Juan's Sweater Shopping Spree A Math Problem

Hey guys! Ever found yourself in a situation where you have a set amount of money and you're trying to figure out how many of something you can buy? Juan's in that exact spot, and it involves some cozy wool sweaters! Let's dive into this fun little mathematical puzzle and see if we can help Juan figure out how many sweaters he can snag.

Unraveling the Sweater Scenario

Our main keyword here is Juan's sweater purchase. Juan has a budget of 600 soles, and he's eyeing some lovely wool sweaters that all have the same price. Now, here's where it gets interesting. If Juan tries to buy the sweaters at 33 soles each, uh oh, he comes up short – he doesn't have enough money! But, if he decides to buy them at a slightly lower price of 30 soles each, he ends up with some cash to spare. The big question we need to answer is: how many sweaters did Juan actually buy?

To crack this, we need to think about what this information tells us. The fact that he's short on cash at 33 soles per sweater means he can't afford the number of sweaters he'd get at that price. On the flip side, having money left over at 30 soles per sweater means he could buy that many, and maybe even a few more if the price was right. This tells us the number of sweaters Juan bought is somewhere between the maximum he could buy at 33 soles (if he had just a little more money) and the minimum he could buy at 30 soles (if he spent almost all his money).

Setting Up the Mathematical Framework

Let's translate this real-world problem into math. Let's use the variable 'x' to represent the number of sweaters Juan bought. We know two crucial things:

1. 33x > 600 (Because buying 'x' sweaters at 33 soles each costs more than 600 soles)
2. 30x < 600 (Because buying 'x' sweaters at 30 soles each costs less than 600 soles)

These two inequalities are the key to solving our problem. They create a boundary, a range within which the actual number of sweaters Juan bought must fall. Now, let's solve these inequalities to narrow down our possibilities. Solving inequalities is similar to solving equations, but with a slight twist – we need to pay attention to how the inequality sign behaves when we multiply or divide by a negative number (which isn't an issue in this case, thankfully!).

Solving the Inequalities Step-by-Step

First, let's tackle the inequality 33x > 600. To isolate 'x', we'll divide both sides of the inequality by 33:

x > 600 / 33 x > 18.18 (approximately)

This tells us that Juan bought more than 18.18 sweaters. Since you can't buy a fraction of a sweater, this means Juan bought at least 19 sweaters.

Now, let's look at the second inequality: 30x < 600. Again, we'll isolate 'x' by dividing both sides by 30:

x < 600 / 30 x < 20

This tells us that Juan bought fewer than 20 sweaters.

Finding the Sweet Spot

Okay, so we know Juan bought more than 18.18 sweaters (meaning at least 19) and fewer than 20 sweaters. But wait a minute… This seems like a contradiction! How can he buy more than 18 sweaters and less than 20 sweaters at the same time? This is where we need to take a closer look at the original problem statement and see if we missed any crucial information.

Hold on! We've made a slight misinterpretation. The problem states that if Juan buys at 33 soles, he lacks money, and if he buys at 30 soles, he has money left over. This doesn't necessarily mean he can't afford any number of sweaters at 33 soles or that he'll always have money left over at 30 soles. It just sets the boundaries for the exact number of sweaters he bought and the exact price he paid.

Let's reframe our inequalities to reflect this more precisely. Let 'C' be the exact cost per sweater and 'x' be the number of sweaters Juan bought.

• The fact that 33 soles is too expensive means: 33 > C
• The fact that 30 soles leaves him with money means: 30 < C

So, the price per sweater 'C' is somewhere between 30 and 33 soles. We also know that the total cost (Cx) must be less than or equal to 600 soles (since that's his budget) and that buying one more sweater at 30 soles would exceed his budget.

Refining Our Approach with a New Insight

The key insight here is that the total amount Juan spent on the sweaters must be a whole number (since we're dealing with money). This means that when we multiply the number of sweaters ('x') by the price per sweater ('C'), we should get a whole number that is less than or equal to 600. This gives us a new avenue to explore: trying out whole number values for 'x' within the range we've established (19 seems like a good place to start) and seeing if we can find a price 'C' between 30 and 33 that works.

Let's try x = 19 sweaters. If Juan bought 19 sweaters, the price per sweater would be 600 / 19 = 31.58 soles (approximately). This price falls neatly between 30 and 33 soles! So, it looks like we've found our solution.

Confirming the Solution

To be absolutely sure, let's check if this solution makes sense in the context of the original problem.

• If Juan bought 19 sweaters at 31.58 soles each, the total cost is 19 * 31.58 = 600.02 soles. This is practically 600 soles, which fits his budget perfectly.
• If the price was 33 soles, 19 sweaters would cost 19 * 33 = 627 soles, which is more than he has.
• If the price was 30 soles, 19 sweaters would cost 19 * 30 = 570 soles, leaving him with some money left over.

Our solution checks out! Juan bought 19 wool sweaters.

The Takeaway: Problem-Solving Power!

This problem is a fantastic example of how math can be used to solve real-world scenarios. We started with a word problem, translated it into mathematical inequalities, and then used logical reasoning and a bit of trial-and-error to arrive at the answer. Remember guys, even if a problem seems tricky at first, breaking it down into smaller parts and using the right tools can lead you to the solution! The key here was not just blindly applying formulas, but really thinking about what the problem was telling us and using that information to guide our approach. And remember the main keyword: Juan's sweater purchase which led us to the final answer of 19 sweaters.