Solving For Sandal Price Using Substitution Haidar's Purchase Problem

Hey guys! Let's break down this math problem together. It's about Haidar buying sandals and hats, and we need to figure out how much a pair of sandals costs. Don't worry, we'll use a cool method called substitution to solve it. So, grab your thinking caps, and let's dive in!

Understanding the Problem

Okay, so the core of the problem is this: Haidar spent Rp345,000.00 on 5 pairs of sandals and 4 hats. We also know that the price of 2 pairs of sandals is the same as the price of 3 hats. Our mission, should we choose to accept it, is to find the price of one pair of sandals. To make things easier, we're going to use something called the substitution method. This method is super handy for solving problems with two unknowns – in our case, the price of sandals and the price of hats. We'll start by turning this word problem into mathematical equations. Let's use 'x' to represent the price of a pair of sandals and 'y' to represent the price of a hat. This makes it easier to manipulate the values and find our answers. So, the first piece of information we have is about the total cost. Haidar bought 5 pairs of sandals, which would cost 5 times the price of one pair (5x), and 4 hats, which would cost 4 times the price of one hat (4y). The total cost of these items is Rp345,000.00. We can write this as an equation: 5x + 4y = 345,000. This equation is the first key to unlocking our problem. It tells us the relationship between the number of sandals and hats Haidar bought and the total amount he spent. Next, we have another crucial piece of information: the price of 2 pairs of sandals is equal to the price of 3 hats. This gives us another equation. If one pair of sandals costs 'x', then 2 pairs cost 2x. Similarly, if one hat costs 'y', then 3 hats cost 3y. The problem states that these two amounts are equal, so we can write this as: 2x = 3y. This equation is super important because it gives us a direct relationship between the price of sandals and the price of hats. Now that we have these two equations, we can use them together to solve for the unknowns. The beauty of math is that it provides us with the tools to solve complex problems step by step. So, we've transformed the word problem into mathematical equations, and we're one step closer to finding the price of a pair of sandals. In the next section, we'll dive into the substitution method and see how we can use these equations to find our answer.

Setting Up the Equations

Alright, let's get those equations set up! As we discussed, the main goal is to translate the problem's information into mathematical language. This involves identifying the unknowns (the price of sandals and hats) and expressing the given relationships as equations. This step is crucial because it transforms a word problem into a form that we can solve using algebraic techniques. First, let’s define our variables. We'll use 'x' for the price of a pair of sandals and 'y' for the price of a hat. This is a standard practice in algebra – using letters to represent unknown quantities. It makes the equations cleaner and easier to work with. Remember, the first piece of information we have is that Haidar bought 5 pairs of sandals and 4 hats for a total of Rp345,000.00. This can be written as an equation. The cost of 5 pairs of sandals is 5 times the price of one pair, which is 5x. Similarly, the cost of 4 hats is 4 times the price of one hat, which is 4y. The total cost is the sum of these two amounts, so we have: 5x + 4y = 345,000. This is our first equation, and it represents the total amount Haidar spent. It's a linear equation with two variables, and it tells us the relationship between the number of sandals and hats purchased and the total cost. Now, let's look at the second piece of information: the price of 2 pairs of sandals is equal to the price of 3 hats. This can also be written as an equation. The cost of 2 pairs of sandals is 2x, and the cost of 3 hats is 3y. Since these two amounts are equal, we can write: 2x = 3y. This is our second equation, and it gives us a direct relationship between the price of sandals and the price of hats. It's a crucial piece of information because it allows us to relate the two unknowns. So, we now have two equations: 5x + 4y = 345,000 and 2x = 3y. These equations form a system of linear equations, and we can use them together to solve for x and y. The next step is to choose a method to solve this system. In this case, we're going to use the substitution method, which involves solving one equation for one variable and substituting that expression into the other equation. This method is particularly useful when one equation can be easily solved for one variable in terms of the other. Before we jump into the substitution, let's recap what we've done. We've successfully translated the word problem into two mathematical equations, each representing a piece of information from the problem. We've defined our variables clearly, and we've written the equations in a standard algebraic form. Now, we're ready to use the substitution method to find the price of a pair of sandals.

