Solving Inequalities Felicity's Dog Food Consumption Problem
Introduction
This article delves into the mathematical inequality that represents the amount of dog food Felicity's dog consumes in relation to Martin's dog. We'll explore the problem step-by-step, breaking down the given information and translating it into a mathematical expression. This exercise highlights how inequalities are used to describe real-world scenarios involving constraints and relationships between quantities. Understanding this type of problem is crucial for developing problem-solving skills in mathematics and applying them to practical situations. Our primary focus revolves around deciphering the conditions stated in the problem and transforming them into a precise mathematical inequality. We will carefully examine each part of the statement, ensuring we accurately capture the constraints on Felicity's dog's food consumption. This process involves identifying key phrases, assigning variables, and combining them to form the desired inequality. The final result will provide a clear and concise representation of the relationship between the food intake of Felicity's dog and Martin's dog.
Problem Statement Breakdown
To effectively tackle this problem, we need to dissect the information provided and identify the key components. The problem states that Felicity's dog eats no more than two cups of dog food per day. This establishes an upper limit on Felicity's dog's food intake. In mathematical terms, this means the amount of food Felicity's dog eats is less than or equal to two cups. The problem further states that Felicity's dog eats at least one-quarter cup more than one-half of the amount Martin's dog eats. This part establishes a relationship between the food consumption of Felicity's dog and Martin's dog. It tells us that Felicity's dog eats a certain fraction of Martin's dog's food plus an additional amount. The amount of food that Martin's dog eats is represented by the variable m. This is a crucial piece of information as it allows us to express the relationship between the two dogs' food consumption in terms of a variable. By carefully analyzing these statements, we can start to build the mathematical inequality that represents the situation. This involves translating the verbal descriptions into symbolic expressions and combining them to form a cohesive mathematical statement. The breakdown of the problem statement is the foundation for constructing the inequality and ultimately solving the problem.
Translating the Information into Mathematical Expressions
Now, let's translate the information we've broken down into mathematical expressions. First, we know that Felicity's dog eats no more than two cups of food. If we let 'f' represent the amount of food Felicity's dog eats, we can write this as: f ≤ 2. This inequality states that the amount of food Felicity's dog eats is less than or equal to 2 cups. Next, we need to represent the statement that Felicity's dog eats at least one-quarter cup more than one-half of the amount Martin's dog eats. We know that Martin's dog's food intake is represented by m. One-half of the amount Martin's dog eats can be written as (1/2)m. One-quarter cup more than this amount would be (1/2)m + (1/4). Since Felicity's dog eats at least this amount, we can write this as: f ≥ (1/2)m + (1/4). This inequality states that the amount of food Felicity's dog eats is greater than or equal to one-half the amount Martin's dog eats plus one-quarter cup. By combining these two inequalities, we can create a comprehensive mathematical representation of the problem. This translation process is essential for bridging the gap between the verbal description and the symbolic language of mathematics. The ability to accurately translate real-world scenarios into mathematical expressions is a fundamental skill in problem-solving.
Combining Inequalities
We now have two inequalities: f ≤ 2 and f ≥ (1/2)m + (1/4). To fully represent the problem, we need to combine these inequalities. Since we are interested in the relationship between m (the amount Martin's dog eats) and the constraints on Felicity's dog's food consumption, we need to eliminate f from the inequalities. We can do this by recognizing that the two inequalities provide both an upper and a lower bound for f. The first inequality, f ≤ 2, tells us that f is at most 2. The second inequality, f ≥ (1/2)m + (1/4), tells us that f is at least (1/2)m + (1/4). Since f must satisfy both of these conditions, we can combine them into a single inequality: (1/2)m + (1/4) ≤ f ≤ 2. This combined inequality states that (1/2)m + (1/4) is less than or equal to f, and f is less than or equal to 2. To express the relationship solely in terms of m, we can focus on the left-hand side of the inequality. We know that (1/2)m + (1/4) must be less than or equal to 2. This gives us the inequality: (1/2)m + (1/4) ≤ 2. This inequality is the key to answering the problem. It represents the constraint on the amount Martin's dog eats based on the information provided about Felicity's dog's food consumption. Combining inequalities is a powerful technique in mathematics that allows us to express multiple constraints in a single, concise statement. This step is crucial for simplifying the problem and arriving at the final solution.
Isolating the Variable
To find the inequality that represents the amount of food Martin's dog eats (m), we need to isolate m in the inequality (1/2)m + (1/4) ≤ 2. The first step in isolating m is to subtract (1/4) from both sides of the inequality. This gives us: (1/2)m ≤ 2 - (1/4). Simplifying the right-hand side, we have: (1/2)m ≤ (8/4) - (1/4), which becomes (1/2)m ≤ (7/4). Now, to get m by itself, we need to multiply both sides of the inequality by 2. This gives us: m ≤ (7/4) * 2, which simplifies to m ≤ (7/2). Therefore, the inequality that represents the amount of food Martin's dog eats is m ≤ 7/2. This inequality tells us that Martin's dog eats no more than 7/2 cups of food. Isolating the variable is a fundamental algebraic technique used to solve equations and inequalities. By performing valid operations on both sides of the inequality, we can isolate the variable of interest and determine its possible values. This process is essential for understanding the constraints on the variable and finding solutions to the problem.
Final Inequality and Interpretation
The final inequality that represents the amount of food Martin's dog eats is m ≤ 7/2. This inequality signifies that Martin's dog consumes a maximum of 7/2 cups, or 3.5 cups, of food per day. This result is derived from the given conditions that Felicity's dog eats no more than two cups of food daily and consumes at least one-quarter cup more than half the amount Martin's dog eats. This inequality provides a clear and concise mathematical representation of the relationship between the food intake of the two dogs. It allows us to understand the constraints on Martin's dog's food consumption based on the information provided about Felicity's dog. The ability to interpret mathematical inequalities in the context of real-world problems is a crucial skill. It allows us to translate abstract mathematical concepts into meaningful information and make informed decisions. In this case, the inequality tells us the maximum amount of food Martin's dog can eat while still satisfying the given conditions. Understanding the meaning of the inequality is just as important as deriving it. The final inequality, m ≤ 7/2, encapsulates the solution to the problem, providing a clear and quantifiable upper limit for Martin's dog's daily food consumption. This conclusion underscores the power of mathematical inequalities in modeling and solving real-world scenarios involving constraints and relationships between variables.
Conclusion
In conclusion, by carefully breaking down the problem statement, translating the information into mathematical expressions, combining inequalities, and isolating the variable, we have successfully derived the inequality that represents the amount of food Martin's dog eats. The final inequality, m ≤ 7/2, provides a clear and concise answer to the problem. This exercise demonstrates the importance of understanding mathematical concepts such as inequalities and their application in real-world scenarios. The ability to translate verbal descriptions into mathematical expressions, manipulate inequalities, and interpret the results is a valuable skill in problem-solving and critical thinking. This process not only helps us solve specific problems but also enhances our understanding of the underlying mathematical principles. By working through this problem, we have gained a deeper appreciation for the power of mathematics in modeling and analyzing real-world situations. The systematic approach we have used, from breaking down the problem statement to arriving at the final inequality, can be applied to a wide range of mathematical problems. This highlights the versatility and importance of mathematical skills in various fields and aspects of life.