Comic Book Collection Problem Solving A Mathematical Approach

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In the realm of mathematical puzzles, work-rate problems often present intriguing challenges. This article delves into a classic scenario involving Charlene and Gina organizing a comic book collection. The problem states that working together, they can complete the task in 18 minutes. However, Gina, when working independently, requires 15 minutes more than Charlene to finish the same collection. Our goal is to dissect this problem, understand the underlying concepts, and arrive at a solution. We will explore the fundamental principles of work-rate problems, set up equations to model the given information, and employ algebraic techniques to solve for the unknowns. This journey will not only provide a solution to the specific problem but also equip readers with the tools to tackle similar mathematical puzzles. The mathematical concepts involved, such as rates, time, and their relationships, are crucial in various real-world applications, making this exploration both academically enriching and practically relevant. Let's embark on this mathematical adventure and unravel the comic book collection conundrum, understanding each step of the process and highlighting the elegance of mathematical problem-solving. We will start by carefully defining the variables and translating the word problem into a set of equations, which is the cornerstone of solving any mathematical puzzle. The beauty of mathematics lies in its ability to transform seemingly complex scenarios into manageable equations, and this problem is a perfect example of that. So, fasten your seatbelts as we dive deep into the world of rates, time, and collaborative work, all within the context of a comic book collection waiting to be organized.

Problem Statement

Let's formally state the problem we aim to solve. Charlene and Gina, when working together, can organize Charlene's comic book collection in 18 minutes. If Gina were to work alone, she would take 15 minutes longer than Charlene would take working alone. The question is to determine the individual time each person would take to organize the collection. This type of problem falls under the category of work-rate problems, where we analyze the rates at which individuals or entities perform tasks, either independently or collaboratively. The key to solving such problems is to express the work done as a fraction of the total work, and then use the relationships between work, rate, and time to set up equations. In this particular scenario, the "work" is organizing the comic book collection, and the "rate" is the amount of the collection organized per minute. We are given information about the combined rate and the difference in individual rates, which allows us to formulate a system of equations. To effectively solve this, we need to carefully define our variables. Let's denote the time it takes Charlene to organize the collection alone as $x$ minutes. Then, the time it takes Gina to organize the collection alone will be $x + 15$ minutes, as she takes 15 minutes longer. This initial setup is crucial, as it translates the word problem into mathematical language, paving the way for algebraic manipulation and solution. The elegance of mathematics lies in its ability to represent real-world scenarios with concise and precise equations, and this problem is a testament to that. We will now proceed to express the rates of work in terms of these variables and then formulate the equations that govern their collaborative and individual efforts. This step is the bridge between the problem statement and the mathematical solution, and it requires a clear understanding of the relationships between work, rate, and time. So, let's move forward and translate this word problem into a set of solvable equations.

Defining Variables and Rates

To begin solving this problem, we need to define our variables clearly. Let's denote the time it takes Charlene to organize the comic book collection alone as $x$ minutes. Consequently, Gina would take $x + 15$ minutes to complete the same task working independently. This is because the problem states that Gina takes 15 minutes longer than Charlene. Now, let's consider the rates at which Charlene and Gina work. The rate of work is the amount of work done per unit of time. In this context, the "work" is organizing the entire comic book collection, which we can represent as 1 (or 100%). Therefore, Charlene's rate of work is $1/x$ (the fraction of the collection she organizes per minute), and Gina's rate of work is $1/(x + 15)$ (the fraction of the collection she organizes per minute). When they work together, their rates add up. The problem states that working together, they can complete the collection in 18 minutes. This means their combined rate is $1/18$ (the fraction of the collection they organize together per minute). This understanding of individual and combined rates is crucial for setting up the equation. We are essentially translating the word problem into mathematical expressions, which is the cornerstone of problem-solving in mathematics. The beauty of this approach lies in its ability to break down a complex scenario into manageable components. By defining variables and expressing rates, we have transformed the problem into a mathematical framework that we can now manipulate algebraically. The next step is to formulate the equation that represents their combined work rate and then solve for the unknown variable, $x$. This will reveal the time it takes Charlene to organize the collection alone, and subsequently, we can find the time it takes Gina. So, let's proceed to construct the equation that embodies the essence of their collaborative effort.

Setting up the Equation

Now that we have defined the variables and expressed the rates of work, we can set up the equation that represents the given scenario. When Charlene and Gina work together, their individual rates combine to achieve the overall rate of work. We know that Charlene's rate is $1/x$, Gina's rate is $1/(x + 15)$, and their combined rate is $1/18$. Therefore, we can express the relationship as follows:

1x+1x+15=118 \frac{1}{x} + \frac{1}{x + 15} = \frac{1}{18}

This equation is the heart of the problem. It encapsulates the essence of their collaborative work and provides the key to unlocking the solution. The left side of the equation represents the sum of their individual rates, while the right side represents their combined rate. This equation is a rational equation, meaning it involves fractions with variables in the denominator. To solve it, we will need to eliminate the fractions by finding a common denominator and then simplifying the equation into a more manageable form, likely a quadratic equation. The process of setting up the equation is a crucial step in problem-solving. It requires a deep understanding of the relationships between the given information and the variables we have defined. In this case, we have successfully translated the word problem into a concise mathematical statement. The elegance of this equation lies in its simplicity and its ability to capture the complexity of the scenario. It is a testament to the power of mathematics in representing real-world situations. The next step is to solve this equation for $x$, which will reveal the time it takes Charlene to organize the comic book collection alone. This will be followed by finding the time it takes Gina. So, let's move forward and embark on the algebraic journey of solving this equation.

