Algebraic Expressions And Equations Solving For The Cost Of Trousers And Tie

In this article, we'll delve into a real-world scenario involving a purchase made by Mana – a pair of trousers and a tie. By carefully dissecting the given information, we'll embark on a mathematical journey to unravel the cost of each item and express their relationship algebraically. This exploration will not only enhance our understanding of algebraic expressions but also demonstrate how mathematics can be applied to everyday situations, such as managing personal finances and making informed purchasing decisions. Get ready to put on your thinking caps and unravel the mysteries hidden within Mana's wardrobe expenses!

1. Understanding the Problem A Breakdown of the Given Information

Before we dive into the algebraic expressions, let's take a moment to thoroughly understand the problem at hand. Mana spent a total of £390.60 on a pair of trousers and a tie. This is our key piece of information, as it represents the combined cost of the two items. We are also told that the cost of the trousers is represented by £x, which means that 'x' is a variable that holds the numerical value of the trousers' price. The final crucial detail is the relationship between the cost of the trousers and the tie: the trousers cost five times as much as the tie. This establishes a proportionality that we can use to express the cost of the tie in terms of 'x'. By carefully extracting these pieces of information, we lay the foundation for building our algebraic expressions and ultimately solving for the individual costs.

To further solidify our understanding, let's rephrase the problem in simpler terms. Imagine you're shopping for clothes, and you know the total amount you've spent. You also know the price of one item and how it relates to the price of another. Your goal is to figure out the individual prices of each item. This is essentially the challenge we face with Mana's purchase. By breaking down the problem into smaller, more manageable parts, we can approach it with clarity and confidence. Remember, understanding the problem is the first crucial step towards finding the solution. Once we have a firm grasp of the information, we can move on to translating it into mathematical language.

2. Expressing the Cost Algebraically A Step-by-Step Guide

Now that we have a clear understanding of the problem, it's time to translate the given information into an algebraic expression. This involves using variables and mathematical operations to represent the relationships between the costs of the trousers and the tie. The problem states that the cost of the trousers is £x, which serves as our starting point. The crucial piece of information is that the trousers cost five times as much as the tie. This means that the cost of the tie is one-fifth (1/5) of the cost of the trousers. To express this algebraically, we divide the cost of the trousers (£x) by 5, resulting in £x/5. This expression, £x/5, represents the cost of the tie in terms of 'x'.

To represent the total cost of the trousers and the tie, we simply add their individual costs together. The cost of the trousers is £x, and the cost of the tie is £x/5. Therefore, the expression for the total cost is £x + £x/5. This is the algebraic representation of the total amount Mana spent. To simplify this expression further, we can find a common denominator for the terms. In this case, the common denominator is 5. We can rewrite £x as £5x/5. Adding this to £x/5, we get the simplified expression £(5x + x)/5, which further simplifies to £6x/5. This final expression, £6x/5, represents the total cost of the trousers and the tie in a concise algebraic form. This is a key step in solving the problem, as it allows us to relate the variable 'x' to the known total cost of £390.60.

3. Forming the Equation Connecting the Expression to the Total Cost

Having derived the algebraic expression for the total cost of the trousers and the tie (£6x/5), our next step is to form an equation that connects this expression to the actual amount Mana paid (£390.60). An equation is a mathematical statement that shows the equality between two expressions. In this case, we know that the expression £6x/5 represents the total cost, and we also know that the total cost is £390.60. Therefore, we can set these two equal to each other, forming the equation: £6x/5 = £390.60. This equation is the heart of our problem-solving process, as it allows us to solve for the unknown variable 'x', which represents the cost of the trousers.

