Garden Surface Calculation Problem

Hey everyone! Today, we're diving into a cool math problem that involves calculating the total area of the city gardens. Imagine you're in charge of making sure all those beautiful green spaces get the right amount of water. Let's break down this problem step by step so you can easily understand how to solve it. This is a fun, practical application of fractions and basic algebra, so stick with me!

Understanding the Problem

Okay, so here’s what we know: The folks in charge of watering the city gardens completed $\frac{3}{5}$ of the total area in the first shift. In the afternoon shift, they watered the remaining part, which covers $1,248 \,\text{m}^2$. Our mission is to find out what the total area of the city gardens is. Basically, we need to figure out what whole area the $\frac{3}{5}$ and $1,248 \,\text{m}^2$ make up together. This is like having a pizza, where someone ate a few slices, and we need to figure out how big the whole pizza was to begin with. To kick things off, let's define our terms clearly and set up the foundation for solving this problem. First, we acknowledge that the total area is what we're trying to find—our unknown. The fraction of the area watered in the morning is $\frac{3}{5}$, a clear piece of information. And the area watered in the afternoon is $1,248 \,\text{m}^2$. The goal here is not just to find a number, but to understand how different parts of a whole relate to each other, so that you can apply these principles to other similar situations. For instance, you might use similar math to plan how much fertilizer you need for a garden or to calculate the amount of paint needed for a project.

Setting Up the Equation

Alright, let's get to the math part! To solve this, we need to set up an equation. Let's use 'x' to represent the total area of the gardens. We know that the first shift covered $\frac{3}{5}$ of x, which can be written as $\frac{3}{5}x$. The remaining area that was watered in the afternoon is $1,248 \,\text{m}^2$. So, the equation looks like this:

$$\frac{3}{5}x + 1248 = x$$

This equation tells us that the fraction of the garden watered in the morning plus the area watered in the afternoon equals the total area of the garden. Think of it like this: if you add the portion of work done in the morning to the portion done in the afternoon, you get the entire work completed. To make this equation more approachable, let's walk through each component step by step. The term $\frac{3}{5}x$ represents a fraction of the total area, showing that not all of it was watered in the morning. The addition of $1,248 \,\text{m}^2$ accounts for the rest of the garden that was watered later. Together, these two parts make up the full area, represented by 'x'. Breaking down the equation like this helps you visualize the relationship between the different parts and how they contribute to the whole, which is essential for understanding and solving the problem effectively.

Solving for 'x'

Okay, now it's time to solve for 'x'. Here’s how we can do it:

1. Get all the 'x' terms on one side:

   Subtract $\frac{3}{5}x$ from both sides of the equation:

   $$1248 = x - \frac{3}{5}x$$

2. Simplify the right side:

   To subtract the fractions, we need a common denominator. We can rewrite 'x' as $\frac{5}{5}x$:

   $$1248 = \frac{5}{5}x - \frac{3}{5}x$$

   $$1248 = \frac{2}{5}x$$

3. Isolate 'x':

   To get 'x' by itself, multiply both sides by $\frac{5}{2}$ (which is the reciprocal of $\frac{2}{5}$):

   $$1248 \cdot \frac{5}{2} = x$$

4. Calculate 'x':

   $$x = \frac{1248 \cdot 5}{2}$$

   $$x = \frac{6240}{2}$$

   $$x = 3120$$

So, the total area of the city gardens is $3,120 \,\text{m}^2$. Remember, each step in this process is about isolating the variable to uncover its value. By understanding these steps, you can tackle similar problems with confidence. For instance, if you knew the total area and wanted to find out how much was watered in the morning, you could rearrange the equation to solve for that missing piece. Math is like a puzzle; each piece fits perfectly to reveal the whole picture!

Checking Our Answer

To make sure we got the right answer, let's plug our value of $x = 3120$ back into the original equation and see if it holds true:

$$\frac{3}{5}(3120) + 1248 = 3120$$

First, calculate $\frac{3}{5}$ of $3120$:

$$\frac{3}{5} \cdot 3120 = \frac{3 \cdot 3120}{5} = \frac{9360}{5} = 1872$$

Now, add this to $1248$:

$$1872 + 1248 = 3120$$

So, our equation checks out! This confirms that the total area of the city gardens is indeed $3,120 \,\text{m}^2$. When solving math problems, checking your work is super important to ensure accuracy. By plugging the solution back into the original equation, you can verify that both sides balance out. This not only gives you confidence in your answer but also reinforces your understanding of the problem and the relationships between the different components. Always double-check—it's a great habit that will help you avoid mistakes and master mathematical concepts!

Practical Implications

Understanding and solving this type of problem isn't just about getting the right answer on a math test. It has practical implications in real-world scenarios. For example, city planners need to accurately calculate areas to manage resources effectively, such as water for irrigation, fertilizer for plant health, and manpower for maintenance. Knowing the exact area helps them allocate resources efficiently, ensuring that all parts of the city gardens receive the necessary care.

Moreover, this type of calculation can be applied to various other fields, such as agriculture, landscaping, and construction. Farmers need to calculate the area of their fields to determine how much seed and fertilizer to use. Landscapers need to estimate the area of a garden to plan the layout and purchase the right amount of materials. Construction workers need to calculate areas to estimate the amount of materials needed for building projects.

By mastering these basic mathematical concepts, you can develop valuable problem-solving skills that are applicable in numerous real-life situations. So, keep practicing and exploring different applications of math—it's a tool that will serve you well in many aspects of life!

Conclusion

So, there you have it! The total area of the city gardens is $3,120 \,\text{m}^2$. By breaking down the problem, setting up the equation, and solving for 'x', we were able to find the answer. Remember, the key to solving math problems is to understand each step and practice regularly. With a bit of patience and effort, you can tackle any math challenge that comes your way. Keep up the great work, and happy problem-solving! Remember, math isn't just about numbers; it's about understanding the world around us and making smart decisions based on that understanding.