Mend's Fashion Expenses A Mathematical Problem Solving

Mend's recent fashion splurge presents an intriguing scenario for mathematical exploration. He spent a total of £390.60 on a stylish new pair of trousers and a tie, and we're tasked with dissecting this expense using the power of algebra. The challenge lies in expressing the total cost in terms of a variable, x, which represents the cost of the tie. Additionally, we know the trousers cost twice as much as the tie. Let's delve into this mathematical puzzle and unravel Mend's sartorial spending.

1. Expressing the Total Cost in Terms of x

To begin, we need to translate the given information into algebraic expressions. Let's break down the problem step by step:

• Cost of the tie: This is directly given as x.
• Cost of the trousers: We know the trousers cost twice as much as the tie. Therefore, the cost of the trousers can be expressed as 2 * x, or simply 2x.
• Total cost: The total cost is the sum of the cost of the tie and the cost of the trousers. So, the total cost can be expressed as x + 2x.

Now, we can simplify this expression by combining like terms. Both x and 2x are terms involving the variable x, so we can add their coefficients (the numbers in front of the x). The coefficient of the first x is 1 (since x is the same as 1 * x), and the coefficient of the second x is 2. Adding these coefficients, we get 1 + 2 = 3. Therefore, the simplified expression for the total cost is 3x.

This expression, 3x, elegantly represents the total cost of Mend's purchase in terms of the cost of the tie (x). It encapsulates the relationship between the tie's price and the trousers' price, allowing us to calculate the total cost if we know the value of x. Furthermore, this expression serves as the foundation for solving the next part of the problem, where we'll likely be asked to find the actual value of x given the total cost.

The power of algebra lies in its ability to represent real-world situations using symbolic expressions. In this case, we've successfully transformed a word problem about Mend's clothing expenses into a concise algebraic expression. This expression not only represents the total cost but also highlights the underlying mathematical relationship between the prices of the tie and the trousers. By understanding this relationship, we can further analyze Mend's spending and gain deeper insights into the problem. The use of the variable x allows for generalization, meaning we can use this expression to calculate the total cost for any value of the tie's price, provided the trousers always cost twice as much. This flexibility is a key advantage of using algebraic representation.

2. Finding the Cost of the Tie and the Trousers

Now that we have an expression for the total cost (3x) and we know the actual total cost (£390.60), we can set up an equation and solve for x. This will tell us the cost of the tie. From there, we can easily calculate the cost of the trousers.

Setting up the equation: We know that 3x represents the total cost, and the total cost is £390.60. Therefore, we can write the equation:

3x = 390.60

Solving for x: To isolate x and find its value, we need to divide both sides of the equation by 3. This is because the equation states that 3 multiplied by x equals 390.60, so to undo the multiplication, we perform the inverse operation, which is division.

Dividing both sides by 3, we get:

x = 390.60 / 3

Performing the division, we find:

x = 130.20

Therefore, the cost of the tie (x) is £130.20.

Calculating the cost of the trousers: We know the trousers cost twice as much as the tie. Since the tie costs £130.20, the trousers cost:

2 * £130.20 = £260.40

So, the trousers cost £260.40.

Verification: To ensure our calculations are correct, we can add the cost of the tie and the cost of the trousers to see if they equal the total cost:

£130.20 (tie) + £260.40 (trousers) = £390.60

This matches the given total cost, so our calculations are accurate.

In conclusion, by using algebraic expressions and equations, we've successfully determined the individual costs of Mend's tie and trousers. We found that the tie cost £130.20 and the trousers cost £260.40. This problem demonstrates the practical application of algebra in everyday scenarios, such as managing personal finances and understanding spending habits. The ability to translate word problems into mathematical equations is a crucial skill in problem-solving and critical thinking.

3. The Power of Algebraic Problem-Solving

This exercise involving Mend's clothing purchase highlights the fundamental principles of algebraic problem-solving. It showcases how we can use variables, expressions, and equations to represent real-world situations and find solutions. The process of translating a word problem into a mathematical equation is a key skill in mathematics and has applications far beyond the classroom.

The Importance of Variables: The variable x played a crucial role in this problem. It allowed us to represent the unknown cost of the tie and, more importantly, to establish a relationship between the cost of the tie and the cost of the trousers. By using a variable, we could express the cost of the trousers (2x) in terms of the cost of the tie, creating a connection between the two unknowns. This connection was essential for setting up the equation that ultimately led us to the solution. Without variables, we would be limited to working with specific numbers and would not be able to represent unknown quantities or relationships between them.

Expressions as Building Blocks: The expression 3x represents the total cost in a concise and general way. It's not just a numerical value; it's a formula that can be used to calculate the total cost for any value of x. This highlights the power of algebraic expressions as building blocks. They allow us to represent complex situations using a combination of variables, constants, and operations. In this case, the expression 3x encapsulates the entire scenario of Mend's purchase, showing how the cost of the tie directly influences the total cost.

Equations as Tools for Solving: The equation 3x = 390.60 was the key to solving the problem. It established an equality between the algebraic expression representing the total cost and the actual total cost. Equations are powerful tools because they allow us to find the value of unknown variables by using known information. By applying algebraic operations to both sides of the equation, we were able to isolate x and determine its value. This process of solving equations is a fundamental skill in mathematics and is used in a wide range of applications, from engineering and physics to economics and computer science.

Real-World Applications: This problem, while seemingly simple, demonstrates the practical relevance of algebra in everyday life. We encounter situations involving unknown quantities and relationships between them all the time. Whether it's calculating the total cost of a shopping trip, determining the optimal dosage of a medication, or analyzing financial data, the principles of algebraic problem-solving are invaluable. By mastering these principles, we can approach complex problems with confidence and find effective solutions.

In conclusion, Mend's fashion expenses provided a perfect context for exploring the power of algebraic problem-solving. By using variables, expressions, and equations, we were able to dissect the problem, find the unknown costs, and understand the relationships between the different quantities. This example underscores the importance of algebra as a tool for representing and solving real-world problems, highlighting its relevance in both academic and practical settings. The ability to translate real-world scenarios into mathematical models is a valuable skill that empowers us to make informed decisions and solve complex challenges.

Conclusion

In summary, we've successfully navigated the mathematical landscape of Mend's clothing purchase. We started by expressing the total cost in terms of x, creating the expression 3x. We then used this expression and the given total cost to form an equation, which we solved to find the cost of the tie. Finally, we calculated the cost of the trousers and verified our results. This problem serves as a testament to the power of algebra in translating real-world scenarios into mathematical models and finding solutions to practical problems. The ability to use variables, expressions, and equations is a valuable skill that extends far beyond the classroom, empowering us to analyze situations, make informed decisions, and solve complex challenges in various aspects of life.