Hey guys! Let's dive into a fun math problem that involves quarters and nickels. We're going to break down a word problem step by step, making sure it's super easy to understand. Our mission? To figure out which equation correctly represents the situation where Alex has a bunch of quarters and nickels adding up to a specific total. So, grab your thinking caps, and let's get started!
Understanding the Problem
Okay, so here’s the deal. Alex has 84 coins in total, and these coins are a mix of quarters and nickels. Now, we all know that quarters are worth 25 cents each, and nickels are worth 5 cents each. The total value of all these coins is $17.80. That’s our puzzle! We need to figure out which equation can help us find out exactly how many quarters Alex has. To solve this, we need to translate the words into math. This means we'll use variables and create an equation that shows the relationship between the number of coins and their values. Remember, the key is to represent the information accurately and in a way that we can solve.
Breaking Down the Information
Let's really break this down. The most important thing here is that we're dealing with two types of coins, each with a different value, and a total amount. So, we need to consider both the number of coins and their value. Think of it like this: if Alex had only quarters, the total would be much higher than $17.80. If Alex had only nickels, the total would be much lower. The mix of these two is what gives us the $17.80. So, we need an equation that captures this balance. We'll use a variable, usually denoted as 'q', to represent the number of quarters because that’s what the question asks us to find. Then, we need to figure out how to represent the number of nickels in terms of 'q'. Since Alex has 84 coins total, the number of nickels can be expressed as 84 minus the number of quarters (84 - q). This is a crucial step because it allows us to write everything in terms of one variable, making the equation solvable. Remember, the goal here is not just to find the answer but to understand the logic behind setting up the equation. Once you get this part, solving similar problems becomes a breeze!
The Value of Each Coin Type
Now, let's talk about the value. Each quarter is worth $0.25, and each nickel is worth $0.05. This is super important because we need to translate the number of coins into their monetary value. If Alex has 'q' quarters, the total value of the quarters is 0.25 multiplied by 'q', or 0.25q. Similarly, if Alex has (84 - q) nickels, the total value of the nickels is 0.05 multiplied by (84 - q), or 0.05(84 - q). The problem tells us that the total value of all the coins is $17.80. This is the final piece of the puzzle. We know the value of the quarters, the value of the nickels, and the total value. The only thing left is to put it all together into an equation. This equation will show that the sum of the value of the quarters and the value of the nickels equals the total value, $17.80. Setting up the equation correctly is the most critical step in solving the problem. It ensures that we're accurately representing the relationships between the quantities involved. Once the equation is set, we can use algebraic techniques to solve for 'q', which will give us the number of quarters Alex has.
Setting Up the Equation
Alright, let’s get to the nitty-gritty of setting up the equation. We know the value of the quarters is 0.25q, and the value of the nickels is 0.05(84 - q). The total value is $17.80. So, we can write this as an equation:
Value of Quarters + Value of Nickels = Total Value
This translates to:
- 25q + 0.05(84 - q) = 17.80
This equation perfectly captures the information given in the problem. It says that the value of the quarters (0.25q) plus the value of the nickels [0.05(84 - q)] equals the total value ($17.80). This is the equation we're looking for! Now, let's take a look at the answer choices and see which one matches our equation.
Examining the Options
Now, let's look at the answer options given. It’s super important to match our equation with the correct option. This is where paying attention to detail really matters. Each option presents a slightly different equation, and only one will accurately reflect the problem's conditions. So, let’s go through each one and see if it matches our equation: 0.25q + 0.05(84 - q) = 17.80.
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Option A:
This equation is incorrect. Notice that it has (q - 84) inside the parentheses, which means it's subtracting 84 from the number of quarters. But we know that the number of nickels is 84 minus the number of quarters, not the other way around. So, this one is not the correct representation of the problem.
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Option B:
This equation is also incorrect. Here, the coefficients are switched around. It looks like the value of the nickels (0.05) is multiplied by the number of quarters, and the value of the quarters (0.25) is multiplied by the number of nickels (84 - q). This doesn’t match our understanding of the problem, so it's not the right equation.
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Option C:
This equation looks promising! If we rewrite it, it becomes 0.25q + 0.05(84 - q) = 17.80, which is exactly the equation we derived. This option correctly represents the value of the quarters (0.25q) added to the value of the nickels [0.05(84 - q)], and it equals the total value ($17.80). So, this looks like our winner!
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Option D:
This equation is incorrect for the same reason as option A. The expression (q - 84) means we're subtracting the total number of coins from the number of quarters, which doesn’t make sense in our problem. This option doesn’t accurately represent the relationship between the number of coins and their values.
Choosing the Correct Equation
After carefully analyzing each option, it's clear that Option C is the equation that correctly represents the problem. It matches the equation we derived: 0.25q + 0.05(84 - q) = 17.80. So, the answer is C!
Why Option C is the Answer
Let’s really nail down why Option C is the correct equation. It all comes down to how we represent the value of the quarters and nickels in relation to the total value. We've established that 'q' stands for the number of quarters. The value of these quarters is simply 0.25q (since each quarter is worth $0.25). Now, the tricky part is the nickels. Since Alex has a total of 84 coins, and 'q' of them are quarters, the remaining coins must be nickels. So, the number of nickels is (84 - q). Each nickel is worth $0.05, so the total value of the nickels is 0.05(84 - q). The total value of all the coins ($17.80) is the sum of the value of the quarters and the value of the nickels. This is exactly what Option C expresses: 0.05(84 - q) + 0.25q = 17.80. Option C correctly adds the value of nickels to the value of quarters to equal the total value. The other options fail because they either mix up the values and quantities or incorrectly represent the number of nickels. Option A and D have the incorrect expression for the number of nickels, and Option B mixes up the coefficients with the variables. This detailed explanation should help clarify why Option C is the one that clicks and accurately reflects the problem's conditions. Remember, in word problems, understanding the relationships between the quantities is the key to setting up the correct equation!
Final Thoughts
And there you have it! We've successfully navigated through a word problem, broken it down piece by piece, and identified the correct equation. The key takeaway here is that understanding the problem and translating it into mathematical language is the most crucial step. Once you have the equation set up correctly, solving it becomes much easier. So, next time you encounter a similar problem, remember to read carefully, identify the variables, and think about how the different pieces of information relate to each other. You’ve got this!
