Solving A Coin Puzzle Finding The Number Of Coins Of Each Denomination

In this article, we'll delve into a fascinating mathematical problem involving Roni's coin collection. This problem combines elements of algebra and ratio, offering a great opportunity to sharpen our problem-solving skills. We'll systematically break down the information provided, set up equations, and ultimately determine the number of coins Roni has in each denomination. Let's embark on this numerical journey and unravel the mystery of Roni's coins!

Our main task is to find the number of coins of each denomination that Roni possesses. Roni has a total of ₹486 in coins, which are of three denominations one rupee, two rupees, and five rupees. We know the total count of all coins is 200. An important piece of information is the ratio of two-rupee coins to five-rupee coins, which is 5:2. This ratio gives us a proportional relationship between the number of these two types of coins, which will be crucial in setting up our equations. To effectively tackle this problem, we need to translate this word problem into a set of algebraic equations. We'll use variables to represent the unknown quantities (the number of coins of each denomination) and formulate equations based on the given information. This will allow us to use algebraic techniques to solve for the unknowns and arrive at our answer. So, let's gear up to translate this information into mathematical expressions and solve the coin puzzle.

The first step in solving this problem is to clearly define our variables. Let's use 'x' to represent the number of one-rupee coins, 'y' for the number of two-rupee coins, and 'z' for the number of five-rupee coins. With these variables defined, we can now translate the given information into equations. We know that the total number of coins is 200, which gives us our first equation: x + y + z = 200. This equation represents the sum of all the coins equaling the total count. Next, we know that the total value of all the coins is ₹486. We can express the value of each denomination in terms of our variables: 1x for one-rupee coins, 2y for two-rupee coins, and 5*z for five-rupee coins. Summing these up, we get our second equation: x + 2y + 5z = 486. This equation represents the total monetary value of the coins. Finally, we have the ratio of two-rupee coins to five-rupee coins, which is 5:2. This can be expressed as the equation y/z = 5/2. To simplify this, we can cross-multiply to get 2y = 5z, or rearranging, 2y - 5z = 0. This equation represents the proportional relationship between the two-rupee and five-rupee coins. Now we have a system of three equations with three variables, which we can solve to find the values of x, y, and z.

With our equations set up, the next challenge is to solve this system and find the values of x, y, and z. We have three equations: 1) x + y + z = 200, 2) x + 2y + 5z = 486, and 3) 2y - 5z = 0. There are several methods to solve such systems, including substitution, elimination, and matrix methods. For this problem, let's use the method of elimination, which is particularly efficient when dealing with linear equations. We can start by subtracting equation (1) from equation (2). This will eliminate x, giving us a new equation in terms of y and z. Subtracting (1) from (2), we get (x + 2y + 5z) - (x + y + z) = 486 - 200, which simplifies to y + 4z = 286. Now we have two equations involving y and z: y + 4z = 286 and 2y - 5z = 0. We can multiply the first of these equations by 2 to align the coefficients of y, giving us 2y + 8z = 572. Now we can subtract the second equation (2y - 5z = 0) from this new equation. This gives us (2y + 8z) - (2y - 5z) = 572 - 0, which simplifies to 13z = 572. Solving for z, we divide both sides by 13 to get z = 44. Now that we have the value of z, we can substitute it back into one of the equations involving y and z to find y. Using 2y - 5z = 0, we get 2y - 5(44) = 0, which simplifies to 2y = 220. Dividing by 2, we find y = 110. Finally, with the values of y and z, we can substitute them back into equation (1) to find x. Using x + y + z = 200, we get x + 110 + 44 = 200, which simplifies to x + 154 = 200. Subtracting 154 from both sides, we get x = 46. Therefore, we have found the values of x, y, and z, representing the number of one-rupee, two-rupee, and five-rupee coins, respectively.

After methodically setting up and solving our system of equations, we've arrived at the solution to our coin puzzle. Our calculations have revealed the following: Roni has 46 one-rupee coins, 110 two-rupee coins, and 44 five-rupee coins. To ensure the accuracy of our solution, it's always a good practice to verify our results against the original problem statement. Let's check if these values satisfy all the given conditions. First, the total number of coins should be 200. Adding the number of each type of coin, we have 46 + 110 + 44 = 200, which confirms the total count. Next, the total value of the coins should be ₹486. Calculating the value, we have (46 * 1) + (110 * 2) + (44 * 5) = 46 + 220 + 220 = 486, which matches the given total value. Finally, the ratio of two-rupee coins to five-rupee coins should be 5:2. The ratio is 110/44, which simplifies to 5/2, as required. Since our solution satisfies all the given conditions, we can confidently conclude that Roni has 46 one-rupee coins, 110 two-rupee coins, and 44 five-rupee coins. This completes our exploration of Roni's coin collection problem. Through this exercise, we've applied our algebraic skills and problem-solving techniques to unravel a real-world scenario. Such mathematical puzzles not only enhance our understanding of algebraic concepts but also sharpen our logical reasoning and analytical abilities.

In conclusion, this coin problem has been a wonderful journey through the realms of algebra and problem-solving. We successfully navigated the challenge by translating the word problem into a system of algebraic equations and methodically solving for the unknowns. The solution revealed the composition of Roni's coin collection, providing us with a tangible answer to our initial question. This exercise underscores the power of mathematics in deciphering real-world scenarios. By applying algebraic principles, we were able to dissect the problem, establish relationships between the variables, and arrive at a precise solution. The ability to translate real-world situations into mathematical models is a fundamental skill that extends far beyond the classroom. It is a tool that empowers us to analyze, interpret, and make informed decisions in various aspects of life. Problems like these enhance our analytical thinking and logical reasoning skills, which are invaluable in both academic and professional pursuits. The satisfaction of successfully solving a complex problem like this also reinforces the importance of perseverance and systematic thinking. Each step, from setting up the equations to solving them, required careful attention and a methodical approach. This highlights the significance of breaking down complex problems into smaller, manageable parts, a strategy that can be applied to numerous challenges we encounter. We've not only found the solution to this specific problem but also honed our problem-solving skills, which are essential for lifelong learning and success. This coin puzzle serves as a testament to the beauty and applicability of mathematics in our daily lives. It encourages us to approach challenges with a structured mindset and to appreciate the power of mathematical tools in unraveling the complexities of the world around us. So, let us continue to embrace the world of numbers and unlock the countless possibilities that mathematics offers.