Book Purchase Problem: Hardcover Vs Paperback

Hey guys! Let's dive into a fun math problem today that involves figuring out how many hardcover and paperback books someone bought. This is a classic type of problem that you might encounter in algebra, and it's a great way to practice setting up and solving systems of equations. We'll break it down step by step, so don't worry if it seems a little tricky at first. Let’s get started and unravel this book-buying mystery!

Understanding the Problem

So, here's the scenario: Gillian went to a library book sale and purchased a total of 25 books. That's a good haul! Now, some of these books were hardcover, and those cost $1.50 each. The rest were paperback, and they were a steal at just $0.50 each. Gillian spent a total of $26.50. The big question we need to answer is: how many of each type of book did she buy? This kind of problem is super practical because it helps us think about how different variables (like the number of books and their prices) relate to each other. Understanding the problem is the first key step to solving it.

First and foremost, Identifying the knowns and unknowns is paramount. We know the total number of books, the price of each type of book, and the total amount spent. What we don't know, and what we need to find out, is the number of hardcover books and the number of paperback books. To tackle this, we'll use a system of equations, which is a fancy way of saying we'll use two equations with two unknowns. This is a standard technique in algebra for solving problems like this, and it's something you'll use again and again in math and science. Setting up these equations carefully is crucial. We need to make sure they accurately represent the information given in the problem. A small mistake here can throw off the whole solution, so it's worth taking your time and double-checking your work. Once we have the equations set up, we can use different methods to solve them, such as substitution or elimination. We'll walk through one of these methods in detail so you can see how it works. Remember, the goal is to find the values that satisfy both equations simultaneously – that is, the values that make both equations true at the same time. Solving word problems like this is not just about finding the right answer; it's also about developing your problem-solving skills. These are skills that will be valuable in all sorts of situations, both in and out of school. So, let's get started and see how we can crack this problem!

Setting Up the Equations

Alright, let's translate this word problem into some math equations. This is where the fun really begins! We're going to use variables to represent the unknowns, and then we'll create equations that show the relationships between those variables. So, let's use 'h' to represent the number of hardcover books Gillian bought and 'p' to represent the number of paperback books. Makes sense, right? Now, we know two key things from the problem. First, Gillian bought a total of 25 books. That means the number of hardcover books plus the number of paperback books must equal 25. We can write that as an equation: h + p = 25. This first equation is all about the quantity of books.

Next, we need to think about the cost. Each hardcover book costs $1.50, so the total cost of hardcover books is 1.50h. Each paperback book costs $0.50, so the total cost of paperback books is 0.50p. Gillian spent a total of $26.50, so we can write another equation: 1.50h + 0.50p = 26.50. This second equation represents the total cost of the books. Now we have a system of two equations:

1. h + p = 25
2. 1. 50h + 0.50p = 26.50

This is where the magic happens! We have two equations with two unknowns, and that means we can solve for h and p. There are a couple of ways we can do this, but one common method is called substitution. The idea behind substitution is to solve one equation for one variable and then substitute that expression into the other equation. This will give us a single equation with just one variable, which we can solve easily. Another method is elimination, where we manipulate the equations to eliminate one variable. We'll focus on substitution for this problem, but it's good to know that there are different tools in your math toolbox! Setting up the equations correctly is half the battle. Once you have the equations, the rest is just algebra. But if you mess up the equations, you're going to get the wrong answer. So, always take your time and double-check your work.

Solving the System of Equations

Okay, guys, we've got our system of equations set up, and now it's time to solve them! We're going to use the substitution method, which is a super handy technique for this kind of problem. Remember, our equations are:

1. h + p = 25
2. 1. 50h + 0.50p = 26.50

The first step in substitution is to solve one of the equations for one of the variables. Looking at equation 1 (h + p = 25), it seems easiest to solve for p. We can do this by subtracting h from both sides: p = 25 - h

Great! Now we have an expression for p in terms of h. This is the key to the substitution method. The next step is to substitute this expression for p into the other equation (equation 2). So, wherever we see p in equation 2, we're going to replace it with (25 - h):

1. 50h + 0.50(25 - h) = 26.50

Now we have a single equation with just one variable, h! This is exactly what we wanted. Now we can solve for h. First, we need to distribute the 0.50:

