The Product Of Two Numbers Is 3312. If The Multiplicand Is Increased By 18, The Product Increases By 1296. Find The Sum Of The Factors.

In the realm of mathematics, the interplay between numbers often presents intriguing puzzles. One such puzzle involves the product of two numbers, their relationship, and how changes to one number affect the final outcome. This article delves into a specific problem: the product of two numbers is 3312, and when the multiplicand is increased by 18 units, the product increases by 1296 units. Our mission is to find the sum of the factors in this multiplication. This seemingly simple problem unlocks a fascinating journey into the core principles of arithmetic, offering a unique opportunity to explore the properties of multiplication and the elegance of mathematical problem-solving. Understanding these fundamental concepts not only helps in solving this particular problem but also lays a strong foundation for tackling more complex mathematical challenges in the future. The ability to decipher such numerical relationships is crucial in various fields, from engineering and finance to computer science and data analysis. This exploration underscores the importance of mathematical literacy in today's world and highlights the power of logical reasoning in uncovering hidden patterns within seemingly ordinary numbers. Solving this requires a strategic approach, combining algebraic representation with careful deduction. We will explore how setting up equations based on the given information can help us unravel the values of the unknown factors. Furthermore, we will emphasize the importance of verifying our solution to ensure accuracy and a thorough understanding of the problem's conditions. Let's embark on this mathematical quest, where the thrill of discovery awaits us at every step.

Decoding the Problem Setting Up the Equations

To decode this mathematical problem, the first crucial step is to translate the word problem into a set of algebraic equations. This allows us to represent the unknown quantities with variables and express the given relationships in a concise and mathematically tractable form. Let's denote the two numbers as 'x' (the multiplicand) and 'y' (the multiplier). According to the problem statement, the product of these two numbers is 3312. This can be expressed as the following equation:

• Equation 1: x * y = 3312

Now, the problem further states that if we increase the multiplicand (x) by 18 units, the product increases by 1296 units. This provides us with another crucial piece of information that we can translate into an equation. The new multiplicand is (x + 18), and the new product is 3312 + 1296 = 4608. Therefore, we can write the second equation as:

• Equation 2: (x + 18) * y = 4608

With these two equations in hand, we now have a system of equations that we can solve to find the values of x and y. This process of converting word problems into algebraic equations is a fundamental skill in mathematics. It allows us to move from a descriptive problem to a symbolic representation that can be manipulated using the tools of algebra. The ability to accurately set up equations is often the most challenging part of solving word problems, as it requires a clear understanding of the relationships between the given quantities. By mastering this skill, we can unlock the power of algebra to solve a wide range of problems in various fields. The next step in our journey is to strategically solve these equations. There are several methods we can employ, such as substitution or elimination. The choice of method often depends on the specific form of the equations and the ease with which one variable can be expressed in terms of the other. In this case, we'll explore how to use these equations to find the individual values of 'x' and 'y'.

Solving the System Unveiling the Factors

With our two equations neatly established, we now embark on the exciting phase of solving the system and unveiling the values of the unknown factors, 'x' and 'y'. Our equations are:

• Equation 1: x * y = 3312
• Equation 2: (x + 18) * y = 4608

One effective approach to solving this system is to use the method of substitution. We can begin by expanding Equation 2:

• Equation 2 Expanded: x * y + 18y = 4608

Now, we notice that the term 'x * y' appears in both Equation 1 and the expanded form of Equation 2. We know from Equation 1 that x * y = 3312. Therefore, we can substitute this value into the expanded Equation 2:

• Substitution: 3312 + 18y = 4608

This substitution simplifies our equation, leaving us with a single equation in one variable, 'y'. Now, we can easily solve for 'y':

• Solving for y: 18y = 4608 - 3312
• 18y = 1296
• y = 1296 / 18
• y = 72

We have successfully found the value of 'y', which is 72. Now that we know 'y', we can substitute this value back into either Equation 1 or Equation 2 to solve for 'x'. Let's use Equation 1:

• Substituting y in Equation 1: x * 72 = 3312
• Solving for x: x = 3312 / 72
• x = 46

Thus, we have determined that 'x' is 46. Therefore, the two numbers are 46 and 72. This process highlights the power of algebraic manipulation in solving systems of equations. By strategically substituting and simplifying, we were able to isolate the variables and determine their values. The solution now brings us closer to our final objective: finding the sum of the factors.

Finding the Sum The Final Calculation

Having diligently deciphered the values of our two numbers, x = 46 and y = 72, we now arrive at the final stage of our mathematical journey: finding the sum of these factors. This step is straightforward, requiring a simple addition of the two numbers.