This equation is a simple algebraic equation that can be solved using basic algebraic techniques. The goal is to isolate 'x' on one side of the equation. To do this, we can first multiply both sides of the equation by 5 to eliminate the fraction. This gives us: £6x = £390.60 * 5, which simplifies to £6x = £1953. Now, to isolate 'x', we divide both sides of the equation by 6. This gives us: x = £1953 / 6, which results in x = £325.50. Therefore, the cost of the trousers (£x) is £325.50. By forming and solving this equation, we have successfully determined the value of 'x' and taken a significant step towards finding the individual costs of the trousers and the tie. This demonstrates the power of algebraic equations in solving real-world problems.

4. Calculating the Individual Costs Unveiling the Prices of Trousers and Tie

With the value of 'x' (the cost of the trousers) now known to be £325.50, we can proceed to calculate the individual costs of both the trousers and the tie. We already know that the cost of the trousers is £325.50. To find the cost of the tie, we recall that the trousers cost five times as much as the tie. This means that the cost of the tie is one-fifth (1/5) of the cost of the trousers. To calculate this, we divide the cost of the trousers (£325.50) by 5: £325.50 / 5 = £65.10. Therefore, the cost of the tie is £65.10.

Now we have successfully determined the individual costs of both items: the trousers cost £325.50, and the tie costs £65.10. To verify our solution, we can add these two costs together to see if they equal the total amount Mana paid (£390.60): £325.50 + £65.10 = £390.60. This confirms that our calculations are correct. We have not only found the individual costs but also verified our solution, ensuring its accuracy. This step-by-step process of calculating the individual costs demonstrates the practical application of algebraic expressions and equations in solving real-world financial problems. Understanding these concepts can empower individuals to make informed purchasing decisions and manage their finances effectively.

5. Real-World Applications Beyond the Classroom

The mathematical concepts we've explored in this article, such as algebraic expressions and equations, extend far beyond the classroom. They are fundamental tools that can be applied to a wide range of real-world situations, particularly in the realm of personal finance and decision-making. Imagine you're planning a budget for the month. You have a fixed income, and you need to allocate it to various expenses, such as rent, utilities, groceries, and entertainment. Algebraic expressions can help you represent these expenses and create a budget that aligns with your income. For example, you can use variables to represent the amounts allocated to each category and form equations to ensure that your total expenses do not exceed your income.

Another practical application is in comparing prices and making informed purchasing decisions. Suppose you're shopping for a new laptop, and you're comparing different models with varying specifications and prices. You can use algebraic expressions to represent the features and costs of each model and create equations to determine the best value for your money. For instance, you might consider factors like processing speed, memory, storage capacity, and screen size, assigning numerical values to each and forming an expression that represents the overall performance and value of each laptop. By comparing these expressions, you can make a rational decision based on your needs and budget. These are just a few examples of how the mathematical skills we've honed in this article can empower us to navigate everyday financial challenges and make sound decisions. The ability to translate real-world scenarios into mathematical language and solve equations is a valuable asset in today's complex world.

6. Conclusion Mastering Algebra for Everyday Life

In conclusion, we've successfully navigated a real-world scenario involving Mana's purchase of trousers and a tie, demonstrating the power and practicality of algebraic expressions and equations. By carefully dissecting the problem, we were able to translate the given information into mathematical language, form an equation, and solve for the unknown variables. This process not only allowed us to determine the individual costs of the trousers and the tie but also highlighted the broader applications of algebra in everyday life. From managing personal finances to making informed purchasing decisions, the ability to think algebraically is a valuable skill that can empower us to solve problems and navigate challenges with confidence.

This exploration serves as a reminder that mathematics is not just an abstract subject confined to textbooks and classrooms. It is a powerful tool that can be used to understand and interact with the world around us. By embracing mathematical thinking and developing our problem-solving skills, we can unlock new opportunities and make informed decisions in all aspects of our lives. The journey through Mana's wardrobe expenses has been a testament to the relevance and versatility of algebra, demonstrating its ability to transform real-world scenarios into solvable mathematical puzzles. As we continue to learn and grow, let us remember the lessons learned here and apply them to the challenges and opportunities that lie ahead.