Finding The Two-Digit Number A Mathematical Puzzle

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Introduction

In the realm of mathematics, we often encounter intriguing puzzles that challenge our problem-solving skills. One such puzzle involves finding a two-digit number that satisfies specific conditions related to the sum and product of its digits. This article delves into the process of solving this mathematical conundrum, providing a step-by-step approach to unravel the mystery behind the elusive two-digit number.

This article will guide you through a mathematical exploration to discover this number. We will begin by defining the problem clearly, then move on to establishing the equations that represent the given conditions. These equations will then be manipulated and solved using algebraic techniques. Throughout this process, we'll highlight the key concepts and strategies involved in solving such problems. Understanding how to tackle this particular problem not only provides a solution but also enhances our general mathematical reasoning abilities. So, let's embark on this intellectual journey and uncover the hidden number using the power of mathematics. Our primary focus will be on translating the word problem into mathematical equations, which is a crucial step in solving many mathematical problems. We will also explore different algebraic manipulations and substitution methods to find the solution. This will not only help in solving this specific problem but will also enhance your problem-solving skills in general. Remember, mathematics is not just about finding the answer; it's about the process of reasoning and applying the right techniques to reach the solution. So, let's dive in and discover the beauty of mathematical problem-solving.

Problem Statement

Let's clearly state the problem we aim to solve: We are searching for a two-digit number that adheres to two specific criteria. First, the number itself must be equal to four times the sum of its individual digits. Second, the same number should also be equal to three times the product of its digits. These two conditions form the core of our puzzle, and finding a number that satisfies both is our primary objective. To clarify further, consider a two-digit number, say 42. The sum of its digits is 4 + 2 = 6, and the product is 4 * 2 = 8. We need to find a number where the number itself is four times the sum of its digits and three times the product of its digits. This means we need to find a number, let's call it 'N', such that N = 4 * (sum of digits) and N = 3 * (product of digits). Both these conditions must be true for the number we are looking for. The complexity of the problem arises from the fact that we need to satisfy two distinct conditions simultaneously. This requires us to carefully set up equations and solve them in a way that both conditions are met. It's a bit like finding a key that fits two different locks. We need to use our mathematical tools and reasoning to find this unique number. This problem is a great example of how mathematics can be used to solve puzzles and challenges in a systematic way. By breaking down the problem into smaller parts, setting up equations, and using algebraic techniques, we can find the solution. So, let's proceed with our journey to find this elusive two-digit number.

Setting up the Equations

To approach this problem systematically, we need to translate the given conditions into mathematical equations. This involves representing the unknown two-digit number and its digits using variables. Let's denote the tens digit as 'x' and the units digit as 'y'. Therefore, the two-digit number can be represented as 10x + y. This is because the tens digit contributes 10 times its value to the number, while the units digit contributes its face value. For instance, if x = 4 and y = 2, the number would be 10*4 + 2 = 42.

Now, let's translate the given conditions into equations. The first condition states that the two-digit number is equal to four times the sum of its digits. This can be written as: 10x + y = 4(x + y). This equation represents the relationship between the number and the sum of its digits. The second condition states that the two-digit number is equal to three times the product of its digits. This can be written as: 10x + y = 3xy. This equation represents the relationship between the number and the product of its digits. We now have a system of two equations with two variables: 10x + y = 4(x + y) and 10x + y = 3xy. Solving this system of equations will give us the values of x and y, which will then allow us to find the two-digit number. The process of setting up equations is a crucial step in solving mathematical problems. It allows us to represent the problem in a symbolic form, which can then be manipulated using algebraic techniques. In this case, by translating the word problem into two equations, we have created a pathway to find the solution. The next step is to solve these equations, which we will discuss in the following sections.

Solving the Equations

Now that we have our equations, the next step is to solve them. We have two equations:

  1. 10x + y = 4(x + y)
  2. 10x + y = 3xy

Let's simplify the first equation: 10x + y = 4x + 4y. Subtracting 4x and y from both sides gives us: 6x = 3y. Dividing both sides by 3, we get: 2x = y. This simplified equation tells us that the units digit (y) is twice the tens digit (x). This is a crucial piece of information that will help us find the solution.

