Cabinet Production: Optimizing Type 1 And Type 2 Output

Hey guys, let's dive into a cool problem that blends manufacturing and a bit of algebra. We're talking about a company that's churning out two types of cabinets: Type 1 and Type 2. They've got a weekly production target of 110 total cabinets. Now, the kicker is how they balance the production of these two types. Last week, things got specific: the number of Type 2 cabinets produced was 20 more than twice the number of Type 1 cabinets. This is where the math kicks in, and we need to figure out exactly how many of each type they made. We'll be using algebra to solve this puzzle, so get ready to put on your thinking caps! This isn't just about numbers; it's about understanding how production goals and specific constraints work together in a real-world scenario. We'll break down the problem, set up our equations, and solve for $x$, which represents the number of Type 1 cabinets. So, stick around as we unravel this production mystery!

Understanding the Production Variables

Alright, let's get down to business with these cabinets, shall we? We've got two stars in our show: Type 1 cabinets and Type 2 cabinets. The total production for the week is a solid 110 cabinets. Think of this as our overall goal, the grand total we need to hit. Now, the relationship between the two types is what makes this problem interesting. We're told that the number of Type 2 cabinets produced last week was 20 more than twice the number of Type 1 cabinets. This is a crucial piece of information that ties everything together. To make sense of this, we need to define our variables. The problem wisely tells us to let '$x$' represent the number of Type 1 cabinets produced. This is our starting point, our unknown hero. If '$x$' is the number of Type 1 cabinets, then we need to express the number of Type 2 cabinets in terms of '$x$'. Based on the condition given, 'twice the number of Type 1 cabinets' would be $2x$. And since the number of Type 2 cabinets exceeded this by 20, the number of Type 2 cabinets is $2x + 20$. See how that works? We've translated a word problem into algebraic expressions. This is a fundamental skill in mathematics and incredibly useful in real-world problem-solving, especially in fields like manufacturing, engineering, and economics. Understanding these variables is the first step to setting up the equations that will lead us to the solution. It's like gathering all the puzzle pieces before you start assembling the picture. So, let's make sure we're clear on this: $x$ = Type 1 cabinets, and $2x + 20$ = Type 2 cabinets. Easy peasy, right? Now that we've got our variables defined, we can move on to the next crucial step: forming the equation that represents the total production.

Setting Up the Algebraic Equation

Now that we've got our players (Type 1 and Type 2 cabinets) and their corresponding algebraic expressions defined, it's time to bring them together to form our main equation. Remember, the company produces a total of 110 cabinets each week. This total is the sum of the Type 1 cabinets and the Type 2 cabinets. So, if we add the number of Type 1 cabinets (which we've defined as $x$) and the number of Type 2 cabinets (which we've defined as $2x + 20$), the result must equal the total production of 110. This gives us our algebraic equation: $x + (2x + 20) = 110$. This equation is the heart of our problem. It encapsulates all the information given in a concise mathematical form. When you're dealing with word problems, the ability to translate the narrative into an equation like this is super important. It's the bridge between understanding the situation and being able to solve it quantitatively. Let's break down why this equation works: the '$x$' on the left side represents all the Type 1 cabinets. The '$(2x + 20)$' represents all the Type 2 cabinets. When we add them together, we are accounting for every single cabinet produced. And since we know the total is 110, setting the sum equal to 110 is the logical next step. This process of translating words into symbols is what makes algebra so powerful. It allows us to model real-world scenarios and find solutions using systematic methods. So, take a moment to appreciate this equation – it's the key that will unlock the number of cabinets produced. We're almost there, guys! The next step is to actually solve this equation to find the value of $x$, which will tell us how many Type 1 cabinets were made.

