Solving Tourist Visit Puzzle How Many Went To Aquarium And Zoo

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Embark on a mathematical journey to solve a captivating tourist puzzle! A grand total of 1800 tourists explored either the mesmerizing depths of an aquarium or the captivating wildlife at a zoo. The intrigue deepens as we learn that the aquarium saw 400 fewer visitors than the zoo. Our mission is to unravel this numerical enigma and determine the individual number of tourists who graced each attraction – the aquarium and the zoo. This problem is a classic example of a system of equations, where we have two unknowns (number of aquarium visitors and number of zoo visitors) and two pieces of information (total visitors and the difference in visitors). Let's delve into the world of algebraic problem-solving and unveil the solution to this intriguing puzzle.

Setting Up the Equations: A Step-by-Step Guide

To effectively tackle this problem, we need to translate the given information into mathematical equations. This will allow us to use the power of algebra to solve for the unknown quantities. Let's break down the process:

  1. Define the Variables: The first crucial step is to assign variables to represent the unknowns. Let's use:

    • A = the number of tourists who visited the aquarium
    • Z = the number of tourists who visited the zoo

    By assigning these variables, we create a symbolic representation of the quantities we are trying to find. This is a cornerstone of algebraic problem-solving, allowing us to manipulate these unknowns within equations.

  2. Formulate the First Equation: The problem states that a total of 1800 tourists visited either the aquarium or the zoo. This information can be directly translated into an equation:

    • A + Z = 1800

    This equation represents the sum of visitors to both attractions equaling the total number of tourists. It captures the first key piece of information provided in the problem statement.

  3. Formulate the Second Equation: We are also told that 400 fewer tourists visited the aquarium compared to the zoo. This can be expressed as:

    • A = Z - 400

    This equation signifies that the number of aquarium visitors is equal to the number of zoo visitors minus 400. It captures the second crucial piece of information, the difference in attendance between the two attractions.

With these two equations, we have a system of equations that accurately represents the problem's conditions. Now, we can employ various algebraic techniques to solve for the values of A and Z, ultimately revealing the number of tourists who visited each location.

Solving the System of Equations: Unveiling the Solution

With our equations established, we can now employ algebraic techniques to find the values of A (aquarium visitors) and Z (zoo visitors). There are a few methods we could use, but the substitution method is particularly well-suited for this problem. Here's how it works:

  1. Isolate a Variable: We already have one equation conveniently set up to isolate a variable: A = Z - 400. This equation expresses A directly in terms of Z.

  2. Substitute: The core of the substitution method lies in replacing a variable in one equation with its equivalent expression from the other equation. We'll substitute the expression Z - 400 for A in the first equation (A + Z = 1800):

    • (Z - 400) + Z = 1800

    This substitution eliminates A from the equation, leaving us with a single equation containing only the variable Z.

  3. Solve for the Remaining Variable: Now we can solve the simplified equation for Z:

    • Combine like terms: 2Z - 400 = 1800
    • Add 400 to both sides: 2Z = 2200
    • Divide both sides by 2: Z = 1100

    We've successfully determined that 1100 tourists visited the zoo!

  4. Back-Substitute: With the value of Z known, we can now find the value of A. We'll substitute Z = 1100 back into either of our original equations. The equation A = Z - 400 is the simpler choice:

    • A = 1100 - 400
    • A = 700

    Therefore, 700 tourists visited the aquarium.

By carefully applying the substitution method, we've successfully solved the system of equations and unveiled the number of visitors to each attraction. This demonstrates the power of algebra in tackling real-world problems.

Verification and Interpretation: Ensuring Accuracy and Understanding

Before we declare victory, it's crucial to verify our solution to ensure accuracy and ensure our answer makes sense in the context of the problem. This step helps prevent errors and deepens our understanding of the results.

  1. Verification: Let's check if our values for A and Z satisfy both original equations:

    • Equation 1: A + Z = 1800
      • Substitute: 700 + 1100 = 1800
      • Result: 1800 = 1800 (This equation holds true)
    • Equation 2: A = Z - 400
      • Substitute: 700 = 1100 - 400
      • Result: 700 = 700 (This equation also holds true)

    Since our values satisfy both equations, we can be confident in the correctness of our solution. This verification step is a cornerstone of mathematical problem-solving, catching potential errors and solidifying our understanding.

  2. Interpretation: Now, let's translate our mathematical results back into the context of the problem. We found that:

    • A = 700 tourists visited the aquarium.
    • Z = 1100 tourists visited the zoo.

    This means that the zoo was the more popular destination, attracting 400 more visitors than the aquarium, as stated in the problem. The aquarium had fewer visitors and we have accurately calculated these numbers based on the provided information.

By verifying our solution and interpreting the results, we've completed the problem-solving process. This ensures accuracy and allows us to fully understand the answer in the context of the original question.

Conclusion: Unveiling the Mystery of Tourist Destinations

In conclusion, by employing the power of algebraic equations, we successfully deciphered the mystery of tourist destinations. We discovered that 700 tourists explored the captivating underwater world of the aquarium, while a larger crowd of 1100 visitors marveled at the diverse wildlife in the zoo. This mathematical journey highlights the practical application of systems of equations in solving real-world problems. From setting up the equations to applying the substitution method and verifying the results, each step reinforced the importance of logical reasoning and meticulous calculations. This problem serves as a testament to the beauty and utility of mathematics in unraveling the complexities of everyday scenarios. The ability to translate word problems into mathematical expressions empowers us to analyze and solve a wide range of challenges, making mathematics an invaluable tool in our quest for knowledge and understanding.