Solving The Animal Buying Problem A Mathematical Puzzle

How many of each animal (goat, cow, and buffalo) can be bought with 20 rupees if a goat costs 25 paisa, a cow costs 50 paisa, and a buffalo costs 5 rupees, and you need to buy a total of 20 animals?

Introduction: Unraveling the Mystery of the Twenty Animals

This article delves into a fascinating mathematical puzzle centered around purchasing a specific number of animals – goats, cows, and buffaloes – with a limited budget. This classic problem, often encountered in mathematical recreations and introductory algebra courses, challenges us to find a combination of these animals that satisfies both the quantity and cost constraints. The core of the puzzle lies in the varying prices of each animal and the fixed amount of money available. Let's embark on this intriguing journey of numbers and logic, exploring different approaches to crack the code of this engaging animal buying conundrum. Our main keywords include animal buying problem, mathematical puzzle, and algebraic solutions. We aim to provide a comprehensive and insightful exploration of the puzzle, making it accessible to both math enthusiasts and those new to problem-solving.

Problem Statement: Decoding the Animal Purchase

The problem presents a scenario where you have 20 rupees to spend on purchasing 20 animals, comprising goats, cows, and buffaloes. The price of each animal is as follows: a goat costs 25 paisa, a cow costs 50 paisa, and a buffalo costs 5 rupees. The challenge is to determine the exact number of each animal – goats, cows, and buffaloes – that can be bought within the given budget of 20 rupees, while also ensuring that the total number of animals purchased is precisely 20. This puzzle beautifully illustrates the application of mathematical concepts in real-world-like scenarios. It requires careful analysis, logical reasoning, and a systematic approach to arrive at the correct solution. The key here is to translate the given information into mathematical equations and then solve them to find the values that satisfy all the conditions. This section will guide you through the initial setup of the problem, laying the groundwork for the subsequent steps of finding the solution. Understanding the problem statement is crucial before attempting any solution.

Setting Up the Equations: The Foundation of the Solution

To solve this animal buying problem, we need to translate the given information into mathematical equations. Let's denote the number of goats as 'x', the number of cows as 'y', and the number of buffaloes as 'z'. From the problem statement, we can derive two key equations. First, the total number of animals is 20, which gives us the equation: x + y + z = 20. Second, the total cost of the animals is 20 rupees, which, considering the prices in paisa (25 paisa for a goat, 50 paisa for a cow, and 5 rupees or 500 paisa for a buffalo), translates to the equation: 25x + 50y + 500z = 2000 (since 20 rupees is equal to 2000 paisa). These two equations form the foundation for solving the problem. We have two equations with three unknowns, which means we are dealing with a system of Diophantine equations. Solving such systems often involves finding integer solutions, as we cannot buy a fraction of an animal. This section highlights the critical step of equation formulation, which is essential for any mathematical problem-solving endeavor.

Solving the Equations: A Step-by-Step Approach

Now that we have our equations, the next step is to solve them. We have two equations: x + y + z = 20 and 25x + 50y + 500z = 2000. To simplify, we can divide the second equation by 25, which gives us x + 2y + 20z = 80. Now we have a simpler system of equations: x + y + z = 20 and x + 2y + 20z = 80. Subtracting the first equation from the second, we get y + 19z = 60. This equation is crucial as it reduces the problem to two variables. Since x, y, and z represent the number of animals, they must be non-negative integers. We can analyze the equation y + 19z = 60 to find possible values for y and z. If z = 0, then y = 60, which is not possible as it would violate the condition x + y + z = 20. If z = 1, then y = 60 - 19 = 41, also not possible. If z = 2, then y = 60 - 38 = 22, again not possible. If z = 3, then y = 60 - 57 = 3. This gives us a feasible solution. Substituting z = 3 and y = 3 into the equation x + y + z = 20, we get x + 3 + 3 = 20, which gives x = 14. Therefore, one possible solution is 14 goats, 3 cows, and 3 buffaloes. Solving the equations systematically is key to finding the correct solution.

Verification and Alternative Solutions: Ensuring Accuracy and Exploring Possibilities

After finding a solution, it's crucial to verify it. In our case, we found x = 14, y = 3, and z = 3. Let's check if these values satisfy both equations. For the first equation, x + y + z = 14 + 3 + 3 = 20, which is correct. For the second equation, 25x + 50y + 500z = 25(14) + 50(3) + 500(3) = 350 + 150 + 1500 = 2000 paisa, which is also correct. Thus, our solution is verified. Now, let's explore if there are any other possible solutions. From the equation y + 19z = 60, we know that as z increases, y decreases. If we try z = 4, then y = 60 - 19(4) = 60 - 76 = -16, which is not possible as y cannot be negative. Therefore, z = 3 is the only possible value for z that yields non-negative integer solutions for y. Consequently, there is only one unique solution to this problem. This step of verification and exploration is vital in mathematical problem-solving.

Conclusion: The Elegance of Mathematical Problem-Solving

In conclusion, the animal buying problem is a delightful example of how mathematical principles can be applied to solve real-world-like scenarios. We successfully determined that the only possible solution is to buy 14 goats, 3 cows, and 3 buffaloes with the given budget and constraints. The problem not only tests our algebraic skills but also enhances our logical reasoning and analytical thinking. The process of setting up equations, solving them systematically, and verifying the solution demonstrates the elegance and power of mathematical problem-solving. This puzzle serves as an excellent educational tool, encouraging a deeper understanding of mathematical concepts and their applications. It highlights the importance of careful analysis, logical deduction, and a systematic approach in tackling complex problems. By dissecting the problem, formulating equations, and finding integer solutions, we have unlocked the mystery of the twenty animals, showcasing the beauty and practicality of mathematics.