Tropical Fish Puzzle: Solving For Angelfish & Parrotfish!

Let's dive into a mathematical problem involving tropical fish! Carlos, a dedicated curator, has acquired a collection of 405 tropical fish for a museum exhibit. Among these, there are parrotfish and angelfish, with the number of parrotfish being eight times that of angelfish. Our mission is to determine exactly how many of each type of fish Carlos purchased and identify the correct system of equations that models this scenario. It's a classic algebra problem with a splash of marine life!

Setting Up the Equations

To solve this problem, we need to translate the given information into mathematical equations. Let's define our variables:

• Let x represent the number of angelfish.
• Let y represent the number of parrotfish.

From the problem statement, we can derive two crucial equations:

1. Total Number of Fish: The total number of angelfish and parrotfish is 405. This can be expressed as:

   x + y = 405

2. Relationship Between Parrotfish and Angelfish: There are eight times as many parrotfish as angelfish. This can be expressed as:

   y = 8x

Thus, the system of equations that models this problem is:

{x + y = 405
y = 8x
}

This system of equations allows us to solve for the number of angelfish (x) and parrotfish (y) that Carlos bought. Understanding how to set up these equations is the first critical step in solving the problem. It transforms the word problem into a format we can manipulate mathematically to find our answers. The first equation, x + y = 405, represents the total quantity, providing a foundational constraint. The second equation, y = 8x, introduces the proportional relationship between the two types of fish, adding another layer of information that is essential for finding unique solutions. Together, these equations capture the essence of the problem and set the stage for algebraic manipulation. Moreover, recognizing these relationships and accurately translating them into mathematical notation is a fundamental skill in problem-solving, applicable across various domains beyond just mathematics. It’s about understanding the underlying structure of the problem and expressing it in a concise and actionable form. This process not only aids in solving the specific problem at hand but also enhances analytical thinking and the ability to approach complex scenarios with a structured methodology. Remember, the key is to identify the core relationships and express them in a way that facilitates further analysis and solution.

Solving the System of Equations

Now that we have our system of equations, we can solve for x and y. We can use the substitution method since we already have y expressed in terms of x.

1. Substitute y = 8x into the first equation:

   x + 8x = 405

2. Combine like terms:

   9x = 405

3. Divide both sides by 9 to solve for x:

   x = 405 / 9

   x = 45

4. Now that we have the value of x, we can substitute it back into the equation y = 8x to find y:

   y = 8 * 45

   y = 360

Therefore, Carlos bought 45 angelfish and 360 parrotfish. The substitution method is a powerful tool in solving systems of equations, particularly when one equation readily expresses one variable in terms of another. By substituting y = 8x into the equation x + y = 405, we effectively reduced the problem to a single equation with one unknown, making it much easier to solve. The process of combining like terms and isolating the variable is a fundamental algebraic technique that allows us to systematically unravel the relationships defined by the equations. Furthermore, the importance of accurately performing each step in the calculation cannot be overstated; a small error in arithmetic can lead to a completely incorrect answer. After finding the value of x, we then used it to find the value of y, completing the solution. This methodical approach not only helps in finding the correct answers but also enhances our understanding of the underlying mathematical principles. In practice, this skill is invaluable in various fields, including engineering, economics, and computer science, where complex problems often need to be broken down into manageable equations and solved systematically.

Verification

To ensure our solution is correct, let's verify it:

• Total Fish: 45 angelfish + 360 parrotfish = 405 fish (Correct)
• Relationship: 360 parrotfish / 45 angelfish = 8 (Correct)

Our solution satisfies both conditions, confirming that Carlos bought 45 angelfish and 360 parrotfish. Verification is a critical step in any problem-solving process, as it provides a means to confirm the accuracy of the solution and catch any potential errors. In this case, we verified that the sum of the angelfish and parrotfish equals the total number of fish, and that the ratio of parrotfish to angelfish is indeed 8 to 1, as specified in the original problem. This process not only validates the solution but also reinforces our understanding of the relationships between the variables. Moreover, the practice of verification fosters a sense of confidence in our work and promotes a more meticulous approach to problem-solving. It’s a habit that can prevent costly mistakes and ensure that decisions are based on accurate information. In real-world scenarios, where the consequences of errors can be significant, verification becomes even more essential. Whether it's double-checking calculations in financial analysis or validating the results of a scientific experiment, the principle remains the same: always take the time to verify your work to ensure accuracy and reliability.

Choosing the Correct System of Equations

The correct system of equations that models this problem is:

{x + y = 405
y = 8x
}

Let's examine why the other option provided is incorrect:

• A. $\left\{\begin{array}{l}x+4 y=8 \\ y=405\end{array}\right.$ This system is incorrect because the first equation, x + 4y = 8, does not accurately represent the relationship between the number of angelfish and parrotfish, nor does it account for the total number of fish. The second equation, y = 405, incorrectly states that there are 405 parrotfish, regardless of the number of angelfish. Selecting the correct system of equations is paramount to accurately modeling and solving a problem. The system x + y = 405 and y = 8x correctly captures the two key pieces of information provided in the problem: the total number of fish and the proportional relationship between the two types of fish. In contrast, the incorrect option, x + 4y = 8 and y = 405, fails to represent these relationships accurately. The first equation in the incorrect system does not logically connect the number of angelfish and parrotfish, and the second equation makes an unsupported assertion about the number of parrotfish. Understanding why a particular system of equations is correct or incorrect requires a careful examination of how each equation relates to the problem's conditions. This involves not only recognizing the mathematical symbols but also interpreting their meaning in the context of the problem. The ability to critically evaluate and select the appropriate mathematical model is a crucial skill in problem-solving, enabling us to translate real-world scenarios into solvable mathematical expressions. This skill is essential not only in mathematics but also in various fields such as engineering, economics, and computer science, where accurate modeling is fundamental to effective problem-solving.

Conclusion

In summary, Carlos bought 45 angelfish and 360 parrotfish for his museum display. The system of equations that models this problem is:

{x + y = 405
y = 8x
}

This problem highlights the importance of translating word problems into mathematical equations and using algebraic techniques to solve them. By carefully defining variables, setting up the correct equations, and verifying the solution, we can successfully solve complex problems. For further exploration of systems of equations and problem-solving strategies, consider visiting Khan Academy's section on algebra.