Mend's Trousers And Tie Cost Calculation A Mathematical Problem

Mend went shopping and spent £90.60 on a stylish new pair of trousers and a matching tie. We know the tie cost £x, and the trousers were four times the price of the tie. In this article, we'll delve into the mathematics behind Mend's purchase, figuring out the total cost using an algebraic expression.

1. Expressing the Total Cost Algebraically

To begin, let's break down the problem. We know the cost of the tie is £x. The trousers cost four times as much as the tie, which translates to 4 * £x = £4x. Now, to find the total cost, we simply add the cost of the tie and the trousers together.

So, the expression for the total cost in terms of x is: £x (tie) + £4x (trousers). Combining like terms, we get the expression £5x. This algebraic expression, £5x, represents the total amount Mend spent on his new outfit. This foundational expression allows us to connect the unknown cost of the tie, represented by 'x', to the overall expenditure. It serves as a building block for further analysis, enabling us to determine the individual cost of each item if we have additional information or constraints.

Understanding how to construct and manipulate algebraic expressions is a core skill in mathematics. It allows us to model real-world situations and solve problems in a systematic way. In this case, we've successfully translated the information about Mend's purchase into a concise algebraic form. This process involves identifying the key variables, establishing relationships between them, and using mathematical operations to combine them into a meaningful expression. The expression £5x encapsulates the total cost in a way that can be easily used for calculation or further analysis. For instance, if we knew the total cost was a specific amount, we could then solve for 'x' to determine the cost of the tie. The power of algebra lies in its ability to represent complex scenarios in a simplified and manageable form, making it an indispensable tool in various fields beyond just mathematics.

Furthermore, this initial expression lays the groundwork for more complex scenarios. We could introduce additional factors, such as sales tax or discounts, and incorporate them into the algebraic representation. For example, if there was a 10% sales tax, we could multiply the total cost expression by 1.10 to account for the added expense. Similarly, if Mend had a coupon for a certain percentage off, we could subtract that percentage from the original cost before adding the sales tax. The flexibility of algebraic expressions allows us to adapt to changing conditions and model a wide range of real-world situations. The key is to carefully identify each component of the problem and represent it appropriately in the expression. This systematic approach ensures that the final expression accurately reflects the scenario being modeled.

2. Finding the Value of x

We know that the total cost, £5x, is equal to £90.60. Now, we can set up an equation: 5x = 90.60. To solve for x (the cost of the tie), we need to isolate x. We can do this by dividing both sides of the equation by 5.

So, x = 90.60 / 5, which gives us x = 18.12. This means the cost of the tie is £18.12. This step is crucial in translating the algebraic representation into a concrete numerical value. By applying the principles of equation solving, we've effectively untangled the relationship between the total cost and the unknown variable, 'x'. The process of isolating the variable involves performing the same operations on both sides of the equation, maintaining the balance and ensuring that the solution remains valid. In this case, dividing both sides by 5 allowed us to isolate 'x' and determine its value. The result, x = 18.12, provides a specific answer to the initial question about the cost of the tie. This value can then be used to calculate other related quantities, such as the cost of the trousers. The ability to solve equations is a fundamental skill in mathematics, enabling us to find solutions to a wide variety of problems across different disciplines.

Moreover, understanding the concept of inverse operations is key to solving equations effectively. Division is the inverse operation of multiplication, which is why we divided both sides of the equation by 5 to undo the multiplication by 5 on the left side. Similarly, if the equation involved addition, we would use subtraction to isolate the variable. The ability to recognize and apply inverse operations is crucial for manipulating equations and arriving at the correct solution. This understanding also extends to more complex equations involving exponents, logarithms, and other mathematical functions. The underlying principle remains the same: use inverse operations to systematically isolate the variable and determine its value. This methodical approach ensures accuracy and efficiency in the problem-solving process.

3. Calculating the Cost of the Trousers

Since the trousers cost four times the price of the tie, we can calculate the cost of the trousers by multiplying the value of x (which is £18.12) by 4. So, the cost of the trousers is 4 * £18.12 = £72.48. This calculation demonstrates how the value of 'x', determined in the previous step, can be used to find other related quantities. By substituting the numerical value of 'x' into the expression for the cost of the trousers (4x), we arrive at a specific answer. This process highlights the interconnectedness of mathematical concepts and how solving for one variable can unlock the solution to other parts of the problem. The ability to apply previously calculated values in subsequent calculations is essential for problem-solving in various contexts. It allows us to build upon our findings and progressively unravel complex scenarios.

Furthermore, this step underscores the importance of careful attention to detail in mathematical calculations. Multiplying £18.12 by 4 requires accuracy and precision to arrive at the correct answer of £72.48. Errors in calculation can propagate through the problem and lead to incorrect conclusions. Therefore, it's crucial to double-check calculations and ensure that each step is performed correctly. The use of calculators can aid in this process, but it's equally important to understand the underlying mathematical principles and the logic behind each calculation. This ensures that the final answer is not only numerically correct but also makes sense within the context of the problem. A strong foundation in arithmetic and the ability to perform calculations accurately are fundamental skills in mathematics and beyond.

4. Verification: Confirming the Total Cost

To ensure our calculations are correct, we can add the cost of the tie (£18.12) and the cost of the trousers (£72.48) together. £18.12 + £72.48 = £90.60. This matches the total amount Mend paid, so our calculations are accurate. This final step is crucial for ensuring the validity of our solution. By independently verifying the results, we can catch any errors that may have occurred during the calculation process. In this case, adding the individual costs of the tie and trousers should equal the total amount spent. If the sum does not match the given total, it indicates that there is an error in one or more of the preceding steps. This verification process reinforces the importance of accuracy and attention to detail in mathematical problem-solving.

Moreover, the verification step highlights the concept of reverse operations in mathematics. We used addition to verify the total cost, which is the reverse operation of the subtraction that would have been used if we were given the total cost and the cost of one item and asked to find the cost of the other. Understanding these relationships between operations can help to check the reasonableness of an answer. For instance, if the calculated cost of the trousers was significantly higher or lower than expected, it would signal a potential error in the calculations. Therefore, verification is not just about confirming the numerical result but also about ensuring that the solution aligns with the logical context of the problem.

Conclusion: Mend's Outfit Cost Analyzed

In conclusion, we successfully expressed the total cost of Mend's new trousers and tie as an algebraic expression, £5x. By solving the equation 5x = 90.60, we determined that the tie cost £18.12 and the trousers cost £72.48. We verified our results by confirming that the sum of the individual costs matched the total amount paid. This exercise demonstrates the power of algebra in solving real-world problems. This comprehensive analysis of Mend's purchase demonstrates the practical application of algebraic principles in everyday scenarios. By translating the word problem into an algebraic equation, we were able to systematically solve for the unknown variable and determine the cost of each item. This process highlights the importance of mathematical literacy in navigating real-world situations and making informed decisions.

Furthermore, the problem-solving approach used in this example can be applied to a wide range of other scenarios. The ability to break down complex problems into smaller, manageable steps, identify key variables, and establish relationships between them is a valuable skill in various fields, including finance, engineering, and science. The methodical approach of setting up an equation, solving for the unknown, and verifying the solution ensures accuracy and efficiency in problem-solving. Therefore, mastering these fundamental algebraic concepts and techniques is essential for success in both academic and professional pursuits.

Keywords

Algebraic expression, equation solving, cost calculation, word problems, mathematics.