Tricky Percentage Problems: Which Aquarium Has More Fish?

Hey guys! Are you ready to dive into some tricky percentage problems? We've got a fun one today that involves aquariums and fish. It might seem a little confusing at first, but don't worry, we'll break it down step by step. So, grab your thinking caps, and let's get started!

The Aquarium Challenge

Let's dive straight into our percentage problem! Imagine we have two aquariums: a big one and a small one. Initially, the big aquarium has twice as many fish as the small one. Now, here's where it gets interesting. After some time, the big aquarium's fish population decreases by 25%, while the small aquarium's fish population increases by 1.5 times its original amount. The big question is: which aquarium has more fish now? This is a classic example of a percentage problem that requires careful thinking and a solid understanding of how percentages work. You can't just jump to a conclusion; you need to work through the math to find the correct answer. These kinds of problems are not just about the numbers; they're about understanding the relationships between the quantities involved. This involves setting up the problem correctly, choosing the right operations, and interpreting the results. This is why percentage problems are often used in math education – they help develop critical thinking and problem-solving skills. We need to figure out how the changes in fish populations affect the overall number of fish in each aquarium. To do that, we'll need to use percentages to calculate the increases and decreases accurately. So, let's move on and figure out a clear method to solve this.

Breaking Down the Problem

Okay, guys, to solve this percentage increase and decrease problem, let's break it down into manageable steps. This will help us avoid confusion and make sure we're on the right track. First, we need to represent the initial number of fish in each aquarium using variables. This is a common strategy in algebra, where we use letters to stand for unknown quantities. Let's say the small aquarium initially has 'x' number of fish. Since the big aquarium has twice as many fish, it initially has '2x' fish. Using variables makes it easier to perform calculations and track changes. Now, let's think about the changes that occur. The big aquarium's fish population decreases by 25%. To calculate this decrease, we need to find 25% of '2x' and then subtract it from the original amount. Remember, a percentage is just a fraction out of 100, so 25% is the same as 25/100, or 0.25. On the other hand, the small aquarium's fish population increases by 1.5 times its original amount. This means we need to multiply 'x' by 1.5 and add that to the original number of fish. It's super important to keep track of whether we need to add or subtract when dealing with percentage increases and decreases. Once we've calculated the new number of fish in each aquarium, we can directly compare them to see which one has more. So, are you guys ready to put these steps into action? Let's move on to the calculations and see what we find!

Step-by-Step Calculations

Alright, let's get those calculators ready and crunch some numbers! This is the heart of solving our fishy percentage problem. First, let's tackle the big aquarium. We know it started with '2x' fish, and the population decreased by 25%. To find the decrease, we calculate 25% of 2x, which is 0.25 * 2x = 0.5x. This means the fish population decreased by 0.5x fish. To find the new number of fish in the big aquarium, we subtract this decrease from the original amount: 2x - 0.5x = 1.5x. So, the big aquarium now has 1.5x fish. Remember, it's crucial to understand what each step represents in the context of the problem. We're not just doing math for the sake of it; we're modeling a real-world situation. Now, let's move on to the small aquarium. It started with 'x' fish, and the population increased by 1.5 times. This means we need to add 1.5 times the original amount to the original amount. So, the increase is 1.5 * x = 1.5x. To find the new number of fish, we add this increase to the original number: x + 1.5x = 2.5x. This calculation shows that the small aquarium now has 2.5x fish. Do you guys see how we're using basic arithmetic operations – multiplication, subtraction, and addition – to work with percentages? This is a fundamental skill in many areas of math and everyday life. Now that we have the new fish populations for both aquariums, we can finally compare them and answer the question.

Comparing the Results

Okay, mathletes, we've done the hard work of calculating the new fish populations. Now comes the moment of truth: comparing the results and figuring out which aquarium has more fish. We found that the big aquarium now has 1.5x fish, and the small aquarium has 2.5x fish. To compare these two quantities, we simply look at the coefficients of 'x'. In this case, we're comparing 1.5 and 2.5. It's pretty clear that 2.5 is greater than 1.5. This means that 2.5x is greater than 1.5x. So, what does this tell us in the context of our problem? It tells us that the small aquarium has more fish than the big aquarium after the changes in population. Even though the big aquarium started with twice as many fish, the 25% decrease and the 1.5 times increase in the small aquarium led to the small aquarium having more fish in the end. This kind of result might seem counterintuitive at first, which is what makes percentage problems so interesting. They often challenge our initial assumptions and require us to think carefully about the relationships between quantities. It's important to remember that percentages are relative changes. A 25% decrease from a larger number can be smaller than a 1.5 times increase from a smaller number. This is a crucial concept to grasp when working with percentages. So, are we ready to state our final answer with confidence? Let's do it!

The Final Verdict

Drumroll, please! We've reached the end of our aquarium adventure! After carefully calculating and comparing the fish populations, we've arrived at our final answer. The aquarium with more fish after the changes is… the small aquarium! That's right, even though it started with fewer fish, the 1.5 times increase in its population outweighed the 25% decrease in the big aquarium's population. This problem is a great example of how percentages can sometimes be tricky. It's not enough to just look at the percentage changes; you need to consider the initial values as well. A large percentage decrease from a big number can still result in a larger number than a smaller percentage increase from a small number. Remember, guys, math isn't just about getting the right answer. It's about understanding the process and the logic behind the solution. Breaking down complex problems into smaller, manageable steps is a key skill that will help you not only in math but in many other areas of life. So, what did we learn today? We learned how to tackle percentage increase and decrease problems, how to use variables to represent unknown quantities, and how to compare results to draw conclusions. We also learned that sometimes the obvious answer isn't the correct one, and it's important to think critically and carefully. Great job, everyone! You've successfully navigated this tricky percentage problem. Keep practicing, and you'll become percentage problem-solving pros in no time!