life and various professional fields.

Conclusion

The inequality 0.85x + 1.29y < 5 accurately represents Dan's purchase of potatoes and grapes, where the total cost was less than $5. By carefully breaking down the problem, identifying the components, and translating the information into mathematical notation, we were able to arrive at the correct solution. This exercise highlights the practical application of inequalities in modeling real-world scenarios.

• Inequality: The core mathematical concept used in the problem.
• Algebra: The branch of mathematics dealing with symbolic relationships.
• Word Problem: The type of problem that requires translating a real-world scenario into mathematical terms.
• Cost: The total expense of the purchase.
• Pounds: The unit of weight used for the produce.
• Price per Pound: The cost of each item per unit of weight.
• Total Cost: The sum of the costs of all items.
• Less Than: The inequality symbol (<) used to represent the constraint.
• Mathematical Representation: Expressing a real-world situation using mathematical symbols and equations.
• Problem Solving: The process of finding a solution to a given problem.
• Real-World Application: The practical use of mathematical concepts in everyday situations.
• Budgeting: Managing expenses within a specific limit.
• Resource Allocation: Distributing resources efficiently.
• Optimization: Finding the best solution to a problem within given constraints.
• Coefficients: The numerical values multiplying the variables in an equation or inequality (e.g., 0.85 and 1.29).
• Variables: Symbols representing unknown quantities (e.g., x and y).
• Mathematical Modeling: The process of creating a mathematical representation of a real-world situation.
• Algebraic Expression: A combination of variables, constants, and mathematical operations.