Coins And Banknotes: A Math Problem!

Nov 17, 2025 by Admin 37 views

Hey guys! Let's dive into a super fun math problem that involves coins and banknotes. Imagine you're helping a friend named Ana figure out how much of her money is in coins and how much is in banknotes. This is a classic problem that mixes a bit of algebra with real-world thinking, making it both useful and interesting.

Setting Up the Problem

So, Ana has 25 lei. That's the total amount we're working with. She has this 25 lei in two forms: 2 lei coins and 5 lei banknotes. Now, here's the kicker: she has a total of 8 items, which means she has a mix of both coins and banknotes adding up to eight. Our mission, should we choose to accept it, is to figure out exactly how many 2 lei coins and how many 5 lei banknotes Ana has.

Why is this important? Well, these kinds of problems aren't just academic exercises. They help build your problem-solving skills, which are super handy in everyday life. Whether you're splitting a bill with friends, managing your budget, or even figuring out discounts while shopping, the ability to break down a problem and solve it step-by-step is invaluable. Plus, it's kinda fun, right?

Breaking Down the Knowns

Let's get organized. We know a few key things:

Total Amount: Ana has 25 lei.
Coin Value: Each coin is worth 2 lei.
Banknote Value: Each banknote is worth 5 lei.
Total Items: Ana has a total of 8 coins and banknotes.

With these pieces of information, we can start building our strategy to solve the puzzle. The trick is to use variables to represent the unknowns and then create equations that link everything together. Trust me; it's easier than it sounds!

Diving into Algebraic Representation

Alright, let’s get a little algebraic! This might sound intimidating, but it's just a fancy way of using letters to represent numbers we don't know yet. In our case, we don't know how many coins Ana has, and we don't know how many banknotes she has. So, let's give them names. We'll use 'x' for the number of 2 lei coins and 'y' for the number of 5 lei banknotes. Simple enough, right?

Now, we can create equations based on the information we already have. Remember, Ana has a total of 8 coins and banknotes. This gives us our first equation:

x + y = 8

This equation tells us that the number of coins (x) plus the number of banknotes (y) equals 8. Easy peasy!

Next, we know that the total value of all the coins and banknotes is 25 lei. We can express this as another equation. Since each coin is worth 2 lei, the total value of the coins is 2x. Similarly, since each banknote is worth 5 lei, the total value of the banknotes is 5y. Add those together, and you get the total value of 25 lei. So, our second equation looks like this:

2x + 5y = 25

This equation tells us that the value of the coins (2x) plus the value of the banknotes (5y) equals 25. Now we have two equations with two unknowns, which means we can solve for x and y!

Solving the System of Equations

Now comes the fun part: solving the system of equations! We have two equations:

x + y = 8
2x + 5y = 25

There are a couple of ways to tackle this. One common method is called substitution. The idea here is to solve one equation for one variable and then substitute that expression into the other equation. This leaves us with a single equation with a single variable, which is much easier to solve.

Using the Substitution Method

Let's start with the first equation, x + y = 8. We can easily solve for x:

x = 8 - y

Now, we take this expression for x and substitute it into the second equation:

2(8 - y) + 5y = 25

See what we did there? We replaced 'x' in the second equation with '(8 - y)'. Now we just have one equation with one variable, 'y'. Let's simplify and solve for 'y':

16 - 2y + 5y = 25 3y = 9 y = 3

So, we've found that y = 3. This means Ana has 3 banknotes of 5 lei each. Awesome!

Finding the Value of x

Now that we know the value of 'y', we can easily find the value of 'x'. Remember our equation x = 8 - y? Just plug in the value of 'y':

x = 8 - 3 x = 5

So, Ana has 5 coins of 2 lei each. We've cracked the code!

Verifying the Solution

Before we declare victory, let's make sure our solution actually works. We can plug the values of x and y back into our original equations to see if they hold true.

Equation 1: x + y = 8 5 + 3 = 8 8 = 8 (Yep, that checks out!)

Equation 2: 2x + 5y = 25 2(5) + 5(3) = 25 10 + 15 = 25 25 = 25 (Fantastic, this one checks out too!)

Since both equations are true with our values for x and y, we can be confident that our solution is correct.

The Final Answer

Alright, drumroll please! After all that math, we've finally arrived at the answer. Ana has:

5 coins of 2 lei each
3 banknotes of 5 lei each

Isn't that satisfying? We took a real-world problem, broke it down into smaller parts, used a little bit of algebra, and solved it. You're basically a math detective now!

Real-World Applications

Now, you might be wondering,