Solving The Cake Sales Puzzle: A Math Problem

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Hey guys! Let's dive into a fun little math problem. This one's all about cakes, money, and figuring out how many of each type were sold. Sounds yummy, right? We're going to break down the problem step-by-step so you can easily understand how to solve it. It’s like a delicious recipe, but for math! So grab your calculators (or your thinking caps) and let's get started. This is a common type of problem you might find in a math test, so understanding the approach will definitely come in handy. We'll be using some basic algebra concepts, but don't worry, it's nothing too complicated. The goal here is to clarify the question and guide you through the solution. This is not just about getting the answer; it's about learning how to find the answer. We'll show you the thought process. Now let's explore this problem together and become math problem-solving pros!

Understanding the Problem: The Cake Shop Scenario

Alright, let's set the scene. Imagine a cake shop. This shop is selling two sizes of cakes: small and large. The small cakes cost R$15 each, and the large cakes cost R$25 each. Now, here's the kicker: we know that a total of 14 cakes were sold, and the total earnings for the day were R$290. Our mission, should we choose to accept it, is to figure out exactly how many large cakes were sold. This kind of problem often appears in math contests or school exercises, and the key is to translate the words into equations. It's like decoding a secret message! The problem gives us two important pieces of information: the total number of cakes sold and the total amount of money earned. We have two unknowns: the number of small cakes and the number of large cakes. Our goal is to use the given information to create equations that we can solve to find the answer. It's like a puzzle where we're trying to figure out which pieces fit together. Understanding the components is the key to solving the cake problem.

Breaking Down the Information

Let's organize the data to make it easier to digest. We know the following:

  • Small cake price: R$15
  • Large cake price: R$25
  • Total cakes sold: 14
  • Total earnings: R$290

Now, how do we use this? Well, the first thing is to represent our unknowns with variables. Let's say:

  • 'x' = the number of small cakes
  • 'y' = the number of large cakes

Using these variables, we can form two equations based on the information provided. This is like creating a roadmap to solve our mystery! From the total number of cakes sold, we can write our first equation: x + y = 14. This tells us the combined number of small and large cakes is 14. Then, from the total earnings, we can make our second equation: 15x + 25y = 290. This means the total money from selling the cakes is R$290. Remember, each small cake contributes R$15 and each large cake R$25 to the total.

Setting Up the Equations: Translating Words to Math

Okay, now that we've got our variables defined and our information organized, it's time to create those equations. As we mentioned, we have two key pieces of information we can turn into equations. The first equation represents the total number of cakes, and the second represents the total earnings. This is where the magic of algebra really comes in handy! We're transforming word problems into manageable mathematical statements. Now, let’s get those equations down:

  • Equation 1 (Total Cakes): x + y = 14 This equation simply states that the number of small cakes (x) plus the number of large cakes (y) equals 14.
  • Equation 2 (Total Earnings): 15x + 25y = 290 Here, we multiply the number of small cakes (x) by their price (15) and the number of large cakes (y) by their price (25). The sum of these two amounts gives us the total earnings of R$290.

With these two equations, we can start solving for our unknowns (x and y). This process is an essential part of the problem. Remember, the goal is to find out how many large cakes (y) were sold. We'll solve these equations step by step to determine the value of y. This is where the fun begins, as we start to unravel the problem. Let’s tackle the system of equations and get the correct answers. We are now ready to put on our detective hats and solve for y, the number of large cakes.

Method: Solving the Equations

We have a few methods at our disposal to solve these equations. We can use substitution or elimination. For this problem, let's use the substitution method. This involves solving one equation for one variable and substituting that expression into the other equation. It simplifies our work! Here's how to do it:

  1. Solve for x in Equation 1: From the equation x + y = 14, we can easily find that x = 14 - y. This gives us an expression for x in terms of y.
  2. Substitute into Equation 2: Now, substitute x = 14 - y into the second equation, 15x + 25y = 290. This gives us 15(14 - y) + 25y = 290.
  3. Simplify and Solve:
    • Expand the equation: 210 - 15y + 25y = 290
    • Combine like terms: 10y = 80
    • Solve for y: y = 8

Voila! We have solved for y, which represents the number of large cakes. The answer is 8. Now we know exactly how many large cakes were sold, thanks to our equation-solving skills. Remember that this process will show you the number of large cakes (y) that were sold. After completing the equation, we can conclude that the number of large cakes sold is 8. The beauty of these problems is in the process.

Finding the Answer: The Grand Finale

We've crunched the numbers, and the answer is in! By using the substitution method, we've determined that y = 8. This means that 8 large cakes were sold. Isn't that awesome? We started with a word problem and, through the power of equations, found our answer. Now that we have calculated the amount of large cakes sold, we can also find the amount of small cakes sold. We can substitute the value of y back into the first equation: x + 8 = 14. Solving for x, we get x = 6. Thus, the cake shop sold 6 small cakes and 8 large cakes. This is great for us because we've answered our initial question: how many large cakes were sold? The answer: 8. The whole process demonstrates how simple algebra can be used to solve real-world problems. We can now be proud of knowing how to solve the cake shop problem! And to celebrate our success, maybe we could treat ourselves to a delicious cake!

Quick Recap and Conclusion

Let’s recap what we did:

  1. We understood the problem and identified the key information.
  2. We defined our variables: x for small cakes and y for large cakes.
  3. We formed two equations based on the total cakes and total earnings.
  4. We used the substitution method to solve for y.
  5. We found that 8 large cakes were sold.

And that's it! You've successfully solved the cake problem. Congratulations on your hard work! Keep practicing these types of problems, and you'll become a math whiz in no time. If you got the answer right, great job! If not, don’t worry. Review the steps and practice similar problems. Math is all about practice and understanding. It's like any skill: the more you do it, the better you get. You are now equipped with the knowledge to tackle similar problems. Keep enjoying math, guys! It is a fun adventure. You'll be ready for your next math challenge with these awesome techniques.