Solving Cat's Coin Puzzle How Many Quarters Does She Have

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Introduction: Unraveling Cat's Coin Mystery

In this intriguing mathematical puzzle, we delve into the world of Cat, a feline with a penchant for collecting coins. Cat possesses a total of 19 coins, a mix of quarters and dimes, which together amount to a grand total of $4. Our mission, should we choose to accept it, is to decipher the exact number of quarters Cat has in her collection. To embark on this quest, we'll employ the power of algebraic equations, carefully dissecting the given information to arrive at the solution. This puzzle not only tests our mathematical prowess but also highlights the practical application of algebra in everyday scenarios. Coin problems like these are a staple in mathematical education, fostering critical thinking and problem-solving skills. Understanding how to approach such problems can be invaluable in various real-life situations, from managing personal finances to calculating discounts and even in fields like accounting and economics. So, let's sharpen our pencils and dive into the world of quarters and dimes, ready to uncover the secrets of Cat's coin stash.

Setting Up the Equations: The Foundation for Our Solution

The key to solving this coin conundrum lies in translating the given information into a set of algebraic equations. This process allows us to represent the unknown quantities—the number of quarters and dimes—with variables, creating a mathematical framework for our solution. We are presented with two crucial pieces of information: the total number of coins Cat has and the total value of those coins. Let's break this down step by step. Firstly, we know that Cat has 19 coins in total, a combination of quarters and dimes. We can represent the number of quarters with the variable q and the number of dimes with the variable d. This leads us to our first equation: q + d = 19. This equation simply states that the sum of the number of quarters and the number of dimes equals 19. It's a straightforward representation of the total number of coins. Secondly, we know that the total value of Cat's coins is $4. To express this in an equation, we need to consider the value of each coin. A quarter is worth $0.25, and a dime is worth $0.10. Therefore, the total value of the quarters is 0.25q, and the total value of the dimes is 0.10d. This gives us our second equation: 0.25q + 0.10d = 4. This equation represents the total value of the coins, taking into account the individual value of each quarter and dime. With these two equations in hand, we have established a system of equations that accurately represents the problem. This system forms the foundation for our solution, allowing us to use algebraic techniques to determine the values of q and d, and ultimately, the number of quarters Cat possesses.

Solving the System of Equations: Unveiling the Number of Quarters

Now that we have our system of equations, the next step is to solve it. This will reveal the values of q (the number of quarters) and d (the number of dimes). There are several methods we can employ to solve a system of equations, including substitution, elimination, and graphing. For this particular problem, the substitution method is a straightforward and efficient approach. Let's revisit our equations:

  1. q + d = 19
  2. 0.25q + 0.10d = 4

Our first step in using the substitution method is to solve one of the equations for one variable in terms of the other. Let's choose the first equation and solve for d: d = 19 - q. Now we have an expression for d in terms of q. The next step is to substitute this expression for d into the second equation: 0.25q + 0.10(19 - q) = 4. This substitution eliminates the variable d from the second equation, leaving us with an equation in terms of q only. Now we can simplify and solve for q:

    1. 25q + 1.9 - 0.10q = 4
    1. 15q + 1.9 = 4
    1. 15q = 2.1
  • q = 14

Therefore, we have found that Cat has 14 quarters. To ensure our solution is correct, we can substitute this value back into our original equations to find the number of dimes and verify the total value. This process of solving the system of equations demonstrates the power of algebra in unraveling real-world problems, allowing us to determine unknown quantities from given information.

Verification: Confirming Our Solution and Ensuring Accuracy

With the value of q (number of quarters) determined to be 14, it's crucial to verify our solution. This step ensures that our answer is accurate and consistent with the given information. Verification involves substituting the value of q back into our original equations and checking if they hold true. Let's start with the first equation, q + d = 19. We know that q = 14, so we can substitute this value into the equation: 14 + d = 19. Solving for d, we get: d = 19 - 14 = 5. This tells us that Cat has 5 dimes. Now, let's move on to the second equation, which represents the total value of the coins: 0.25q + 0.10d = 4. We'll substitute the values we found for q and d into this equation: 0.25(14) + 0.10(5) = 4. Simplifying the equation:

    1. 5 + 0.5 = 4
  • 4 = 4

The equation holds true! This confirms that our solution is correct. We have successfully determined that Cat has 14 quarters and 5 dimes. The total number of coins is 14 + 5 = 19, which matches the given information. The total value of the coins is (14 * $0.25) + (5 * $0.10) = $3.50 + $0.50 = $4.00, which also matches the given information. This thorough verification process not only confirms the accuracy of our solution but also reinforces the importance of checking our work in mathematical problem-solving. It's a crucial step in ensuring that we have arrived at the correct answer and haven't made any errors along the way. By verifying our solution, we can confidently conclude that Cat has 14 quarters in her coin collection.

Conclusion: The Power of Algebra in Everyday Puzzles

In conclusion, we have successfully navigated the mathematical maze of Cat's coin collection, employing the power of algebra to unravel the mystery of her quarters and dimes. By setting up a system of equations and utilizing the substitution method, we determined that Cat has 14 quarters and 5 dimes, a combination that perfectly aligns with the given total of 19 coins and a value of $4. This exercise demonstrates the practical application of algebra in solving real-world problems, even those as seemingly simple as counting coins. The ability to translate a word problem into a mathematical equation is a fundamental skill that extends far beyond the classroom. It's a valuable tool for problem-solving in various aspects of life, from managing finances and making purchasing decisions to planning budgets and understanding statistical data. Moreover, this coin problem highlights the importance of critical thinking and logical reasoning. By carefully analyzing the given information, identifying the unknowns, and developing a systematic approach, we were able to arrive at the solution. This process of problem-solving fosters valuable cognitive skills that are essential for success in academics, careers, and everyday life. The satisfaction of solving a mathematical puzzle like this lies not only in finding the correct answer but also in the journey of discovery and the strengthening of our analytical abilities. So, the next time you encounter a mathematical challenge, remember the lessons learned from Cat's coin conundrum: break down the problem, set up your equations, and embrace the power of algebra to unlock the solution.