Provision Duration For 15 People: Calculation & Solution

Hey guys! Let's dive into a classic math problem involving provisions and people. We've got a situation where a warehouse has enough supplies for a certain number of people for a specific duration, and we need to figure out how long those supplies will last if the number of people changes. This type of problem is a great example of an inverse proportion, and understanding it can help you solve similar real-world scenarios.

Understanding the Problem

So, our core question revolves around understanding how long provisions will last when the number of consumers changes. This is a typical problem involving inverse proportionality. In simpler terms, if you have more people consuming the supplies, the provisions will last for a shorter time, and vice versa. To really grasp this, let's break down the problem step-by-step and make sure we're all on the same page. Think of it like this: you have a fixed amount of food, and the more mouths you have to feed, the quicker the food runs out. This concept is super important in various fields, from logistics and supply chain management to even everyday life situations like planning a camping trip.

When tackling problems like these, it's crucial to first identify the relationship between the quantities involved. In our case, the quantities are the number of people and the number of days the provisions will last. It's pretty intuitive that these two are inversely proportional. This means that as one quantity increases, the other decreases, assuming the total amount of provisions remains constant. Imagine you're planning a hike, and you've packed a certain amount of trail mix. If more friends join you, that same amount of trail mix won't last as long, right? This is the core concept of inverse proportionality in action. So, before we even start crunching numbers, understanding this relationship is key to setting up the problem correctly and getting to the right solution.

Now, let's really nail down what we know and what we need to find out. We're given that the provisions are enough for 12 people for 30 days. This is our baseline, our starting point. We also know that the number of people is increasing to 15. This is the change we need to account for. And what we're trying to figure out, the big question we're solving, is: how many days will the provisions last for these 15 people? Identifying these knowns and unknowns is like laying the foundation for a building. It helps us organize our thoughts and see the problem clearly. Once we have a solid understanding of what we're working with and what we're looking for, the actual calculation becomes much more straightforward. So, let's keep these pieces in mind as we move forward and explore the solution.

Setting Up the Proportion

Alright, let's get into the math! Since we know this is an inverse proportion, we need to set up our equation carefully. With inverse proportions, we multiply the related quantities and set them equal to each other. This might sound a bit technical, but it's actually a pretty simple concept once you see it in action. Think of it this way: the total amount of "consumption" (people multiplied by days) stays the same, regardless of how many people there are. This constant consumption is what allows us to equate the two scenarios. So, let's break down how we translate this idea into a mathematical equation.

Our initial scenario gives us 12 people lasting for 30 days. This means the total "provision units" can be represented as 12 multiplied by 30. We can write this as 12 * 30. This calculation gives us a sort of "total provision capacity" in terms of person-days. It's like saying we have enough supplies for 360 person-days. Now, this number is crucial because it remains constant. No matter how many people we have, the total amount of provisions doesn't change. This is the foundation of our inverse proportion. Understanding this constant total is key to setting up the equation correctly. So, let's keep this "360 person-days" in mind as we move to the next part of setting up the problem.

Now, let's bring in the changed scenario: 15 people. We don't know how many days the provisions will last, so let's call that "x". Our equation will then be 15 * x. Remember, the total "provision units" must remain the same. So, we can set up the equation: 12 * 30 = 15 * x. This equation is the heart of our solution. It beautifully captures the inverse relationship between the number of people and the duration the provisions will last. By setting the two scenarios equal to each other, we're essentially saying that the total amount of provisions consumed is the same in both cases. Now that we have this equation, we're just one step away from finding our answer. All that's left is to solve for x, which will tell us exactly how many days the provisions will last for 15 people. Let's dive into that next and crack this problem wide open!

Solving for the Unknown

Okay, guys, it's time to put our algebra hats on and solve for "x"! We've got our equation: 12 * 30 = 15 * x. The goal here is to isolate x on one side of the equation so we can figure out its value. Don't worry; it's a pretty straightforward process. We just need to follow a few simple steps, and we'll have our answer in no time. Think of it like unwrapping a present – each step gets us closer to the solution hidden inside. So, let's get started and unwrap this problem!

First, let's simplify the left side of the equation. We need to multiply 12 by 30. If you do the math, 12 * 30 equals 360. So, our equation now looks like this: 360 = 15 * x. We've taken a little step forward, and the equation is looking a bit cleaner already. This is often the case in math – breaking down a problem into smaller, more manageable steps makes it much easier to solve. Now that we've simplified one side, we're one step closer to isolating x and finding our answer. So, let's keep going and tackle the next part of the process!

Now, we need to get that "x" all by itself. It's currently being multiplied by 15, so to undo that, we need to do the opposite operation: division. We're going to divide both sides of the equation by 15. This is a crucial step because it keeps the equation balanced. Whatever we do to one side, we have to do to the other to maintain the equality. So, when we divide both sides by 15, we get: 360 / 15 = x. This division is the key to unlocking the value of x. It's like using a special tool to open a lock. Once we perform this division, we'll know exactly how many days the provisions will last for 15 people. So, let's do that final calculation and reveal the solution!

Time for the final calculation! 360 divided by 15 equals 24. So, we've found our answer: x = 24. This means that the provisions will last for 24 days if they are sold for 15 people. Woohoo! We've successfully solved the problem. This final step is like the grand finale of a performance – it's the moment when everything comes together and the answer is revealed. So, let's take a moment to appreciate the journey we've been on, from understanding the problem to setting up the equation, and finally, arriving at this solution.

Final Answer and Interpretation

So, after all that math, we've arrived at the answer: The provisions will last for 24 days if they are sold for 15 people. That's it! We cracked the code! But it's not just about getting a number; it's also about understanding what that number means in the context of the problem. Think of it like this: the answer is the destination, but the interpretation is the treasure we find there. So, let's take a moment to reflect on what our answer tells us about the situation.

Our answer of 24 days is a direct result of the inverse relationship we discussed earlier. Remember, because we increased the number of people consuming the provisions, the duration those provisions last has decreased. This makes perfect sense, right? If more people are eating the same amount of food, it's going to run out faster. This concept is fundamental to understanding inverse proportions and how they work in the real world. So, by not only solving for the answer but also interpreting it, we're solidifying our understanding of the underlying principles.

To put it another way, we started with provisions that lasted 30 days for 12 people. By increasing the number of people to 15, we saw the duration decrease to 24 days. This change highlights the practical implications of inverse proportionality. It's not just an abstract mathematical concept; it's something that affects our everyday lives, from planning meals to managing resources. So, understanding this relationship can be incredibly valuable in various situations. And with that, we've not only solved the problem but also gained a deeper understanding of the principles at play. Great job, guys!

In conclusion, we've successfully navigated a problem involving inverse proportions. We've seen how to identify the relationship between quantities, set up the appropriate equation, and solve for the unknown. Remember, these skills are transferable to many other scenarios, so keep practicing and you'll become a math whiz in no time!