Solving the Equation

With the equation set up, the next step is to solve it for $x$. The equation we have is:

1x+1x+15=118 \frac{1}{x} + \frac{1}{x + 15} = \frac{1}{18}

To eliminate the fractions, we find the least common denominator (LCD), which is $18x(x + 15)$. Multiplying both sides of the equation by the LCD, we get:

18(x+15)+18x=x(x+15) 18(x + 15) + 18x = x(x + 15)

Expanding the terms, we have:

18x+270+18x=x2+15x 18x + 270 + 18x = x^2 + 15x

Combining like terms, we get:

36x+270=x2+15x 36x + 270 = x^2 + 15x

Rearranging the equation to form a quadratic equation, we have:

x2−21x−270=0 x^2 - 21x - 270 = 0

Now, we need to solve this quadratic equation. We can do this by factoring, using the quadratic formula, or completing the square. In this case, factoring seems to be the most straightforward approach. We are looking for two numbers that multiply to -270 and add up to -21. These numbers are -30 and 9. Therefore, we can factor the quadratic equation as follows:

(x−30)(x+9)=0 (x - 30)(x + 9) = 0

Setting each factor equal to zero, we get:

x−30=0orx+9=0 x - 30 = 0 \quad \text{or} \quad x + 9 = 0

Solving for $x$, we have:

x=30orx=−9 x = 30 \quad \text{or} \quad x = -9

Since time cannot be negative, we discard the solution $x = -9$. Therefore, the time it takes Charlene to organize the comic book collection alone is $x = 30$ minutes. This algebraic process demonstrates the power of mathematical manipulation in solving real-world problems. We transformed a word problem into an equation, and through careful algebraic steps, we arrived at a solution. The beauty of this process lies in its precision and its ability to provide a definitive answer. Now that we have found the time it takes Charlene, we can easily find the time it takes Gina. So, let's proceed to calculate Gina's time and complete the solution to the problem.

Finding the Solution

We have determined that it takes Charlene 30 minutes to organize the comic book collection alone. Now, we need to find the time it takes Gina. Recall that Gina takes 15 minutes longer than Charlene. Therefore, the time it takes Gina is:

x+15=30+15=45 x + 15 = 30 + 15 = 45

So, Gina takes 45 minutes to organize the comic book collection alone. Now we have the individual times for both Charlene and Gina. Charlene takes 30 minutes, and Gina takes 45 minutes. It's always a good practice to check if our solution makes sense in the context of the original problem. We can verify this by calculating their combined rate and comparing it to the given combined time of 18 minutes. Charlene's rate is $1/30$, and Gina's rate is $1/45$. Their combined rate is:

130+145 \frac{1}{30} + \frac{1}{45}

To add these fractions, we need a common denominator, which is 90. So, we have:

390+290=590=118 \frac{3}{90} + \frac{2}{90} = \frac{5}{90} = \frac{1}{18}

This confirms that their combined rate is indeed $1/18$, which means they would take 18 minutes to organize the collection together, as stated in the problem. This verification step is crucial as it ensures that our solution is consistent with the given information. The problem is now completely solved. We have found the individual times for Charlene and Gina, and we have verified our solution. The journey from the initial word problem to the final answer has been a testament to the power of mathematical problem-solving. We have used concepts of rates, time, and algebraic manipulation to unravel this puzzle. The beauty of mathematics lies not just in finding the answer, but also in the process of logical deduction and problem-solving. So, let's summarize our findings and conclude this mathematical exploration.

Conclusion

In conclusion, we have successfully solved the comic book collection problem. We found that it takes Charlene 30 minutes to organize the collection alone, while it takes Gina 45 minutes. We arrived at this solution by carefully defining variables, expressing rates of work, setting up an equation, solving the equation, and verifying our answer. This problem exemplifies the power of mathematics in modeling and solving real-world scenarios. The concepts of rates, time, and work are fundamental in various fields, making the understanding of such problems not only academically valuable but also practically relevant. The process we followed, from translating the word problem into mathematical expressions to employing algebraic techniques, is a testament to the elegance and precision of mathematics. We started with a seemingly complex scenario and, through logical steps, arrived at a clear and concise solution. The ability to break down a problem into smaller, manageable parts and then solve each part systematically is a crucial skill in mathematics and in life. This problem also highlights the importance of verification. By checking our solution against the original problem statement, we ensured that our answer was not only mathematically correct but also logically consistent with the given information. The journey through this problem has been a rewarding one, demonstrating the beauty and power of mathematical problem-solving. From the initial challenge to the final solution, we have explored the intricacies of rates, time, and collaborative work. The comic book collection is now organized, and we have gained valuable insights into the world of mathematical puzzles. The skills and techniques we have employed here can be applied to a wide range of similar problems, making this exploration a valuable addition to our problem-solving toolkit. So, let's carry forward this knowledge and continue to unravel the mysteries that mathematics has to offer.