What Are The Train Fares From Delhi To Station A And Station B, Given That 2 Tickets To A And 3 To B Cost ₹ 77, And 3 Tickets To A And 5 To B Cost ₹ 124?

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Introduction: Unraveling the Train Ticket Puzzle

In this article, we embark on a mathematical journey to decipher the cost of train tickets from Delhi station to two distinct destinations, station A and station B. We will dissect a word problem, translate it into a system of linear equations, and employ algebraic techniques to unveil the individual fares. This exercise not only sharpens our problem-solving skills but also demonstrates the practical application of mathematics in everyday scenarios. Our primary focus is to determine the fares from Delhi to station A and station B by meticulously analyzing the given information and applying appropriate mathematical methods. This involves setting up a system of equations based on the total costs of different ticket combinations and then solving these equations to find the individual ticket prices. Understanding these methods is crucial for anyone looking to tackle similar mathematical problems efficiently.

The core challenge lies in converting the word problem into a mathematical model. We'll start by defining variables to represent the unknown fares, and then construct equations based on the provided conditions. This process highlights the importance of accurate translation from verbal statements to algebraic expressions. The subsequent steps involve applying techniques such as substitution or elimination to solve the system of equations. Each method has its advantages, and we will explore why a particular approach might be more efficient in this context. The solution will provide us with the exact fares to both stations, offering a clear and concise answer to the original question. Moreover, this problem serves as an excellent example of how linear algebra can be used to solve real-world financial questions, emphasizing the interdisciplinary nature of mathematics and its relevance to practical situations.

Furthermore, we will not only focus on finding the numerical answers but also on understanding the underlying concepts and the reasoning behind each step. This approach is essential for building a strong foundation in algebra and for developing critical thinking skills that are transferable to other areas of study and life. The problem's structure encourages us to think systematically, breaking down a complex problem into smaller, manageable parts. By carefully considering each piece of information and its relationship to the overall question, we can arrive at a solution that is both accurate and well-understood. This holistic approach to problem-solving is what truly empowers us to tackle challenges with confidence and competence.

Setting Up the Equations: Translating Words into Math

To solve this intriguing problem, our initial step involves translating the provided information into a mathematical framework. We need to define variables that will represent the unknown fares we are trying to find. Let's designate the fare from Delhi to station A as x (in ₹) and the fare from Delhi to station B as y (in ₹). This crucial step allows us to convert the word problem into a symbolic representation, making it easier to manipulate and solve. Defining these variables is the cornerstone of our algebraic approach, transforming the narrative into a solvable mathematical puzzle.

Now, we can use the given information to construct a system of linear equations. The first statement tells us that buying 2 tickets for station A and 3 tickets for station B costs ₹ 77 in total. This translates directly into the equation 2x + 3y = 77. Similarly, the second statement informs us that 3 tickets for station A and 5 tickets for station B cost ₹ 124. This gives us the equation 3x + 5y = 124. We now have a system of two linear equations with two unknowns:

  1. 2x + 3y = 77
  2. 3x + 5y = 124

This system of equations represents the core of our problem. Each equation captures the relationship between the fares to stations A and B, based on the given total costs. The challenge now is to solve this system, finding the values of x and y that satisfy both equations simultaneously. To effectively solve this system, we will explore methods such as substitution or elimination. Each method involves manipulating the equations in a way that allows us to isolate one variable and then solve for the other. The choice of method often depends on the specific structure of the equations; in this case, we'll consider both options to determine the most efficient approach.

By setting up these equations, we have effectively transformed a real-world scenario into an abstract mathematical problem. This transformation is a fundamental skill in mathematical problem-solving, as it allows us to apply powerful algebraic tools to find solutions. The next step will be to choose a method to solve this system, which will lead us to the individual fares for stations A and B. This process highlights the beauty of mathematics – its ability to simplify complex situations into manageable, solvable forms.

Solving the Equations: Finding the Fares

With our system of equations established, the next crucial step is to solve for the unknowns, x and y, which represent the fares to station A and station B, respectively. We have two primary methods at our disposal: substitution and elimination. For this particular problem, the elimination method might prove to be more efficient due to the structure of the equations. However, we will briefly discuss both methods to illustrate their applicability.

Elimination Method

The elimination method involves manipulating the equations so that the coefficients of one of the variables are the same (or additive inverses) in both equations. This allows us to eliminate that variable by adding or subtracting the equations. To apply this method, we can multiply the first equation (2x + 3y = 77) by 3 and the second equation (3x + 5y = 124) by 2. This will make the coefficients of x equal in both equations:

  1. Multiplying the first equation by 3, we get: 6x + 9y = 231
  2. Multiplying the second equation by 2, we get: 6x + 10y = 248

Now, we subtract the first modified equation from the second modified equation to eliminate x:

(6x + 10y) - (6x + 9y) = 248 - 231

This simplifies to:

y = 17

So, the fare to station B (y) is ₹ 17. Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the first original equation:

2x + 3(17) = 77 2x + 51 = 77 2x = 26 x = 13

Thus, the fare to station A (x) is ₹ 13.

