Cost Of Pen And Notebook: A Math Problem Solved

Hey guys! Today, we're diving into a cool math problem that involves figuring out the cost of pens and notebooks. It's a classic example of how we can use equations to solve everyday problems. So, let's get started and break it down step by step. The main goal here is to help you understand how to determine the cost of a pen and a notebook when given a couple of different scenarios. This involves setting up equations and using some basic algebra to find the answers. Stick with me, and you'll see it's not as tricky as it might seem!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We know two things:

1. Two pens and five notebooks cost 46 lei.
2. Two pens and eight notebooks cost 70 lei.

Our mission, should we choose to accept it (and we do!), is to find out the individual cost of one pen and one notebook. To do this effectively, understanding the core problem is essential. This means identifying the unknowns (the cost of a pen and a notebook) and recognizing the relationships between them as described in the problem. This initial comprehension forms the foundation for setting up the equations correctly and solving them accurately. So, take a moment to really grasp what we're trying to find out – it'll make the rest of the process much smoother.

Setting Up the Equations

This is where we turn words into math! Let's use some variables to represent the unknowns:

• Let 'x' be the cost of one pen.
• Let 'y' be the cost of one notebook.

Now, we can translate the given information into equations:

• Equation 1: 2x + 5y = 46 (Two pens and five notebooks cost 46 lei)
• Equation 2: 2x + 8y = 70 (Two pens and eight notebooks cost 70 lei)

Setting up these equations is a critical step because it converts the word problem into a mathematical form that we can actually work with. The accuracy of these equations directly impacts the correctness of our final answer. So, it's super important to double-check that each part of the equation correctly represents the information given in the problem. Think of it like building a house – if the foundation (our equations) isn't solid, the rest of the structure won't be either!

Solving the Equations

We've got our equations, now it's time to solve them. There are a few ways to do this, but the elimination method is super handy here. Notice that both equations have '2x' in them. This means we can subtract one equation from the other to eliminate 'x'.

Let's subtract Equation 1 from Equation 2:

(2x + 8y) - (2x + 5y) = 70 - 46

This simplifies to:

3y = 24

Now, we can solve for 'y' by dividing both sides by 3:

y = 8

So, one notebook costs 8 lei! Yay, we're halfway there. Solving the equations is the heart of this problem. The elimination method, which we used here, is particularly effective when we have terms that are the same (or easily made the same) in both equations. The key is to perform the same operation on both sides of the equation to maintain the balance. Once we've isolated one variable, we can then substitute it back into one of the original equations to find the other variable. This step-by-step approach helps us break down the problem into manageable chunks, making it much less daunting.

Finding the Cost of a Pen

Now that we know the cost of a notebook (y = 8), we can plug it back into either Equation 1 or Equation 2 to find the cost of a pen ('x'). Let's use Equation 1:

2x + 5y = 46

Substitute y = 8:

2x + 5(8) = 46

Simplify:

2x + 40 = 46

Subtract 40 from both sides:

2x = 6

Divide by 2:

x = 3

So, one pen costs 3 lei!

Finding the cost of a pen is the final step in our problem-solving journey. Once we've determined the value of one variable (in this case, the cost of a notebook), substituting it back into one of the original equations allows us to solve for the remaining unknown. This process highlights the interconnectedness of the equations and how each piece of information contributes to the overall solution. By carefully substituting and simplifying, we can confidently arrive at the cost of a pen, completing our task.

The Answer

We did it! We found that:

• One pen costs 3 lei.
• One notebook costs 8 lei.

The final answer is the culmination of all our hard work. It's not just about getting the numbers right; it's also about understanding what those numbers represent in the context of the original problem. In this case, we've successfully determined the individual costs of a pen and a notebook, satisfying the conditions outlined in the problem. This sense of accomplishment is a key part of learning and mastering math skills. So, take a moment to appreciate the journey and the result we've achieved together!

Checking Our Work

It's always a good idea to check our answers to make sure they make sense. Let's plug our values back into the original equations:

• Equation 1: 2(3) + 5(8) = 6 + 40 = 46 (Correct!)
• Equation 2: 2(3) + 8(8) = 6 + 64 = 70 (Correct!)

Our answers check out! We can be confident that we've solved the problem correctly.

Checking our work is a crucial habit to develop in problem-solving. It's like a safety net that catches any potential errors and ensures the accuracy of our solution. By substituting our calculated values back into the original equations, we can verify that they satisfy the conditions of the problem. This not only gives us confidence in our answer but also reinforces our understanding of the relationships between the variables. So, always take that extra step to double-check – it's worth the peace of mind!

Real-World Applications

This kind of problem-solving isn't just for textbooks. We use these skills in real life all the time! Imagine you're buying school supplies or groceries and want to figure out the best deal. Understanding how to set up and solve equations can help you make smart decisions and save money. Real-world applications of math skills are everywhere, and this is what makes learning math so valuable. The ability to translate a practical situation into a mathematical model, solve it, and interpret the results is a powerful tool in many aspects of life. Whether you're budgeting, planning a project, or even just figuring out how much of each ingredient you need for a recipe, the problem-solving skills you develop in math class can come in handy.

Conclusion

So there you have it! We've successfully solved a problem involving the cost of pens and notebooks by setting up equations and using the elimination method. Remember, the key is to break down the problem, define variables, create equations, solve them, and then check your work. Keep practicing, and you'll become a math whiz in no time! This example illustrates how solving mathematical problems is not just about finding the right answer; it's about developing a systematic approach and critical thinking skills. By breaking down the problem into smaller, manageable steps, we can tackle even the most challenging questions. And remember, practice makes perfect! The more you apply these skills, the more confident and proficient you'll become in your problem-solving abilities.