Unlocking The Riddle: Finding Nickels In 3odhi's Coin Collection

Hey guys! Let's dive into a fun math problem involving 3odhi's awesome coin collection. This isn't just about numbers; it's about using those numbers to crack a real-world puzzle. We're going to learn how to find the number of nickels in a collection of dimes and nickels, given the total value and the total number of coins. Ready to become coin detectives? Let's go!

The Core Problem: Dimes, Nickels, and Dollars

So, here's the deal: 3odhi has a stash of 175 coins. These coins are a mix of dimes and nickels. Now, if you didn't know, a dime is worth 10 cents ($0.10), and a nickel is worth 5 cents ($0.05). The grand total value of the entire collection is $13.30. Our mission, should we choose to accept it, is to figure out the number of nickels in the collection. This is where those equations come in handy. This question focuses on how to model a real-world scenario with an algebraic equation. Understanding this process can be super helpful in a lot of different situations. We'll be using the provided options to determine which equation correctly represents the relationships between the number of nickels, the number of dimes, and the total value. The key is to remember that the total value is a combination of the value of the dimes and the value of the nickels.

Breaking Down the Components

Before we jump into the equations, let's break down the elements:

• Dimes: Worth $0.10 each.
• Nickels: Worth $0.05 each.
• Total Coins: 175 (dimes + nickels).
• Total Value: $13.30.

We need to build an equation that links all these things together. Specifically, we're looking to express the total value in terms of the number of nickels (which we'll call n).

The Importance of Variable Representation

Choosing the right variable to represent the unknown is super important. In this case, the problem specifically asks us to find n, the number of nickels. That means our equations should be designed to solve for n. The beauty of algebra is its ability to use variables to stand in for things we don't know yet. By carefully assigning variables, we can translate word problems into mathematical equations that we can then solve. For example, if n represents the number of nickels, then we can figure out how many dimes there are in terms of n since we know the total number of coins. It's like having a secret code! Mastering this skill is a cornerstone of algebra, allowing us to model and solve a huge variety of real-world problems. This is exactly what we're going to do here – find that secret code and unlock the number of nickels.

Deciphering the Equations

Alright, let's analyze those equations and see which one does the trick. Remember, we're looking for an equation that accurately represents the total value of the coins. Each equation attempts to relate the value of the nickels and dimes to the total value of the collection. We need to examine each equation closely, paying attention to how it represents the number of dimes and nickels, and how it calculates the total monetary value. Let's break down each option to find the one that correctly models 3odhi's coin situation.

Equation A: A Closer Look

The first equation provided is: $0.1n + 0.05(n - 175) = 13.30$. Let's unpack this one.

• $0.1n$: This term is trying to represent the value of the dimes. However, it's multiplying the number of nickels (n) by $0.10, which is the value of a dime. This suggests a misunderstanding of which coin's value is being calculated. It assumes the total value of dimes is based on the number of nickels. That's a red flag, right?
• $0.05(n - 175)$: This part is where it tries to account for nickels. It subtracts the total number of coins (175) from the number of nickels (n). This results in a negative value because n (the number of nickels) is unlikely to be greater than 175. This is unlikely to be correct, as it would lead to a negative number of dimes, which isn't possible in this scenario.
• $= 13.30$: This correctly sets the equation equal to the total value of the collection.

Based on these observations, Equation A doesn't seem to correctly represent the relationship between the number and value of dimes and nickels. The structure of the equation does not align with the known values in the problem, and there are logical inconsistencies. So, we can eliminate this option.

Equation B: Examining the Components

Here's equation B: $0.1(175 - n) + 0.05n = 13.30$. Let's take a look:

• $0.1(175 - n)$: This represents the value of the dimes. It correctly calculates the number of dimes by subtracting the number of nickels (n) from the total number of coins (175). Then, it multiplies that by $0.10 (the value of a dime). This seems like a promising start. The expression (175 - n) accurately calculates the number of dimes, since you subtract the number of nickels from the total number of coins.
• $0.05n$: This represents the value of the nickels. It multiplies the number of nickels (n) by $0.05 (the value of a nickel). This also looks good.
• $= 13.30$: The total value of the collection.

Equation B appears to accurately model the situation. It correctly calculates the total value of the dimes and nickels separately, and then sums them to equal the total value of the collection. This equation is structured logically, and all components align with the problem's details.

Equation C: The Final Analysis

And now for the final contender, Equation C: $0.1n + 0.05(175 - n) = 13.30$. Let's break it down:

• $0.1n$: This part is intended to calculate the value of the dimes, similar to equation A. However, it is multiplying the number of nickels (n) by the value of a dime ($0.10). This does not accurately represent the value of the dimes.
• $0.05(175 - n)$: This correctly calculates the value of the nickels by first finding the number of dimes by subtracting the number of nickels from the total and then multiplying that amount by $0.05. This seems correct. It takes the total number of coins (175) and subtracts the number of nickels to get the number of dimes.
• $= 13.30$: The total value of the collection.

Even though parts of Equation C appear to be correct, the equation as a whole does not fully reflect the real-world scenario. The calculation of the value of dimes is incorrect, which makes this equation flawed. So, we're likely to rule this one out.

The Verdict: The Winning Equation

After carefully evaluating all the equations, it is clear that Equation B: $0.1(175 - n) + 0.05n = 13.30$ is the one that correctly represents 3odhi's coin collection. This equation accurately calculates the total value of the dimes and nickels. It does this by considering the value of the dimes and the value of the nickels separately, and then summing them to equal the total value of the collection. This is a perfect example of how to translate a word problem into an equation and solve it.

Key Takeaways and Helpful Tips

• Understanding the Problem: Always read the problem carefully and identify the knowns and unknowns. What is being asked of you? What values are provided? This is like the foundation of your equation-building.
• Defining Variables: Choose your variables wisely. Make sure each variable represents a clear quantity in the problem. For instance, in our case, we used n for the number of nickels.
• Formulating Equations: Build your equations step by step. Break down the problem into smaller parts and then translate each part into a mathematical expression. Remember to use the correct values of each coin.
• Checking Your Work: It's always a good idea to check your answers. Plug your solution back into the equation to see if it makes sense. If you find n, you can substitute the value of n into the original equation to see if the equation holds true.
• Practice Makes Perfect: The more you practice solving word problems, the better you'll become at them. Don't be afraid to try different problems, and don't worry if you don't get it right the first time. The more you work with these types of problems, the easier it will become.

So there you have it, guys! We've successfully navigated the world of dimes and nickels and found the perfect equation to solve 3odhi's coin conundrum. Keep practicing, keep learning, and remember that math can be fun and rewarding! With a little bit of practice, you'll become a pro at these problems in no time. Keep those critical thinking skills sharp and happy coin-counting!