Workshop Earnings: Calculating Days Worked & Total Pay

Hey guys! Ever wondered how to figure out a problem where someone works at two different places with different pay rates? It might sound tricky, but we're going to break down a classic math problem step-by-step. Let's dive into a real-world scenario and learn how to solve it together. This guide is designed to help you understand the underlying concepts and apply them to similar situations. So, let's put on our thinking caps and get started!

Understanding the Problem

Let's start by really understanding the problem. A worker has worked in two workshops for a total of 30 days. At the first workshop, they earned 30 euros per day, and at the second workshop, they earned 35 euros per day. The worker's total earnings were 990 euros. The main question here is: How many days did the worker work in each workshop? This type of problem falls under the category of linear equations, and we're going to use some algebra to crack it. The key here is to identify the unknowns, set up the equations, and then solve them systematically.

Before we jump into the solution, let's highlight why understanding the problem is crucial. Many people make the mistake of skimming through the details and immediately trying to apply formulas. However, a solid grasp of the problem's context ensures that we're using the right approach and interpreting the results correctly. In our case, we need to know the total days worked, the daily wages at each workshop, and the total earnings. Missing any of these details can lead to an incorrect solution. So, take your time to read the problem carefully and identify all the pieces of information. Doing so will make the rest of the process much smoother. Remember, math isn't just about plugging numbers into equations; it's about understanding the story behind the numbers!

Setting Up the Equations

Alright, let's get to the fun part: setting up the equations! This is where we translate the word problem into mathematical expressions. First, we need to define our variables. Let's say 'x' represents the number of days the worker worked in the first workshop and 'y' represents the number of days worked in the second workshop. We know two crucial pieces of information that we can turn into equations:

1. The total number of days worked is 30. So, we can write this as: x + y = 30
2. The total earnings are 990 euros. The worker earned 30 euros per day in the first workshop (30x) and 35 euros per day in the second workshop (35y). So, the equation becomes: 30x + 35y = 990

Now we have a system of two linear equations with two variables. This is perfect because we can use various methods, such as substitution or elimination, to solve for 'x' and 'y'. Think of these equations as a puzzle. Each equation gives us a different piece of the puzzle, and when we solve them together, we get the complete picture. The first equation tells us about the relationship between the days worked in each workshop, while the second equation tells us about the relationship between the days worked and the total earnings. By combining these two pieces of information, we can find the exact number of days the worker spent at each location. Remember, setting up the equations correctly is half the battle. Once you have the equations, the rest is just a matter of applying your algebraic skills. So, let's move on to solving these equations and finding out the values of 'x' and 'y'!

Solving the Equations

Okay, guys, let's solve these equations and find out how many days the worker spent at each workshop! We've got two equations:

1. x + y = 30
2. 30x + 35y = 990

Let's use the substitution method. From equation (1), we can express 'y' in terms of 'x':

y = 30 - x

Now, substitute this expression for 'y' into equation (2):

30x + 35(30 - x) = 990

Next, we simplify and solve for 'x':

30x + 1050 - 35x = 990 -5x = -60 x = 12

Great! We've found that x = 12, which means the worker worked 12 days in the first workshop. Now we can plug this value back into the equation y = 30 - x to find 'y':

y = 30 - 12 y = 18

So, the worker worked 18 days in the second workshop. To make sure we're on the right track, it's always a good idea to verify our solution. We can plug the values of 'x' and 'y' back into both original equations. For equation (1), 12 + 18 = 30, which is correct. For equation (2), 30(12) + 35(18) = 360 + 630 = 990, which also checks out. This confirms that our solution is correct. Solving equations like these involves careful manipulation and a bit of algebraic skill. But with practice, you'll get the hang of it. The key is to take it step by step, simplify each expression, and double-check your work along the way. Now that we've solved for 'x' and 'y', we have the answer to our original question: the number of days the worker spent at each workshop.

Checking the Solution

Alright, before we declare victory, let's check our solution to make absolutely sure we've nailed it. We found that the worker worked 12 days in the first workshop and 18 days in the second workshop. To verify this, we need to plug these values back into our original equations and see if they hold true.

Our equations were:

1. x + y = 30
2. 30x + 35y = 990

Let's substitute x = 12 and y = 18 into these equations:

For equation (1):

12 + 18 = 30 30 = 30 (This checks out!)

For equation (2):

30(12) + 35(18) = 990 360 + 630 = 990 990 = 990 (This also checks out!)

Since both equations are satisfied, we can confidently say that our solution is correct. The worker indeed worked 12 days in the first workshop and 18 days in the second workshop. Checking your solution is a crucial step in problem-solving, especially in math. It's like having a safety net that catches any potential errors. By plugging the values back into the original equations, you can quickly identify if you've made a mistake in your calculations. This not only ensures accuracy but also reinforces your understanding of the problem and the solution process. So, never skip this step! It's the final piece of the puzzle that gives you the confidence to say, "I've got this!"

Real-World Applications

Okay, so we've solved the problem, but let's think about real-world applications. Why is this kind of math useful in everyday life? Well, this type of problem-solving comes up more often than you might think! For instance, imagine you're managing a project with a budget and different team members working at different rates. Knowing how to set up and solve these equations can help you allocate resources effectively and ensure you stay within budget. Or, think about investing money in different accounts with varying interest rates. By using similar mathematical principles, you can calculate how much to invest in each account to reach your financial goals.

These problems also help in logistics and supply chain management, where you might need to optimize transportation costs with different shipping rates or delivery times. In the realm of nutrition and dietetics, you might use similar equations to plan meals with specific nutritional targets, balancing different food groups and calorie counts. The skills you learn in solving these types of problems extend far beyond the classroom. They teach you how to break down complex situations into manageable parts, identify key variables, and make informed decisions based on quantitative data. So, while it might seem like a simple word problem, the underlying principles are powerful tools that can be applied in a variety of fields and scenarios. Keep practicing, and you'll find these skills becoming second nature!

Conclusion

So, there you have it! We've successfully tackled a problem involving a worker's earnings in two workshops. By understanding the problem, setting up the equations, solving them systematically, and checking our solution, we've found that the worker worked 12 days in the first workshop and 18 days in the second workshop. This wasn't just about getting the right answer; it was about understanding the process and applying mathematical concepts to real-world scenarios.

Remember, guys, math isn't just a subject you learn in school; it's a powerful tool that can help you solve problems and make better decisions in all aspects of life. Whether you're managing your finances, planning a project, or even just trying to figure out the best deal at the grocery store, the skills you've learned here will come in handy. Keep practicing, keep asking questions, and most importantly, keep exploring the world of mathematics. The more you engage with it, the more you'll discover its beauty and its usefulness. And who knows? Maybe the next challenging problem you solve will lead to your next great achievement. So, keep your thinking caps on and keep solving!