Math Problem: Calculate Huang's Change

Hey guys! Let's dive into a fun math problem today that involves calculating change. We've got our friend Huang here, who's doing some shopping. He's feeling good and decides to buy three shirts, and guess what? They all cost the same amount. Plus, he grabs a pair of pants that sets him back $12. Now, Huang is smart and pays with a crisp $100 bill. The big question is, which expressions can we use to figure out the change Huang should get back? We need to find three correct ways to represent this! This isn't just about getting the right answer; it's about understanding different ways to set up the problem, which is super important in math. Think about how you'd approach this if you were in Huang's shoes at the checkout. What's the total cost of his items, and how do you subtract that from the money he paid? We'll break down each option and see why it works (or doesn't work!). Get ready to flex those math muscles!

Understanding the Problem: Breaking Down Huang's Purchase

Alright, let's get real about Huang's shopping spree. He's buying three shirts and one pair of pants. The key piece of information is that the shirts each cost the same amount. We don't know the exact price of a single shirt, so we're going to use a variable to represent that unknown cost. In algebra, we often use 'x' for these kinds of unknowns, so let's say the cost of one shirt is '$x'. Since Huang buys three shirts, the total cost for all the shirts combined will be 3 times the cost of one shirt, which we can write as $3x$. Now, the pants are a bit simpler – they cost a fixed amount of $12. So, to find the total cost of everything Huang is buying, we need to add the cost of the shirts and the cost of the pants. That gives us $3x + 12$. This expression, $3x + 12$, represents the total amount of money Huang has to pay for his items. He's paying with a $100 bill, and we want to know how much change he'll get back. Change is calculated by taking the amount paid and subtracting the total cost of the items. So, the change Huang should receive is $100 - (3x + 12)$. This is the core of the problem, and understanding how we got here is crucial. We've identified the cost of each item type, calculated the total cost of the shirts, added the cost of the pants to get the grand total, and then set up the subtraction for the change. Pretty straightforward, right? But the cool part is that math often gives us multiple ways to express the same idea, and that's what we'll explore next.

Evaluating the Options: Which Expressions Work?

Now for the fun part, guys – let's look at the expressions provided and see which ones correctly represent the change Huang should receive. Remember, we figured out that the total cost of the items is $3x + 12$, and the change is $100 - (3x + 12)$. We need to find three expressions that are mathematically equivalent to this. Let's break them down one by one.

• Option 1: $100-(12(3x))$ Let's analyze this one. It seems to suggest that the total cost is $12(3x)$. This doesn't make sense because $12 is the cost of the pants, and $3x$ is the total cost of the shirts. Multiplying these two together doesn't represent the combined cost of the shirts and pants. This expression implies Huang is paying $100 minus the product of the pants' cost and the shirts' total cost, which is totally off. So, this expression is incorrect.

• Option 2: $100-(3x+12)$ Look at this one! This expression takes the $100 bill and subtracts the sum of $3x$ (the total cost of the shirts) and $12$ (the cost of the pants). This is exactly what we derived as the correct way to calculate the change: $100 - (total cost)$. Since the total cost is indeed $3x + 12$, this expression is a perfect match and therefore correct.

• Option 3: $(100-12)-3x$ Let's think about this. This expression first subtracts the cost of the pants ($12$) from the $100 bill, leaving $100 - 12$. Then, it subtracts the cost of one shirt ($3x$ is the cost of three shirts, so this must be a typo in the original and should be $3$ or $x$). Assuming it means subtracting $3x$ from the remaining amount, it would be $(100 - 12) - 3x$. This is like saying, "I paid with $100, I used $12 for pants, so I have $88 left. Now I need to subtract the cost of the shirts from that $88." This is a valid way to calculate the change because it breaks down the subtraction sequentially. So, this expression is correct (assuming the intention was to subtract the total cost of the shirts).

• Option 4: $100-3x-12$ This expression starts with $100$ and then subtracts $3x$ (the cost of the shirts) and then subtracts $12$ (the cost of the pants). This is equivalent to $100 - (3x + 12)$ because of the distributive property of negation. When you subtract a sum, it's the same as subtracting each term individually. So, $100 - 3x - 12$ is indeed the same as $100 - (3x + 12)$. This is another correct way to represent the change.

• Option 5: $100-12-x-x-x$ This expression takes the $100$ and subtracts the $12$ for the pants. Then, it subtracts $x$, $x$, and $x$ individually. Since $x$ is the cost of one shirt, subtracting it three times is the same as subtracting $3x$. So, $100 - 12 - x - x - x$ is mathematically equivalent to $100 - 12 - 3x$, which is also equivalent to $100 - (3x + 12)$. This is a very explicit way of showing the subtraction for each item, and it is absolutely correct.

• Option 6: $100 - (12 + 3x)$ This is just a reordering of the terms inside the parentheses compared to option 2. Since addition is commutative ($3x + 12$ is the same as $12 + 3x$), this expression is also mathematically equivalent to $100 - (3x + 12)$. Therefore, this expression is correct.

The Three Correct Expressions

Based on our analysis, we need to select three options that correctly represent the change Huang should receive. The correct expressions are:

1. $100-(3x+12)$: This directly subtracts the total cost ($3x+12$) from the amount paid ($100$). This is the most straightforward representation.
2. $100-3x-12$: This expression subtracts the cost of the shirts and then the cost of the pants from the initial amount. It's equivalent to the first one due to the properties of subtraction.
3. $(100-12)-3x$: This represents subtracting the cost of the pants first, then subtracting the total cost of the shirts from the remaining amount. It shows a step-by-step reduction of the initial payment.

We could also consider $100 - (12 + 3x)$ and $100 - 12 - x - x - x$ as correct, depending on the exact options presented in the original problem. However, sticking to the most common algebraic representations, the first three are usually the ones focused on in these types of problems. The prompt asks for three options. Let's assume the options presented were:

• $100-(12(3x))$
• $100-(3x+12)$
• $(100-12)-3x$
• $100-3x-12$
• $100-12-x-x-x$
• $100 - (12 + 3x)$

Given these, the three most distinct and correct options are:

• $100-(3x+12)$
• $(100-12)-3x$
• $100-3x-12$

These three expressions all accurately model the change Huang should receive, offering different but mathematically sound ways to arrive at the solution. It’s all about understanding how these algebraic expressions translate real-world scenarios into mathematical form, guys! Keep practicing, and you'll be a math whiz in no time.