Simplifying Algebraic Expressions Calculating Cost Of Rice And Beans

In the realm of mathematics, algebraic expressions serve as powerful tools for representing real-world scenarios. These expressions, composed of variables, constants, and mathematical operations, enable us to model and solve problems across various disciplines. In this article, we delve into the simplification of algebraic expressions, using a practical example involving the cost of rice and beans. This example will help illustrate how these expressions can be manipulated to find equivalent forms, making them easier to understand and work with. Understanding algebraic expressions is crucial not only for math students but also for anyone who wants to enhance their problem-solving skills in everyday situations. By mastering the techniques of simplifying these expressions, we gain the ability to break down complex problems into manageable steps, leading to clear and concise solutions. Whether you are calculating the cost of groceries, determining the optimal dimensions for a garden, or analyzing financial investments, the principles of algebraic simplification can be applied to a wide range of contexts.

Problem Statement: Gabriel's Purchase and Return

Let's consider a scenario where Gabriel purchases rice and beans. The total cost of his purchase is represented by the expression $6b + 4r$, where $b$ represents the cost of one bag of beans and $r$ represents the cost of one bag of rice. This expression tells us that Gabriel bought 6 bags of beans and 4 bags of rice. Now, suppose Gabriel decides to return 2 bags of beans and 1 bag of rice. The expression that represents the total cost after the return is given by $(6b + 4r) - (2b + r)$. Our goal is to simplify this expression to determine the new total cost after the return. This problem not only involves basic algebraic concepts but also provides a practical context that makes it easier to grasp the significance of simplifying expressions. The initial expression, $6b + 4r$, gives us a clear picture of Gabriel's original expenditure. The subsequent return of items introduces a change, and the expression $(6b + 4r) - (2b + r)$ captures this adjustment. By simplifying this expression, we can find out the exact cost of the items Gabriel kept, providing a concrete answer to the problem. This exercise highlights the usefulness of algebraic expressions in modeling real-life financial transactions and decision-making processes.

Step-by-Step Simplification

To simplify the expression $(6b + 4r) - (2b + r)$, we need to follow a series of algebraic steps. First, we distribute the negative sign across the terms inside the second parenthesis:

$(6b + 4r) - (2b + r) = 6b + 4r - 2b - r$

This step is crucial because it ensures that we correctly account for the reduction in cost due to the returned items. The negative sign in front of the parenthesis indicates that we are subtracting the entire quantity $(2b + r)$ from the initial cost. By distributing the negative sign, we change the signs of the terms inside the parenthesis, which is essential for the next step. Next, we combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, $6b$ and $-2b$ are like terms, and $4r$ and $-r$ are like terms. Combining them, we get:

$6b - 2b + 4r - r$

This step involves applying the basic arithmetic operations of addition and subtraction to the coefficients of the like terms. It is a fundamental step in simplifying algebraic expressions and helps in reducing the expression to its simplest form. Finally, we perform the subtraction:

$4b + 3r$

This simplified expression, $4b + 3r$, represents the total cost after Gabriel returned 2 bags of beans and 1 bag of rice. This means that Gabriel now has 4 bags of beans and 3 bags of rice. The simplified expression is easier to interpret and use for further calculations. For instance, if we know the cost of one bag of beans and one bag of rice, we can easily substitute those values into the expression to find the total cost. This step-by-step simplification process not only provides the answer but also illustrates the systematic approach to handling algebraic expressions, which is a valuable skill in mathematical problem-solving.

Detailed Explanation of Each Step

Distributing the Negative Sign

The first step in simplifying the expression $(6b + 4r) - (2b + r)$ is distributing the negative sign. This involves multiplying each term inside the second parenthesis by -1. The expression then becomes:

$6b + 4r - 2b - r$

This step is crucial because it correctly reflects the subtraction of the returned items from the initial purchase. The negative sign in front of the parenthesis indicates that we are reducing the total cost, and distributing it ensures that both the cost of the returned beans and the cost of the returned rice are properly subtracted. It is important to remember that when distributing a negative sign, each term inside the parenthesis changes its sign. For example, $+2b$ becomes $-2b$, and $+r$ becomes $-r$. This process is based on the distributive property of multiplication over addition and subtraction, which is a fundamental principle in algebra. The distributive property states that for any numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$ and $a(b - c) = ab - ac$. In our case, we are applying this property with $a = -1$, which results in the change of signs for the terms inside the parenthesis. This step is not just a mechanical operation but a conceptual one, ensuring that we accurately represent the reduction in cost due to the returned items.

