Solving The Money Puzzle: A Math Problem

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Solving the Money Puzzle: A Math Problem

Hey math enthusiasts! Ready to dive into a fun problem that involves some clever thinking and a bit of algebra? This problem comes from GAZETA Senior MATEMATICA, a great resource for challenging math problems. We're going to break down this problem step-by-step so that you can understand it easily. Let's get started!

The Problem Unveiled: Money Matters

GAZETA Senior MATEMATICA XIV.4.251 presents us with a scenario involving three friends: Ioana, Teodora, and Ana. The girls have a total of 896 lei (Romanian currency) between them. Here's where the problem gets interesting: Ioana spends 56 lei on a teddy bear, Teodora buys a puppy for 63 lei, and Ana purchases a kitten for 45 lei. After these purchases, the friends discover that they each have the same amount of money left. The question is: How much money did each girl have initially? This is a classic math problem that combines arithmetic with a bit of algebraic thinking. This particular type of problem is great for developing problem-solving skills and logical reasoning. So, let’s get into the details.

To solve this, we will first create an equation. We will create a sum of Ioana, Teodora, and Ana's money. Then, we will find the sum of their remaining money. This is a common way to solve this type of problem. We will break it down into smaller parts, so follow along! Understanding the question is very important, so let's make sure that we get the grasp of the question. You can use this method to solve any type of problem.

First, let's designate some variables: Let's assume the initial amount of money Ioana had is 'x', Teodora had 'y', and Ana had 'z'. So, according to the problem, x + y + z = 896 lei. Next, we will calculate the money left after each of them bought something. Ioana is left with x - 56, Teodora is left with y - 63, and Ana is left with z - 45. According to the question, these remaining amounts are equal. Thus, we can write x - 56 = y - 63 = z - 45. Now, we are ready to move on. Let's start with simplifying and calculating.

Now, let's break down the problem into smaller parts to make it easier to understand. The key to solving this problem is recognizing that after their purchases, the remaining amounts are equal. This suggests a systematic approach using algebra. This is a great exercise for young mathematicians to hone their skills. We'll start by setting up equations to represent the situation. This part is crucial for any problem.

Setting Up the Equations: The Algebraic Approach

Now, let's formalize our approach by setting up equations. This will make the solution process much more organized and straightforward. We'll use variables to represent the unknowns and construct equations based on the information provided in the problem. This is a great way to represent the problem. Keep in mind that setting up equations is the most important part of solving. We will use the following assumptions to simplify the calculation:

  • Let's denote:

    • Ioana's initial money as I.
    • Teodora's initial money as T.
    • Ana's initial money as A.

  • Total Money:

    • I + T + A = 896 (The total sum of money they have initially).

  • After Purchases:

    • Ioana's remaining money: I - 56
    • Teodora's remaining money: T - 63
    • Ana's remaining money: A - 45

  • Equal Remaining Amounts:

    • I - 56 = T - 63 = A - 45 (This is the critical piece of information. After spending, they all have the same amount).

From the last equation (I - 56 = T - 63 = A - 45), we know that after spending, each girl has the same amount of money. Let's call this common amount 'M'. Thus, we can rewrite the equations as:

  • I - 56 = M => I = M + 56
  • T - 63 = M => T = M + 63
  • A - 45 = M => A = M + 45

Now, we can substitute these values into the total money equation (I + T + A = 896): (M + 56) + (M + 63) + (M + 45) = 896. This simplifies the calculation. From this point, we just have to sum it up. We will continue in the next section.

This is a classic example of an algebra word problem, where the setup of the equations is the most important step. We've translated the word problem into mathematical expressions, which is key to finding a solution. Always take the time to set up equations properly before jumping into calculations; it will make the whole process much easier and less prone to errors.

Solving for the Unknown: Step-by-Step Solution

Now that we've set up our equations, let's solve for the unknown, 'M', which represents the amount of money each girl had remaining after their purchases. This is where we put our algebra skills to work. This process is very important. Let's make sure that we get it right.

We have the equation: (M + 56) + (M + 63) + (M + 45) = 896. Let's simplify this equation:

  • Combine the 'M' terms: 3M + 56 + 63 + 45 = 896
  • Combine the constants: 3M + 164 = 896
  • Subtract 164 from both sides: 3M = 896 - 164
  • 3M = 732
  • Divide both sides by 3: M = 732 / 3
  • M = 244

So, each girl had 244 lei remaining after their purchases. We now have enough information to calculate the initial amount of money each girl had. We'll find out the initial amounts by substituting M = 244 into the equations we derived earlier:

  • Ioana: I = M + 56 = 244 + 56 = 300 lei
  • Teodora: T = M + 63 = 244 + 63 = 307 lei
  • Ana: A = M + 45 = 244 + 45 = 289 lei

Now, let's check our answer to make sure it's correct. We can verify our solution by adding up the initial amounts and confirming that they equal the total amount given in the problem:

300 lei (Ioana) + 307 lei (Teodora) + 289 lei (Ana) = 896 lei

This matches the initial total, confirming the accuracy of our solution! We successfully solved the problem by setting up equations, simplifying them, and solving for the unknowns.

This is a methodical way to solve word problems, ensuring accuracy and understanding. Breaking down the problem into smaller, manageable steps is a key strategy for success.

The Final Answer and Insights: Understanding the Solution

Alright, guys, we've reached the end of our money puzzle! Let's summarize our findings and highlight what we've learned. This part is very important because it concludes the problem. We made it all the way. We will look back and understand what we have done.

The Solution:

  • Ioana initially had 300 lei.
  • Teodora initially had 307 lei.
  • Ana initially had 289 lei.

After their purchases, each girl was left with 244 lei. This problem showcases how algebra can be used to solve real-world scenarios. We've seen how setting up equations correctly can lead to a clear and accurate solution. The key takeaways from this problem are:

  • Understanding the Problem: Carefully reading and understanding the information is the first step. You should get a clear picture of the problem.
  • Variable Assignment: Assigning variables to the unknowns helps translate the problem into mathematical terms.
  • Equation Formulation: Creating equations based on the relationships described in the problem is the most crucial step.
  • Systematic Solving: Using algebraic techniques to solve the equations and find the values of the variables. This step is also very important.
  • Verification: Always double-check your answer to ensure it aligns with the original problem. This is the last part of solving a math problem.

This problem is a great example of how mathematical concepts can be used to analyze and solve problems from everyday life. By applying these steps, you can successfully solve many similar problems and strengthen your problem-solving skills. Keep practicing, and you'll find that these types of problems become easier and more enjoyable. Remember, the journey of solving a problem is just as important as the answer itself. Congrats, guys, you did a great job!

This entire process is a wonderful exercise in logical thinking and problem-solving, which are skills that extend far beyond mathematics and into all aspects of life. It gives us a great sense of satisfaction and enhances our ability to think critically.