Solving The Shopping Spree: Math Problem For Adim, Vito, And Kali

Hey guys! Ever found yourselves scratching your heads over a math problem that seems like it came straight out of a shopping trip? Well, today, we're diving into a classic word problem. This one's all about Adim, Vito, and Kali, and their purchases at a stationery shop. Get ready to flex those math muscles! We're gonna figure out how much Kali needs to pay. Let's break it down, step by step.

The Shopping Scenario

So, here’s the lowdown: Adim buys 2 books, 3 pens, and 1 eraser for Rp17,500.00. Vito then buys 3 books, 2 pens, and 4 erasers for Rp22,500.00. Finally, Kali wants to buy 1 eraser, 1 book, and 1 pen. Our mission? Calculate how much Kali has to pay. This problem is a great example of a system of equations. We have three unknowns: the cost of a book, a pen, and an eraser. We are provided with information on what three different people bought at the same store.

To solve this problem, we need to set up a system of equations based on the information provided. Let’s denote the price of a book as b, the price of a pen as p, and the price of an eraser as e. We can create equations based on the purchases of Adim and Vito. The goal is to find the values of b, p, and e. Then we can figure out the cost of what Kali wants to buy. In the world of math problems, this type is a classic, testing our ability to translate real-world situations into equations and then solve them.

It might look daunting at first, but trust me, with a methodical approach, it's totally manageable. This isn’t just about getting to the answer; it's about the logical journey and how we apply mathematical principles to solve everyday puzzles. The fun is in the process, the aha moments, and finally, the satisfaction of cracking the code. Now, let’s get to work. We will translate the word problem into a system of linear equations. We will then solve for the unknowns. This involves the application of algebraic principles, specifically those related to systems of equations. Ready to solve this?

Breaking Down the Purchases

Let’s write down what we know in equation form, shall we? This is the crucial first step.

• Adim’s purchase: 2 books + 3 pens + 1 eraser = Rp17,500.00. This translates to: 2b + 3p + e = 17,500
• Vito’s purchase: 3 books + 2 pens + 4 erasers = Rp22,500.00. This gives us: 3b + 2p + 4e = 22,500

Our aim is to find the cost of 1 book, 1 pen, and 1 eraser, and then add those costs together to find out what Kali owes. Solving these equations might involve methods like substitution, elimination, or matrices, depending on your preferred approach. Don’t worry; we will take this slowly. Remember, the process of setting up the equations is just as important as solving them. Getting the equations right is half the battle. It’s all about converting the story into mathematical terms. The strategy is to methodically isolate variables, reducing the number of unknowns until we can solve for each one.

We will aim to find the cost of a book, a pen, and an eraser, and then we’ll add these prices up to determine Kali’s total. The key is to remain systematic in our approach. We're not just looking for numbers; we're uncovering the cost of each item. We're building our solution piece by piece, using what we know to uncover the unknown. This part is all about setting the stage for our calculations. So, let’s keep going and crack these equations!

Solving the Equations

Alright, let's dive into the solution! To make things easier, we will start by eliminating one of the variables. Let's use the elimination method, which involves manipulating the equations to eliminate variables. The process involves multiplying the equations by constants, allowing us to eliminate a variable when adding or subtracting the equations. Let's start by multiplying Adim's equation by 4:

8b + 12p + 4e = 70,000

Now, subtract Vito's equation from the new equation:

(8b + 12p + 4e) - (3b + 2p + 4e) = 70,000 - 22,500

This simplifies to:

5b + 10p = 47,500

Now, let’s divide this equation by 5:

b + 2p = 9,500

Now we have a simpler equation! Let’s isolate b:

b = 9,500 - 2p

Great, now we can substitute this value of b back into one of the original equations, like Adim’s equation:

2*(9,500 - 2p) + 3p + e = 17,500

This becomes:

19,000 - 4p + 3p + e = 17,500

Which simplifies to:

-p + e = -1,500

Now we have another equation to work with. And we know that b = 9,500 - 2p.

Let's solve this by expressing e in terms of p:

e = p - 1,500.

We have simplified things considerably. By carefully applying these techniques, we are making progress. We're methodically solving these equations to find the price of each item. We're breaking down a complex problem into smaller parts. And this allows us to move closer to the answer step by step.

Substitution to the Rescue!

Now, we'll substitute b and e in Vito’s equation:

3*(9,500 - 2p) + 2p + 4*(p - 1,500) = 22,500

This expands to:

28,500 - 6p + 2p + 4p - 6,000 = 22,500

Combine the terms, and we get:

22,500 = 22,500

Here, we find that the value of p can't be determined, and the equation simplifies in a way that does not allow us to further solve the variables. But wait! We can still find a relationship with b and e. We know that b = 9,500 - 2p e = p - 1,500

To find Kali’s total, we need b + p + e.

So, let’s add b + p + e together using the relationships we found: (9,500 - 2p) + p + (p - 1,500) = 9,500 - 2p + p + p - 1,500 = 8,000.

So, b + p + e = 8,000.

Calculating Kali's Total

We've got the equations, and we've done the math. Now, let’s find out how much Kali has to pay for her 1 book, 1 pen, and 1 eraser. Remember, we're looking for the sum of b, p, and e. We've already done the work of setting up the equations and solving them. We just need to put it all together. Therefore, Kali has to pay Rp8,000.00. Hooray! We did it!

The Final Answer

So, the answer is: Kali needs to pay Rp8,000.00 for her purchase.

Conclusion

Well, guys, that was a fun ride! We've successfully navigated this word problem, and come out on top. This exercise demonstrates how we can solve real-life scenarios using math. From setting up equations to solving them, we've broken down a complex problem into manageable parts. Keep practicing these problems, and you'll see how your problem-solving skills sharpen over time. Math isn't just about numbers; it's about logic, strategy, and critical thinking. Keep up the great work, and keep solving!