Now, let's substitute y = 2x into the second equation: 10x + y = 3xy. Replacing y with 2x, we get: 10x + 2x = 3x(2x). This simplifies to: 12x = 6x^2. To solve for x, we can rearrange the equation: 6x^2 - 12x = 0. We can factor out 6x from the equation: 6x(x - 2) = 0. This gives us two possible solutions for x: 6x = 0 or x - 2 = 0. If 6x = 0, then x = 0. However, since we are looking for a two-digit number, the tens digit cannot be 0. So, we discard this solution. If x - 2 = 0, then x = 2. This is a valid solution for the tens digit. Now that we have x = 2, we can find y using the equation y = 2x. Substituting x = 2, we get: y = 2 * 2 = 4. So, the units digit is 4. Therefore, the two-digit number is 10x + y = 10 * 2 + 4 = 24. We have found a potential solution, but we need to verify if it satisfies both original conditions. Let's check:

  • Is 24 equal to four times the sum of its digits? The sum of the digits is 2 + 4 = 6, and 4 * 6 = 24. Yes, it satisfies the first condition.
  • Is 24 equal to three times the product of its digits? The product of the digits is 2 * 4 = 8, and 3 * 8 = 24. Yes, it satisfies the second condition.

Since 24 satisfies both conditions, we have found the solution.

The Solution

After carefully setting up the equations and employing algebraic techniques to solve them, we have arrived at the solution. The two-digit number that satisfies the given conditions is 24. This number is indeed four times the sum of its digits (2 + 4 = 6, and 4 * 6 = 24) and three times the product of its digits (2 * 4 = 8, and 3 * 8 = 24). This problem showcases the power of mathematics in solving puzzles and challenges. By breaking down the problem into smaller parts, setting up equations, and using algebraic manipulations, we were able to find the solution. The process of solving this problem not only provides the answer but also enhances our problem-solving skills and mathematical reasoning abilities. It's a testament to the beauty and elegance of mathematics, where logical steps and systematic approaches can lead to the discovery of hidden truths. The solution, 24, is a unique number that embodies the given conditions, making it a satisfying conclusion to our mathematical journey. This exploration also highlights the importance of verifying the solution against the original conditions. It's a crucial step in problem-solving to ensure that the answer we have found is indeed correct and satisfies all the requirements of the problem. In this case, we verified that 24 satisfies both conditions, thus confirming that it is the correct solution. The journey to find this number has been a valuable exercise in mathematical problem-solving, and the solution, 24, stands as a testament to our efforts.

Conclusion

In conclusion, we have successfully found the two-digit number that is equal to four times the sum of its digits and three times the product of its digits. The number is 24. This problem demonstrated the importance of translating word problems into mathematical equations, simplifying those equations, and using algebraic techniques to find the solution. The process involved representing the unknown number and its digits with variables, setting up a system of equations based on the given conditions, and solving those equations using substitution and factoring methods. This exercise highlights the power of mathematical reasoning and problem-solving skills. It also underscores the importance of verifying the solution to ensure its accuracy. The satisfaction of finding the solution, 24, comes from the journey of logical steps and mathematical manipulations that led us to the answer. This type of problem not only enhances our mathematical abilities but also develops our critical thinking and analytical skills. As we conclude this exploration, it's important to remember that mathematics is not just about finding answers; it's about the process of thinking, reasoning, and applying the right techniques to reach the solution. The problem we solved today is just one example of how mathematics can be used to solve puzzles and challenges in a systematic and logical way. By continuing to practice and explore mathematical problems, we can further develop our skills and appreciate the beauty and elegance of this powerful subject. The number 24 will now hold a special place in our understanding of mathematical problem-solving, as it represents the culmination of our efforts and the successful application of mathematical principles. The journey to find this number has been a rewarding experience, and the lessons learned will undoubtedly be valuable in future mathematical endeavors.