Solving for the Number of Type 1 Cabinets ($x$)

Alright, we've done the heavy lifting by setting up the equation: $x + (2x + 20) = 110$. Now comes the satisfying part – solving for '$x$'. This is where we find out just how many Type 1 cabinets were manufactured. First things first, let's simplify the left side of the equation by combining like terms. We have '$x$' and '$2x$', which are like terms. Adding them together gives us $3x$. So, the equation becomes: $3x + 20 = 110$. Our goal now is to isolate '$x$'. To do that, we need to get rid of the '+ 20' on the left side. We can do this by subtracting 20 from both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep the equation balanced. So, subtracting 20 from both sides gives us: $3x + 20 - 20 = 110 - 20$, which simplifies to $3x = 90$. Now we're one step closer! We have $3x$ equaling 90, meaning three times the number of Type 1 cabinets is 90. To find out what just one '$x$' is, we need to divide both sides of the equation by 3. This gives us: $3x / 3 = 90 / 3$. And voilà! We get $x = 30$. So, there you have it! The number of Type 1 cabinets produced last week was 30. This is a huge step in solving our problem. We've successfully used algebraic manipulation to find the value of our primary unknown. It's always a good feeling when you can solve for the variable you started with. But wait, there's more! We're not done yet. We still need to figure out the number of Type 2 cabinets to make sure everything adds up and meets the original conditions. So, let's move on to the next step where we'll verify our answer and calculate the production of Type 2 cabinets.

Calculating Type 2 Cabinets and Verifying the Solution

Awesome job, everyone! We've successfully determined that $x = 30$, which means the company produced 30 Type 1 cabinets. But we're not quite finished. We need to find out how many Type 2 cabinets were made and, crucially, verify that our numbers add up correctly and meet all the conditions of the problem. Remember how we defined the number of Type 2 cabinets? It was $2x + 20$. Now that we know $x = 30$, we can substitute this value into that expression. So, the number of Type 2 cabinets is $2 * (30) + 20$. Let's do the math: $2 * 30 = 60$. Then, $60 + 20 = 80$. So, the company produced 80 Type 2 cabinets. Now, let's check if our solution holds up. First, does the total number of cabinets produced equal 110? We have 30 Type 1 cabinets and 80 Type 2 cabinets. Adding them together: $30 + 80 = 110$. Yes, the total production matches the given constraint! Second, does the number of Type 2 cabinets exceed twice the number of Type 1 cabinets by 20? Twice the number of Type 1 cabinets is $2 * 30 = 60$. And indeed, $80$ (Type 2 cabinets) is $60 + 20$. So, this condition is also met! It's super satisfying when your calculations align perfectly with the problem statement. This verification step is a vital part of problem-solving. It ensures that you haven't made any calculation errors and that your answer is indeed correct. So, to summarize: the company produced 30 Type 1 cabinets and 80 Type 2 cabinets. Together, that's a grand total of 110 cabinets, with the Type 2 production perfectly fitting the described relationship with Type 1 production. This confirms our algebraic solution is spot on!

Conclusion: Optimizing Cabinet Production

So there you have it, folks! We've successfully tackled a word problem involving manufacturing and algebra. We started with a company producing two types of cabinets, Type 1 and Type 2, with a weekly total of 110 cabinets. The key piece of information was that the number of Type 2 cabinets exceeded twice the number of Type 1 cabinets by 20. By defining '$x$' as the number of Type 1 cabinets, we were able to set up the equation $x + (2x + 20) = 110$. Through careful algebraic manipulation, we simplified this to $3x + 20 = 110$, then isolated '$x$' to find $x = 30$. This told us that 30 Type 1 cabinets were produced. We then used this value to calculate the number of Type 2 cabinets, which came out to 80 (using the expression $2x + 20$). Finally, we verified our solution by ensuring that the total cabinets produced (30 + 80) equaled 110 and that the relationship between Type 1 and Type 2 production held true. This problem highlights the power of algebra in solving practical, real-world scenarios, like optimizing cabinet production. Understanding these relationships allows companies to manage their resources effectively and meet their manufacturing targets. Whether you're dealing with cabinets, cars, or cookies, the principles of setting up and solving equations remain the same. Keep practicing these skills, guys, because they're incredibly valuable. We hope you enjoyed this breakdown, and remember, math is all around us, even in the factory!