• Sum of Factors: x + y = 46 + 72
• x + y = 118

Therefore, the sum of the two factors is 118. This concludes our quest to solve the problem. We started with a word problem, translated it into algebraic equations, solved the system of equations to find the individual factors, and finally calculated their sum. This process underscores the power of a systematic approach to problem-solving in mathematics. Each step builds upon the previous one, leading us to the final solution. It's crucial to remember that mathematics is not just about arriving at the correct answer; it's also about understanding the process and the underlying principles. In this case, we've seen how multiplication, algebraic manipulation, and the solving of equations come together to solve a seemingly complex problem. The ability to break down a problem into smaller, manageable steps is a valuable skill that extends beyond mathematics. It's a fundamental approach to problem-solving in various aspects of life. Furthermore, this exercise highlights the importance of verifying our solution. While we have arrived at an answer, it's always prudent to check if our solution satisfies the original conditions of the problem. This ensures that we haven't made any errors along the way and that our answer is indeed correct. In the next section, we will verify our solution to ensure its accuracy and solidify our understanding of the problem.

Verifying the Solution Ensuring Accuracy

Before we confidently declare our solution, it's imperative to verify the solution by ensuring it aligns perfectly with the original conditions stated in the problem. This crucial step acts as a safeguard against potential errors and reinforces our comprehension of the problem's intricacies. Our solution posits that the two numbers are 46 and 72. Let's revisit the problem statement and check if these values satisfy the given conditions.

• Condition 1: The product of the two numbers is 3312.

  Let's multiply our values: 46 * 72 = 3312. This condition is indeed satisfied.

• Condition 2: If the multiplicand is increased by 18 units, the product increases by 1296 units.

  Increasing the multiplicand (46) by 18 gives us 46 + 18 = 64. The new product would be 64 * 72 = 4608. Now, let's check if this new product is 1296 units greater than the original product:

  4608 - 3312 = 1296. This condition is also satisfied.

Since our values satisfy both conditions, we can confidently affirm that our solution is accurate. This process of verification is a cornerstone of mathematical rigor. It not only confirms the correctness of our answer but also deepens our understanding of the problem's underlying structure. By meticulously checking our solution against the original conditions, we ensure that our logical reasoning and calculations are sound. This practice is invaluable in all areas of problem-solving, whether in mathematics, science, or everyday life. It instills a habit of critical thinking and attention to detail, which are essential for success in any endeavor. In conclusion, we have successfully unraveled the puzzle of the two numbers. We transformed a word problem into a system of equations, solved for the unknown factors, calculated their sum, and rigorously verified our solution. This journey exemplifies the power of mathematics in solving real-world problems and highlights the importance of a systematic and analytical approach.

Conclusion The Power of Mathematical Problem-Solving

In conclusion, the journey we undertook to solve this problem – the product of two numbers is 3312, and the effect of increasing the multiplicand – beautifully illustrates the power of mathematical problem-solving. We began with a seemingly complex word problem, carefully deconstructed it into its constituent parts, and then employed the tools of algebra to reveal the hidden solution. This process underscores the importance of translating real-world scenarios into mathematical models, a skill that is invaluable in various fields, from science and engineering to finance and economics. Our step-by-step approach, from setting up equations to solving them and finally verifying the solution, exemplifies the systematic thinking that is at the heart of mathematics. Each step was crucial, and the ability to break down a problem into smaller, manageable parts is a skill that transcends mathematics and applies to problem-solving in all aspects of life. Furthermore, this exercise highlighted the interconnectedness of different mathematical concepts. We utilized multiplication, algebraic manipulation, the solving of systems of equations, and the crucial step of verification. This holistic approach to mathematics is essential for developing a deep understanding of the subject and its applications. The ability to see the connections between different areas of mathematics allows us to tackle more complex problems with confidence and creativity. The final step of verifying our solution is particularly noteworthy. It emphasizes the importance of accuracy and attention to detail in mathematics. By checking our answer against the original conditions, we ensured that our solution was not only correct but also logically sound. This practice instills a sense of rigor and critical thinking that is essential for success in any field. In essence, this problem-solving exercise was not just about finding the sum of the two factors; it was about developing a mindset of logical reasoning, systematic thinking, and a commitment to accuracy. These are the qualities that empower us to tackle challenges, both mathematical and real-world, with confidence and competence. The journey has not only provided us with an answer but also with a deeper appreciation for the elegance and power of mathematics.