Saint Louvent Article

Molly's Money: The Equation

Oct 24, 2025 by SLV Team 28 views

Hey guys, let's dive into a fun little math problem that's super common in everyday life! You know, sometimes we need to figure out how things add up, or in this case, how they multiply. Today, we're going to tackle a scenario involving Molly and her brother's money. We've got Molly, who's feeling pretty good with a cool $45 in her wallet. Now, the key piece of info here is that this $45 is **exactly three times** the amount of cash her brother has. This is where the magic of algebra comes in, helping us translate real-world situations into neat, tidy equations. So, stick around as we break down how to set up an equation to represent this situation, using '$ a

to stand for the amount of money Molly's brother has. It's going to be a breeze, and by the end of it, you'll be a pro at turning word problems into solvable math expressions. We'll make sure to cover all the bases, explaining why we set up the equation the way we do, and how understanding this can help you with all sorts of other problems. Think of it as building a fundamental skill that unlocks a whole world of mathematical understanding. We're aiming to make this super clear, so no one gets left behind. Ready to crunch some numbers and make some sense of Molly's wallet? Let's get this math party started!

Setting Up the Scenario: Understanding the Relationship

Alright, let's get down to business and really understand what's happening with Molly's money. We know for a fact that Molly has $45. That's a solid amount, right? The next crucial piece of information is that this $45 represents **three times the amount of money her brother has**. This is the core relationship we need to capture in our equation. If we let '$ a

be the amount of money Molly's brother has, then we can start building our mathematical representation. Think about it this way: if Molly's brother had, say, $10, then three times that amount would be

3 imes 10 = 30

. If he had $15, then three times that would be

3 imes 15 = 45

. See the pattern? The amount Molly has is always the result of multiplying her brother's amount by three. So, we can express this relationship as: Molly's money = 3 * (Brother's money). Since we know Molly's money is

45, and we've defined the brother's money as '

a , we can substitute these values directly into our relationship. This gives us: $45 = 3 imes a$ . This equation perfectly encapsulates the situation described. It states that the

45 Molly possesses is equal to three times the amount '

a

' that her brother has. We're not solving for '

a just yet; the goal here is simply to represent the situation accurately with an equation. This initial step of translating words into mathematical symbols is often the most important part of solving word problems. It requires careful reading and a solid understanding of how quantities relate to each other. We're building a bridge from the English language to the language of mathematics, and that bridge is our equation. So, when you see a problem like this, always start by identifying the knowns and unknowns, and then figure out the operation that connects them. In this case, the word 'times' clearly points to multiplication. Pretty neat, huh? It’s all about breaking it down into manageable pieces.

Crafting the Equation: The Mathematical Expression

Now that we've got a solid grasp of the scenario, let's formally craft the equation. We've established that Molly has $45, and this amount is three times whatever her brother has. We're using '$ a $' to represent the amount of money Molly's brother has. The phrase "three times as much money as her brother has" directly translates to the mathematical operation of multiplication. So, if the brother has '$ a

dollars, then three times that amount is represented as

3 imes a

, or simply

3a

. Since this amount is equal to the money Molly has, which is $45, we can set these two expressions equal to each other. This leads us directly to the equation: $45 = 3a$ . This equation is the heart of our solution. It’s a concise and accurate representation of the problem statement. It tells us that the quantity $45 is equivalent to the quantity

3a

. When you encounter problems like this, especially in tests or homework, the first step is always to define your variable clearly (which we did with '

a

') and then build the equation that reflects the given relationship. The wording is key here – '3 times as much' is our cue for multiplication. If it said '3 more than', we'd use addition. If it said '3 less than', we'd use subtraction. So, by carefully dissecting the sentence, we arrive at

45 = 3a

. This is the equation that represents the situation. We're not asked to find out how much money the brother has yet, just to write the equation. This skill of translating word problems into algebraic equations is fundamental in mathematics and opens doors to solving a vast array of problems, from simple financial calculations to complex scientific modeling. So, take a moment to appreciate how a few symbols can capture a whole scenario. It’s powerful stuff, guys! This equation is our tool for further analysis, allowing us to isolate '

a

' and find the brother's share if needed. For now, though, we've successfully achieved our goal of representation.

Why This Equation Works: The Logic Behind It

Let's pause for a sec and really dig into why the equation $45 = 3a$ is the perfect fit for this situation. The problem states that Molly's $45 is three times the amount her brother has. The word 'times' is a giant flashing sign pointing towards multiplication. If we didn't know how much the brother had, but we knew Molly had $45, and we were told it was three times that unknown amount, we'd logically say: "Okay, take that unknown amount, multiply it by 3, and you should get $45." This is exactly what the equation $45 = 3a$ does. ' $a$ ' represents that unknown amount (the brother's money), and multiplying it by 3 ( $3a$ ) gives us the amount Molly has. Setting this equal to $45 ( $45 = 3a$ ) completes the representation. It’s like a balance scale: on one side, you have the known quantity ($45), and on the other, you have the expression that describes it in relation to the unknown ( $3a$ ). They must be equal because the problem states they are. This equation structure is crucial for solving for ' $a$ ' later on. If you were to try other operations, like addition ( $45 = a + 3$ ) or subtraction ( $45 = a - 3$ ), it wouldn't make sense in the context of the problem. The phrase "three times as much" unequivocally indicates a multiplicative relationship. Understanding this logical connection between the words in a problem and the mathematical operations used in an equation is key to mastering algebra. It’s not just about memorizing formulas; it’s about understanding the underlying concepts. So, when you see that word 'times', think multiplication. When you see 'more than', think addition. When you see 'less than', think subtraction. And when you see 'divided by', well, you guessed it – division! This problem is a perfect, simple example of how these translations work. We are essentially saying that the value $45 and the value $3a$ describe the exact same quantity of money, just expressed differently. This is why the equals sign ( $=$ ) is so important – it signifies equivalence. The logic is sound, the representation is accurate, and the equation is ready for action!

Beyond the Equation: What's Next?

So, we've successfully written the equation that represents Molly's money situation: $45 = 3a$ . Awesome job, guys! But what can we do with this equation? Well, the most obvious next step is to solve for ' $a$ '. This means finding out exactly how much money Molly's brother has. To do this, we need to isolate ' $a$ ' on one side of the equation. Since ' $a$ ' is currently being multiplied by 3, we need to perform the inverse operation, which is division. We'll divide both sides of the equation by 3 to maintain the balance:

$\frac{45}{3} = \frac{3a}{3}$

This simplifies to:

$15 = a$

So, Molly's brother has $15! Isn't that cool? We started with a word problem, built an equation, and now we've found the answer. This is the power of algebra. But the equation $45 = 3a$ is useful even beyond just finding the brother's money. For instance, if the question was slightly different, say, "Molly has $45, which is 3 times as much as her brother. If their mom gives the brother an extra $5, how much does he have now?" We'd first use our equation to find that the brother has $15, and then add the $5 to get $20. The initial equation serves as the foundation for further calculations. It’s like building blocks; the first block, the equation, allows you to construct more complex answers. This problem also highlights the concept of inverse operations, which is fundamental in solving equations. Multiplication is undone by division, addition by subtraction, and vice versa. Mastering this concept allows you to solve for any unknown variable in an equation, regardless of its complexity. So, while $45 = 3a$ might seem simple, it's a gateway to understanding more advanced mathematical concepts. Keep practicing translating word problems into equations, and you'll find that math becomes much more intuitive and less intimidating. You guys are crushing it!