Solving A Weighted Balls Puzzle Determining The Number Of 120g And 90g Balls

This intriguing problem involves a box filled with balls of two different weights, 120 grams and 90 grams. The total weight of all the balls combined is 2160 grams. We also know that the number of 120-gram balls exceeds the number of 90-gram balls by 4. The challenge is to determine the exact number of balls of each weight. This is a classic example of a problem that can be solved using algebraic equations, allowing us to represent the unknowns and relationships with mathematical symbols. This mathematical puzzle is not just an exercise in algebra, it's a glimpse into how we can use equations to model real-world scenarios and solve practical problems. Understanding these concepts is crucial for developing critical thinking and problem-solving skills applicable in various fields, from science and engineering to finance and everyday life. We can approach this problem by first defining variables to represent the unknown quantities: the number of 120-gram balls and the number of 90-gram balls. Then, we can translate the given information into equations. For instance, the total weight can be expressed as an equation involving the number of balls of each weight and their individual weights. Similarly, the relationship between the number of balls of each weight can be expressed as another equation. Once we have these equations, we can use various algebraic techniques, such as substitution or elimination, to solve for the unknowns. The solution will reveal the specific number of 120-gram balls and 90-gram balls in the box. This process not only provides the answer to the problem but also demonstrates the power of mathematical modeling in deciphering complex situations. This intricate problem requires a methodical approach, blending algebraic manipulation with logical reasoning to arrive at the solution. Solving this problem requires more than just mathematical skills; it necessitates a blend of analytical thinking, logical deduction, and attention to detail. By carefully translating the word problem into mathematical equations, we can create a framework for finding the unknowns. Each equation represents a piece of the puzzle, and by combining them, we can unravel the mystery of the weighted balls. This process highlights the interconnectedness of mathematics and real-world scenarios, showing how abstract concepts can be applied to solve concrete problems.

Setting Up the Equations

To effectively solve this problem, the initial step involves translating the given information into a set of mathematical equations. This process, known as mathematical modeling, is fundamental in applying mathematics to real-world situations. Let's define our variables clearly. Let 'x' represent the number of 120-gram balls and 'y' represent the number of 90-gram balls. These variables will serve as placeholders for the unknown quantities we aim to determine. Now, we can express the information provided in the problem as equations. The first piece of information is the total weight of all the balls, which is 2160 grams. We can express this as an equation by multiplying the number of balls of each weight by their respective weights and summing the results. This gives us the equation: 120x + 90y = 2160. This equation represents the total weight constraint of the problem. The left-hand side represents the combined weight of all the balls, while the right-hand side represents the total weight. The second piece of information relates the number of 120-gram balls to the number of 90-gram balls. We are told that the number of 120-gram balls is 4 greater than the number of 90-gram balls. This can be expressed as the equation: x = y + 4. This equation represents the relationship between the number of balls of each weight. It states that the number of 120-gram balls (x) is equal to the number of 90-gram balls (y) plus 4. With these two equations, we have a system of linear equations. A system of linear equations is a set of two or more linear equations containing the same variables. The solution to a system of linear equations is the set of values for the variables that satisfy all the equations simultaneously. In this case, the solution will give us the values of x and y, which represent the number of 120-gram balls and 90-gram balls, respectively. Solving this system will provide the answer to the problem. This meticulous approach to setting up equations is a cornerstone of problem-solving in mathematics and beyond. By translating word problems into mathematical expressions, we gain a powerful tool for analyzing and solving complex situations. The clarity and precision of these equations lay the foundation for the subsequent steps in finding the solution. The process of translating word problems into mathematical equations is not merely a mechanical exercise; it is a crucial step in problem-solving that requires careful analysis and understanding of the underlying relationships. By meticulously defining variables and expressing the given information as equations, we create a framework for solving the problem using algebraic techniques. The two equations we have established, 120x + 90y = 2160 and x = y + 4, encapsulate the essence of the problem in a concise and mathematical form.