Combining Like Terms

After distributing the negative sign, the expression is $6b + 4r - 2b - r$. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this expression, $6b$ and $-2b$ are like terms because they both contain the variable $b$ raised to the power of 1. Similarly, $4r$ and $-r$ are like terms because they both contain the variable $r$ raised to the power of 1. To combine like terms, we add or subtract their coefficients. The coefficient is the numerical part of the term. For the $b$ terms, we have $6b - 2b$. Subtracting the coefficients, we get $6 - 2 = 4$. So, $6b - 2b = 4b$. For the $r$ terms, we have $4r - r$. Here, $r$ is the same as $1r$, so we have $4r - 1r$. Subtracting the coefficients, we get $4 - 1 = 3$. So, $4r - r = 3r$. By combining these like terms, we simplify the expression to $4b + 3r$. This step is essential because it reduces the expression to its simplest form, making it easier to understand and use for further calculations. Combining like terms is a fundamental technique in algebra and is used extensively in simplifying expressions and solving equations. It helps in organizing the terms and presenting the expression in a more concise and manageable way. This process is based on the commutative and associative properties of addition, which allow us to rearrange and group like terms together without changing the value of the expression.

Final Simplified Expression

After combining like terms, the final simplified expression is $4b + 3r$. This expression represents the total cost of the items Gabriel kept after returning 2 bags of beans and 1 bag of rice. The expression $4b + 3r$ is simpler and more concise than the original expression $(6b + 4r) - (2b + r)$. It clearly shows that Gabriel has 4 bags of beans and 3 bags of rice. This simplified form is not only easier to understand but also more practical for further calculations. For instance, if we know the cost of one bag of beans and one bag of rice, we can easily substitute those values into the expression $4b + 3r$ to find the total cost. The process of simplifying algebraic expressions is crucial in mathematics because it allows us to reduce complex expressions to their simplest forms, making them easier to work with. This is particularly important in problem-solving, where simplifying expressions can often reveal the underlying structure of a problem and lead to a solution. In this case, the simplified expression $4b + 3r$ provides a clear and direct representation of the cost after the return, making it easy to interpret and apply. This final step concludes the simplification process, demonstrating the power of algebraic manipulation in transforming a complex expression into a simpler, more understandable form.

Real-World Applications

The ability to simplify algebraic expressions has numerous real-world applications. Consider scenarios in budgeting, where you might need to calculate the total cost of items after returns or discounts. The principles we've discussed can be directly applied to these situations. For example, if you are planning a party and need to calculate the cost of food and drinks, you might use an algebraic expression to represent the total cost. If some guests cancel and you need to adjust the quantity of items, simplifying the expression will help you determine the new total cost quickly and accurately. In business, algebraic expressions are used extensively for cost analysis, profit calculations, and inventory management. Businesses need to understand how changes in costs, prices, or quantities affect their bottom line. Simplifying algebraic expressions allows them to model these relationships and make informed decisions. For instance, a retailer might use an expression to represent the total revenue from sales, taking into account discounts and returns. Simplifying this expression can help the retailer understand the impact of different pricing strategies or promotional offers. In engineering and science, algebraic expressions are used to model physical phenomena and solve complex problems. Engineers might use expressions to calculate the stress on a bridge or the flow rate in a pipe. Simplifying these expressions is crucial for obtaining accurate results and ensuring the safety and efficiency of designs. Scientists use algebraic expressions to analyze data, develop theories, and make predictions. The ability to simplify these expressions is essential for advancing scientific knowledge and innovation. These examples illustrate that the skill of simplifying algebraic expressions is not just a theoretical exercise but a practical tool that can be applied in various aspects of life. By mastering this skill, individuals can enhance their problem-solving abilities and make more informed decisions in their personal and professional lives.

Conclusion: The Power of Simplification

In conclusion, the simplification of algebraic expressions is a fundamental skill in mathematics with wide-ranging applications. By understanding how to distribute, combine like terms, and simplify expressions, we can solve complex problems more efficiently. In our example, we successfully simplified the expression $(6b + 4r) - (2b + r)$ to $4b + 3r$, representing the total cost after Gabriel's return. This process not only provided us with the answer but also demonstrated the systematic approach to handling algebraic expressions. The ability to simplify expressions is crucial not only for academic success but also for real-world problem-solving. Whether you are calculating the cost of groceries, managing a budget, or analyzing data, the principles of algebraic simplification can help you make informed decisions. By mastering this skill, you can enhance your analytical abilities and approach challenges with confidence. The example we discussed, involving the cost of rice and beans, provides a concrete illustration of how algebraic expressions can be used to model and solve practical problems. The steps involved in simplifying the expression, such as distributing the negative sign and combining like terms, are fundamental techniques that can be applied to a wide variety of algebraic problems. Understanding these techniques is essential for anyone who wants to excel in mathematics and use mathematical principles in their daily lives. In essence, the power of simplification lies in its ability to transform complex expressions into simpler, more manageable forms, making them easier to understand and apply. This skill is a cornerstone of mathematical literacy and a valuable asset in any field that involves quantitative reasoning.