Substitution Method (Alternative Approach)

Alternatively, we could have used the substitution method. This involves solving one equation for one variable and then substituting that expression into the other equation. For example, from the first equation (2x + 3y = 77), we can solve for x:

2x = 77 - 3y x = (77 - 3y) / 2

Now, substitute this expression for x into the second equation:

3[(77 - 3y) / 2] + 5y = 124

This equation involves only y, which we can solve. Multiplying through by 2 to eliminate the fraction gives:

3(77 - 3y) + 10y = 248 231 - 9y + 10y = 248 y = 17

Again, we find that the fare to station B (y) is ₹ 17. Substituting this value back into the expression for x gives:

x = (77 - 3(17)) / 2 x = (77 - 51) / 2 x = 13

We arrive at the same solution: the fare to station A (x) is ₹ 13. Both methods yield the same result, but the elimination method was arguably more straightforward in this case due to the coefficients in the original equations. The efficiency of a method often depends on the specific problem, and choosing the right approach can save time and effort.

The Solution: Fares to Station A and Station B

After meticulously applying algebraic techniques, we have successfully unraveled the mystery of the train fares. Our calculations, using both the elimination and substitution methods, converge on the same definitive answer. The fare from Delhi to station A is ₹ 13, and the fare from Delhi to station B is ₹ 17. This solution provides a clear and concise answer to the original problem, demonstrating the power of linear equations in solving real-world scenarios.

To ensure the accuracy of our solution, it is always prudent to verify the results by substituting the values back into the original equations. This step acts as a crucial check, confirming that our calculated fares satisfy the given conditions. Let's substitute x = 13 and y = 17 into the original equations:

  1. 2x + 3y = 77 becomes 2(13) + 3(17) = 26 + 51 = 77 (Correct)
  2. 3x + 5y = 124 becomes 3(13) + 5(17) = 39 + 85 = 124 (Correct)

Since both equations hold true with our calculated fares, we can confidently conclude that our solution is accurate. This verification step underscores the importance of precision in mathematical problem-solving and reinforces the validity of our findings. The ability to validate solutions is a critical skill that ensures the reliability of our results and enhances our understanding of the problem.

The journey from the initial word problem to the final solution highlights the elegance and practicality of mathematics. We transformed a narrative into a system of equations, applied algebraic methods to solve for the unknowns, and verified our results to ensure accuracy. This process not only provides us with the specific fares but also reinforces the broader applicability of mathematical principles in everyday situations. The ability to approach problems systematically, break them down into manageable components, and apply appropriate techniques is a valuable skill that extends far beyond the realm of mathematics. This example serves as a testament to the power of mathematical thinking in unraveling real-world puzzles.

Conclusion: The Power of Mathematical Problem-Solving

In conclusion, we have successfully navigated the challenge of determining the train fares from Delhi to stations A and B. By translating the word problem into a system of linear equations and employing algebraic techniques, we arrived at the solution: the fare to station A is ₹ 13, and the fare to station B is ₹ 17. This exercise underscores the practical utility of mathematics in solving everyday problems and highlights the importance of developing strong problem-solving skills.

The process we undertook – from defining variables to verifying the solution – illustrates the systematic approach that is crucial for mathematical problem-solving. We began by identifying the unknowns and assigning variables to represent them. This foundational step allowed us to convert the narrative into a symbolic representation, making it amenable to algebraic manipulation. Next, we constructed a system of equations based on the given information, capturing the relationships between the variables. This step requires careful attention to detail and a precise understanding of the problem's constraints.

Once the equations were set up, we explored different methods for solving them, ultimately choosing the elimination method for its efficiency in this particular case. We also discussed the substitution method as an alternative approach, demonstrating the flexibility of algebraic techniques. The key takeaway here is that there is often more than one way to solve a problem, and choosing the most efficient method can save time and effort. Furthermore, the verification step was paramount in ensuring the accuracy of our solution. By substituting the calculated fares back into the original equations, we confirmed that our results satisfied the given conditions, reinforcing the validity of our findings.

This problem serves as a microcosm of the broader application of mathematics in real-world scenarios. Linear equations, in particular, are a powerful tool for modeling and solving problems in various fields, including finance, engineering, and economics. The ability to translate real-world situations into mathematical models and then solve them is a valuable skill that is highly sought after in many professions. Moreover, the critical thinking and problem-solving skills honed through mathematics are transferable to other areas of life, enabling us to approach challenges with confidence and competence. By mastering these skills, we empower ourselves to tackle complex problems, make informed decisions, and navigate the world with a greater understanding of the underlying principles that govern it. The journey through this train fare problem exemplifies the power and versatility of mathematical thinking.