Percy's College Fund: A Math Problem

Hey guys! Let's dive into a cool math problem about Percy, a college student working two part-time jobs. This isn't just about numbers; it's about seeing how math helps us understand real-life situations. Percy's situation is a classic example of how to solve a system of equations, a fundamental concept in algebra. This article will break down the problem step-by-step, making it super easy to follow. We'll find out how much Percy earns per hour at each job, which helps us appreciate how math can solve practical problems. Get ready to flex those brain muscles with a bit of algebra, and see how Percy manages his college finances. This problem teaches us about systems of equations and how they apply to the real world. Let's see how we can use math to figure out Percy's hourly earnings at both jobs. This is more than just a math problem. It’s a sneak peek into how algebra works in practical scenarios, like managing money or understanding work schedules. Ready to unravel the mystery of Percy's earnings? Let’s get started and uncover the math behind Percy's hustle. This is where we break down the problem and make it easy to understand. We’ll show you how to set up the equations and solve for those unknown hourly rates.

Setting Up the Problem

Okay, so the setup is like this: Percy works at the library and a coffee cart. On Mondays, he works 3 hours at the library and 2 hours at the coffee cart, earning $36.50. On Tuesdays, he works 2 hours at the library and 5 hours at the coffee cart, making $50. The core of the problem involves taking these details and translating them into mathematical equations. Each job contributes to Percy's total earnings, and our mission is to calculate his per-hour earnings at each place. We're going to use variables to represent Percy's hourly rates. Let 'x' be the amount Percy earns per hour at the library, and 'y' be the amount he earns per hour at the coffee cart. Then we can translate the information into equations, such as we can then write the equations. We need to translate the word problem into a set of algebraic equations. Here's how we set up the equations: Monday's earnings: 3x + 2y = 36.50. Tuesday's earnings: 2x + 5y = 50. These equations represent Percy's earnings. Each equation is based on the hours he works and his hourly rates. Understanding how to set up these equations is crucial for solving the problem. The goal is to calculate the 'x' and 'y' values, which represent Percy’s hourly wages. Setting up the problem involves translating the details into algebraic equations. We’re turning words into math symbols. We'll use these equations to find out exactly how much Percy earns per hour at each job. This sets the stage for solving the system of equations. We're now ready to use algebra to find out how much Percy earns per hour at each job. Remember, 'x' is Percy's hourly rate at the library, and 'y' is his rate at the coffee cart.

Solving the System of Equations

Alright, it's time to solve those equations and find out Percy's hourly rates. We have two main methods for solving this: substitution and elimination. I'll show you how to use the elimination method, which is pretty straightforward. The elimination method involves manipulating the equations to eliminate one of the variables. The idea is to make the coefficients of either 'x' or 'y' match in both equations, but with opposite signs. This way, when you add the equations together, one variable cancels out. To do this, let's multiply the first equation (3x + 2y = 36.50) by -2 and the second equation (2x + 5y = 50) by 3. This will help us eliminate 'x'. Multiplying the first equation by -2 gives us -6x - 4y = -73. Multiplying the second equation by 3 gives us 6x + 15y = 150. Now, add these two new equations together: (-6x - 4y) + (6x + 15y) = -73 + 150. This simplifies to 11y = 77. To isolate 'y', divide both sides by 11. y = 7. So, Percy earns $7 per hour at the coffee cart. To find 'x', substitute 'y' back into one of the original equations. Let's use the first one: 3x + 2(7) = 36.50. This simplifies to 3x + 14 = 36.50. Subtract 14 from both sides: 3x = 22.50. Divide both sides by 3: x = 7.50. Thus, Percy earns $7.50 per hour at the library. By using the elimination method, we systematically reduced the equations. This is a clear demonstration of how algebra helps us solve real-world problems. We've found the values of 'x' and 'y', which represent Percy's hourly earnings. These steps illustrate the core of solving a system of equations, a handy tool for many real-life scenarios.

Checking the Solution

Before we wrap things up, let's double-check our answers to make sure they're correct. This is a super important step in any math problem. We'll plug the values of 'x' and 'y' back into the original equations to see if they hold true. Remember, we found that x = 7.50 and y = 7. So, for Monday: 3(7.50) + 2(7) = 22.50 + 14 = 36.50. This checks out perfectly! For Tuesday: 2(7.50) + 5(7) = 15 + 35 = 50. This also checks out, confirming our solution. We can confirm the accuracy of our answers by plugging the values of 'x' and 'y' back into the original equations. This confirms the solution and gives us confidence in our calculations. This step verifies that our solutions are consistent with the problem's details, making our results accurate. It is an important step to ensure the integrity of our calculations. Double-checking our work is a great habit to have when solving any math problem.

Conclusion

So, guys, what have we learned? We've successfully used algebra to figure out Percy's hourly earnings at both his jobs. Percy earns $7.50 per hour at the library and $7 per hour at the coffee cart. This problem perfectly illustrates how systems of equations are used in everyday situations, from managing finances to figuring out work schedules. Using math, we've solved a real-world problem and learned a valuable skill. It's a great example of applying math concepts. Through this exercise, we've demonstrated how to set up equations, solve them using the elimination method, and verify our solutions. By understanding this, you're not just solving a math problem. You're getting a practical skill that can be useful in various situations. It shows how we can use math to solve practical problems and gain insights into everyday scenarios. This process highlights the practical application of algebra, showing how it can be used to solve real-world problems. Remember, math isn’t just about numbers; it’s a powerful tool for understanding and solving problems. Keep practicing, and you'll find that math can be a helpful tool in all sorts of situations. Thanks for joining me! Keep up the great work, and remember that math is more than just equations; it's a way to understand the world around us. Keep on practicing, and you will become proficient in solving math problems.