Chairs And Tables A Mathematical Puzzle To Solve

Hey there, math enthusiasts! Ever found yourself scratching your head over a word problem that seems to hide the answer in plain sight? Well, today, we're diving into a classic scenario involving the cost of chairs and tables. It's like being a detective, piecing together clues to crack the case. So, grab your thinking caps, and let's unravel this mathematical puzzle together!

Decoding the Problem: Chairs, Tables, and a Price Tag

Cost analysis is crucial when dealing with problems like this. Let's break down the problem step by step. We're given two key pieces of information: first, the combined cost of three chairs and four tables is GH$320.00. Think of it as our first equation, a vital clue in our quest. Second, we know that two chairs and five tables of the same type will set you back GH$330.00. This is our second equation, another crucial piece of the puzzle. Our mission, should we choose to accept it, is to figure out the total cost of four chairs and three tables. It's like we're running a furniture store and need to price our items just right. Now, how do we transform these words into a solvable equation? That's where the magic of algebra comes in. Algebraic equations are the key to solving this problem. We'll use variables to represent the unknown costs, and then we'll manipulate those equations until we find our answers. So, let's roll up our sleeves and get algebraic!

Setting Up the Equations: The Language of Math

To begin solving this problem, the first crucial step is translating the word problem into mathematical equations. It's like learning a new language, but instead of words, we're using numbers and symbols. Let's represent the cost of one chair with the variable 'c' and the cost of one table with the variable 't'. This is like giving each item a secret code name. Now, let's look at our first piece of information: "The cost of three chairs and four tables is GH$320.00." We can translate this into the equation 3c + 4t = 320. See how we're turning words into math? It's pretty cool, right? Next, let's tackle our second clue: "Two chairs and five tables of the same kind cost GH$330.00." This translates to the equation 2c + 5t = 330. We've now got two equations, a system of equations just waiting to be solved. These equations are the foundation of our solution, the roadmap that will lead us to the answer. Now that we have our equations, the real fun begins: solving them. We'll use a method called elimination, which is like a mathematical magic trick that makes the unknowns reveal themselves. So, let's move on to the next step and watch the math unfold!

Cracking the Code: Solving the System of Equations

Now that we've got our system of equations – 3c + 4t = 320 and 2c + 5t = 330 – it's time to put on our detective hats and solve for 'c' and 't'. We're going to use a method called elimination, which is like a mathematical ninja move. The goal is to eliminate one of the variables, either 'c' or 't', so we can solve for the other. To do this, we need to make the coefficients of either 'c' or 't' the same (but with opposite signs) in both equations. Let's focus on eliminating 'c'. To do this, we'll multiply the first equation by 2 and the second equation by -3. This will give us 6c in the first equation and -6c in the second, setting them up for elimination. So, multiplying the first equation (3c + 4t = 320) by 2, we get 6c + 8t = 640. Then, multiplying the second equation (2c + 5t = 330) by -3, we get -6c - 15t = -990. Now, we have two new equations: 6c + 8t = 640 and -6c - 15t = -990. The coefficients of 'c' are ready to cancel each other out. Next, we'll add these two equations together. When we do this, the '6c' and '-6c' cancel out, leaving us with an equation in terms of 't' only. It's like a magic trick where one variable disappears, making the problem much easier to solve. Let's see what happens when we add the equations together!

Unveiling the Cost of a Table: The First Piece of the Puzzle

After our strategic multiplication and elimination, we're left with a simplified equation that's ready to reveal the cost of a table. When we add the equations 6c + 8t = 640 and -6c - 15t = -990, the 'c' terms cancel out beautifully, leaving us with -7t = -350. It's like watching a perfectly executed math ballet! Now, to find the value of 't', which represents the cost of a table, we need to isolate 't'. We can do this by dividing both sides of the equation by -7. This is a simple but crucial step, like turning a key to unlock a treasure chest. When we divide -350 by -7, we get t = 50. Eureka! We've discovered that the cost of one table is GH$50.00. This is a major breakthrough in our quest, a significant piece of the puzzle falling into place. But our journey isn't over yet. We've found the cost of a table, but we still need to uncover the cost of a chair. Now that we know 't', we can substitute it back into one of our original equations to solve for 'c'. It's like using a secret code to unlock the next level of the game. So, let's move on and find out the cost of a chair!

Decoding the Chair's Price: Completing the Duo

Now that we've triumphantly discovered the cost of a table (t = GH$50.00), it's time to set our sights on finding the cost of a chair. To do this, we'll use a technique called substitution. It's like plugging a known value into a formula to reveal the unknown. We'll take the value of 't' and substitute it into one of our original equations. It doesn't matter which one we choose; both will lead us to the same answer. For this example, let's use the first equation: 3c + 4t = 320. We'll replace 't' with 50, giving us 3c + 4(50) = 320. Now, we have an equation with only one variable, 'c', which makes it much easier to solve. It's like simplifying a complex puzzle into a single, straightforward step. Let's simplify the equation further. 4 multiplied by 50 is 200, so our equation becomes 3c + 200 = 320. To isolate 'c', we need to get rid of the 200 on the left side. We can do this by subtracting 200 from both sides of the equation. This keeps the equation balanced, a fundamental principle in algebra. Subtracting 200 from both sides gives us 3c = 120. We're almost there! Now, to find the value of 'c', we simply divide both sides of the equation by 3. This will give us the cost of one chair. Let's perform this final step and unveil the price of a chair!

The Final Revelation: Cost of a Chair and the Ultimate Answer

We've reached the final stretch of our mathematical journey, and the answer is within sight! We've arrived at the equation 3c = 120, where 'c' represents the cost of a chair. To find 'c', we need to isolate it by dividing both sides of the equation by 3. This is the last piece of the puzzle, the final calculation that will reveal the chair's price. When we divide 120 by 3, we get c = 40. So, the cost of one chair is GH$40.00! We've successfully navigated the maze of equations and unearthed the individual costs of both a table and a chair. But our quest isn't quite complete. Remember, the original question asked us to find the cost of 4 chairs and 3 tables. Now that we know the individual costs, we can easily calculate the total. It's like using the ingredients we've gathered to bake the final cake. To find the total cost, we'll multiply the cost of a chair by 4 and the cost of a table by 3, then add those two amounts together. This will give us the grand total, the answer we've been searching for. So, let's put these values together and find the final cost!

To calculate the final cost of 4 chairs and 3 tables, we simply plug in the values we've found. The cost of 4 chairs is 4 * GH$40.00 = GH$160.00. And the cost of 3 tables is 3 * GH$50.00 = GH$150.00. Now, we add these two amounts together: GH$160.00 + GH$150.00 = GH$310.00. Therefore, the cost of 4 chairs and 3 tables is GH$310.00. We've done it! We've successfully solved the problem, navigating through the equations and uncovering the final answer. Give yourselves a pat on the back; you've earned it!

Conclusion: Math is an Adventure!

So, there you have it, guys! We've tackled a classic math problem involving chairs and tables, and we've emerged victorious. We've seen how to translate word problems into algebraic equations, how to solve a system of equations using elimination and substitution, and how to apply our findings to answer the original question. This journey wasn't just about finding the cost of furniture; it was about honing our problem-solving skills, sharpening our minds, and discovering the power of math. Problem-solving skills are essential in mathematics. Remember, math isn't just about numbers and formulas; it's about critical thinking, logical reasoning, and the thrill of the chase. Every problem is a puzzle waiting to be solved, and every solution is a triumph. So, keep exploring, keep questioning, and keep embracing the adventure that is mathematics!