1. 50h + 12.5 - 0.50h = 26.50

Next, we combine like terms:

1h + 12.5 = 26.50

Now, subtract 12.5 from both sides:

h = 14

Awesome! We've found that h = 14. That means Gillian bought 14 hardcover books. But we're not done yet! We still need to find the number of paperback books, p. This is where we use our expression for p that we found earlier: p = 25 - h. We know h = 14, so we can substitute that in:

p = 25 - 14

p = 11

So, Gillian bought 11 paperback books. We've solved the system of equations! We found that h = 14 and p = 11. But it's always a good idea to check your answer to make sure it makes sense in the context of the original problem. We'll do that in the next section.

Checking the Solution

Alright, let's make sure our solution makes sense! We found that Gillian bought 14 hardcover books and 11 paperback books. To check our answer, we need to see if these numbers satisfy both of our original equations. First, let's check the total number of books:

h + p = 25

14 + 11 = 25

25 = 25

Great! The numbers add up to the correct total number of books. Now, let's check the total cost:

1. 50h + 0.50p = 26.50

2. 50(14) + 0.50(11) = 26.50

21 + 5.50 = 26.50

3. 50 = 26.50

Perfect! The total cost is also correct. Since our solution satisfies both equations, we can be confident that we've found the right answer. Gillian bought 14 hardcover books and 11 paperback books. Checking your solution is a crucial step in any math problem. It helps you catch any mistakes you might have made along the way. It's also a good way to build your confidence in your answer. If you can check your solution and see that it makes sense, you can be sure that you've done the problem correctly.

Real-World Applications

So, why is solving problems like this important? Well, it turns out that systems of equations have all sorts of real-world applications! They're used in everything from engineering to economics to computer science. Anytime you have multiple variables and multiple constraints, you can use a system of equations to model the situation and find a solution. For example, engineers might use systems of equations to design bridges or buildings. They need to make sure that the structure can support a certain weight and withstand certain forces. This involves solving equations that relate the dimensions of the structure, the materials used, and the loads it will carry. In economics, systems of equations are used to model supply and demand. Economists might want to know how the price of a product will affect the quantity that consumers are willing to buy and the quantity that producers are willing to sell. This involves solving equations that relate price, quantity, and other factors like income and production costs.

In computer science, systems of equations are used in computer graphics and simulations. For example, if you're creating a video game, you might need to simulate the movement of objects in the game world. This involves solving equations that describe the forces acting on the objects and their resulting motion. Even in everyday life, you might encounter situations where systems of equations are useful. For example, if you're planning a road trip and you want to figure out how much gas you'll need, you might need to consider factors like the distance you'll be driving, the fuel efficiency of your car, and the price of gas. This can be modeled as a system of equations. So, the next time you're faced with a problem that involves multiple variables and constraints, remember that systems of equations can be a powerful tool for finding a solution! And remember, practice makes perfect. The more you work with systems of equations, the more comfortable you'll become with them, and the easier it will be to apply them to real-world situations.

Conclusion

Alright guys, we made it! We successfully solved the book purchase problem using a system of equations. We broke down the problem, set up the equations, solved for the unknowns, and checked our solution. That's a pretty impressive feat! We saw how to translate a word problem into mathematical equations, which is a super important skill. We used the substitution method to solve the system, and we saw how to check our answer to make sure it made sense. And we even talked about some real-world applications of systems of equations. So, what are the key takeaways from this problem? First, understanding the problem is crucial. Read the problem carefully, identify the knowns and unknowns, and think about the relationships between them. Second, setting up the equations correctly is half the battle. Make sure your equations accurately represent the information given in the problem. Third, there are different methods for solving systems of equations, so choose the one that seems easiest for the particular problem. Fourth, always check your solution to make sure it makes sense. And finally, remember that practice makes perfect! The more you work with these kinds of problems, the better you'll become at solving them.

I hope this walkthrough was helpful! Remember, math is like any other skill – the more you practice, the better you'll get. So, keep practicing, keep asking questions, and don't be afraid to tackle challenging problems. You've got this! And who knows, maybe you'll even use systems of equations to solve a real-world problem someday. Keep up the great work, and I'll see you next time! Now, go grab a book (maybe even a hardcover and a paperback!) and relax – you've earned it!