Solving the System of Equations

Now that we have successfully set up our system of equations, the next crucial step is to solve for the unknowns, 'x' and 'y'. This can be achieved using various algebraic techniques, but one common and efficient method is the substitution method. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, leaving us with a single equation in one variable, which can be easily solved. In our case, we already have the second equation, x = y + 4, solved for x. This makes the substitution method particularly convenient. We can substitute the expression 'y + 4' for 'x' in the first equation, 120x + 90y = 2160. This gives us: 120(y + 4) + 90y = 2160. Now we have a single equation in terms of 'y'. The next step is to simplify this equation by distributing the 120 and combining like terms: 120y + 480 + 90y = 2160. Combining the 'y' terms, we get: 210y + 480 = 2160. Now, we can isolate the 'y' term by subtracting 480 from both sides of the equation: 210y = 1680. Finally, we can solve for 'y' by dividing both sides of the equation by 210: y = 1680 / 210. This gives us: y = 8. So, we have determined that the number of 90-gram balls (y) is 8. Now that we have the value of 'y', we can easily find the value of 'x' by substituting it back into the equation x = y + 4: x = 8 + 4. This gives us: x = 12. Therefore, the number of 120-gram balls (x) is 12. We have successfully solved the system of equations using the substitution method. This process demonstrates the power of algebraic techniques in unraveling complex problems. By systematically manipulating equations, we can isolate the unknowns and determine their values. The solution to this system provides the answer to the original problem: there are 12 balls weighing 120 grams each and 8 balls weighing 90 grams each. This method highlights the elegance and efficiency of algebraic solutions, where a series of logical steps leads to a definitive answer. The process of solving a system of equations is not merely about finding the numerical values of the variables; it is a demonstration of the power of algebraic manipulation and logical reasoning. By carefully applying the substitution method, we have successfully isolated the unknowns and determined their values. This systematic approach to problem-solving is applicable in a wide range of contexts, from scientific research to engineering design. The solution we have obtained, x = 12 and y = 8, provides a clear and concise answer to the original question.

Verifying the Solution

After solving a system of equations, it is always prudent to verify the solution to ensure its accuracy. This step is crucial in problem-solving as it helps to identify any potential errors in the calculations or the setup of the equations. Verification involves substituting the obtained values of the variables back into the original equations and checking if they hold true. In our case, we found that x = 12 (the number of 120-gram balls) and y = 8 (the number of 90-gram balls). To verify our solution, we will substitute these values into the two original equations we established: 120x + 90y = 2160 and x = y + 4. First, let's substitute the values into the equation 120x + 90y = 2160: 120(12) + 90(8) = 1440 + 720 = 2160. The left-hand side of the equation equals the right-hand side, which confirms that the first equation is satisfied. Next, let's substitute the values into the equation x = y + 4: 12 = 8 + 4. This equation also holds true, as 12 is indeed equal to 8 + 4. Since both equations are satisfied by our solution, we can confidently conclude that our solution is correct. This verification process not only confirms the accuracy of the solution but also reinforces our understanding of the problem and the relationships between the variables. By systematically checking our work, we increase the reliability of our results and develop a deeper appreciation for the mathematical principles involved. The act of verification is a cornerstone of sound problem-solving practice. It is not merely a formality but an integral step in ensuring the accuracy and reliability of the solution. By substituting the obtained values back into the original equations, we subject our solution to a rigorous test. This process can reveal errors that might have gone unnoticed during the solving phase. The successful verification of our solution instills confidence in the results and demonstrates the power of mathematical rigor. In this particular problem, the verification process has confirmed that there are indeed 12 balls weighing 120 grams each and 8 balls weighing 90 grams each, which aligns perfectly with the conditions outlined in the problem statement.

Conclusion: The Weighty Revelation

In conclusion, by employing algebraic techniques, we have successfully unraveled the mystery of the weighted balls. We determined that there are 12 balls weighing 120 grams each and 8 balls weighing 90 grams each. This solution not only satisfies the total weight constraint of 2160 grams but also adheres to the condition that the number of 120-gram balls is 4 greater than the number of 90-gram balls. The process of solving this problem involved several key steps, each contributing to the final answer. First, we carefully defined variables to represent the unknown quantities: the number of 120-gram balls and the number of 90-gram balls. This step laid the foundation for translating the word problem into mathematical language. Next, we translated the given information into a system of two linear equations. The first equation represented the total weight of the balls, while the second equation represented the relationship between the number of balls of each weight. This step transformed the problem from a descriptive narrative into a precise mathematical formulation. Then, we employed the substitution method to solve the system of equations. This involved solving one equation for one variable and substituting that expression into the other equation. This eliminated one variable, allowing us to solve for the remaining variable. Once we found the value of one variable, we could easily find the value of the other variable. Finally, we verified our solution by substituting the obtained values back into the original equations. This step ensured the accuracy of our solution and reinforced our understanding of the problem. This problem serves as a compelling illustration of the power of algebra in solving real-world problems. By using mathematical modeling, we can represent complex situations with equations and then use algebraic techniques to find solutions. The skills developed in solving this problem, such as defining variables, setting up equations, and using algebraic methods, are transferable to a wide range of other problems in mathematics, science, and engineering. The successful resolution of this puzzle underscores the beauty and utility of mathematical thinking. The problem of the weighted balls, though seemingly simple on the surface, provides a rich context for exploring fundamental mathematical concepts. By systematically applying algebraic techniques, we have not only found the solution but also gained a deeper appreciation for the power and elegance of mathematics. The process of translating a word problem into a mathematical framework, solving the equations, and verifying the solution is a valuable exercise in logical reasoning and critical thinking. The final answer, revealing the exact number of balls of each weight, brings a sense of satisfaction and reinforces the importance of a methodical approach to problem-solving.