Using the Substitution Method

Okay, time to put on our substitution hats! The substitution method is a clever way to solve systems of equations. The basic idea is to isolate one variable in one equation and then plug that expression into the other equation. This way, we reduce the problem to a single equation with one variable, which is much easier to solve. In our case, we have two equations: 5x + 4y = 345,000 and 2x = 3y. Looking at these, the second equation (2x = 3y) seems easier to manipulate. We can solve it for one of the variables relatively easily. Let's solve the second equation for x. To do this, we divide both sides of the equation by 2: 2x / 2 = 3y / 2. This simplifies to x = (3/2)y. Now we have an expression for x in terms of y. This is the key to the substitution method. We can now take this expression for x and substitute it into the first equation. Remember, the first equation is 5x + 4y = 345,000. Instead of writing 'x', we're going to replace it with the expression we just found, (3/2)y. So, the first equation becomes: 5 * (3/2)y + 4y = 345,000. Notice what we've done here. We've eliminated one of the variables (x) and now we have an equation with only one variable (y). This equation is much easier to solve. Let's simplify the equation. First, multiply 5 by (3/2)y, which gives us (15/2)y. So, the equation becomes: (15/2)y + 4y = 345,000. To combine the terms with 'y', we need a common denominator. We can rewrite 4y as (8/2)y. So, the equation becomes: (15/2)y + (8/2)y = 345,000. Now we can add the fractions: (15/2 + 8/2)y = 345,000. This simplifies to (23/2)y = 345,000. We're getting closer! Now, to isolate 'y', we need to multiply both sides of the equation by the reciprocal of (23/2), which is (2/23). So, we have: (2/23) * (23/2)y = 345,000 * (2/23). The left side simplifies to 'y', and the right side is a calculation we need to do. 345,000 multiplied by 2 is 690,000. Then, we divide 690,000 by 23. Doing the division, we find that y = 30,000. So, we've found the value of 'y', which is the price of a hat. A hat costs Rp30,000.00. But we're not done yet! We still need to find the value of 'x', which is the price of a pair of sandals. Now that we know the value of 'y', we can plug it back into one of our original equations to solve for 'x'. The easiest equation to use is 2x = 3y. We substitute y = 30,000 into this equation: 2x = 3 * 30,000. This simplifies to 2x = 90,000. To solve for 'x', we divide both sides by 2: 2x / 2 = 90,000 / 2. This gives us x = 45,000. Woohoo! We've found it! The price of a pair of sandals is Rp45,000.00. We've successfully used the substitution method to solve the problem.

Finding the Price of Sandals

Alright, let's wrap this up and find the price of those sandals! We've done the heavy lifting with the substitution method, and now it's time to state our answer clearly. Remember, we were trying to figure out the price of a pair of sandals, which we represented with the variable 'x'. Through our calculations, we found that x = 45,000. This means that the price of a pair of sandals is Rp45,000.00. So, there you have it! We've successfully solved the problem. We started with a word problem about Haidar buying sandals and hats, we translated it into mathematical equations, we used the substitution method to solve for the unknowns, and we found the price of a pair of sandals. It's like we're math detectives, cracking the code! To make sure we're on the right track, it's always a good idea to check our answer. We can plug the values we found for x and y back into our original equations to see if they hold true. Let's start with the first equation: 5x + 4y = 345,000. Substitute x = 45,000 and y = 30,000: 5 * 45,000 + 4 * 30,000 = 345,000. This simplifies to 225,000 + 120,000 = 345,000, which is true. Now let's check the second equation: 2x = 3y. Substitute x = 45,000 and y = 30,000: 2 * 45,000 = 3 * 30,000. This simplifies to 90,000 = 90,000, which is also true. Since our values for x and y satisfy both equations, we can be confident that our answer is correct. The price of a pair of sandals is indeed Rp45,000.00. This problem demonstrates the power of algebra in solving real-world problems. By translating the problem into equations and using algebraic techniques, we were able to find the solution in a systematic and logical way. The substitution method is a valuable tool in algebra, and it can be used to solve a wide variety of problems. So, next time you encounter a problem with two unknowns, remember the substitution method! It might just be the key to unlocking the solution.

Final Answer

Okay, folks, we've reached the finish line! After all our calculations and problem-solving, the final answer is crystal clear. The price of a pair of sandals is Rp45,000.00. We used the substitution method to break down the problem step by step, and we successfully found the solution. Remember, we started by setting up the equations, then we used the substitution method to isolate one variable, and finally, we plugged the value back in to find the other variable. It's like a puzzle, and we put all the pieces together! This whole exercise shows how useful math can be in everyday situations. Whether you're shopping, budgeting, or even just trying to figure out the best deal, understanding these concepts can really help. And the substitution method? It's a powerful tool to keep in your math toolkit. So, congratulations! You've tackled this problem like a champ. Keep practicing, keep exploring, and remember, math can be fun! If you ever stumble upon another math mystery, just remember the steps we took today: understand the problem, set up the equations, choose a method (like substitution), solve, and always double-check your answer. You've got this! And hey, if you're ever stuck, don't hesitate to ask for help. Math is a team sport, and there are plenty of people who are happy to lend a hand. Now, go forth and conquer more math challenges!