Factory Production: January & February Performance

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Hey guys! Let's break down this math problem about a factory's production performance over two months. We'll figure out how much the factory overfulfilled its production plan. This is a classic percentage problem, and we'll tackle it step-by-step so you can totally nail it. We will use a friendly tone, so it's easier to follow. Get ready to flex those math muscles!

Decoding the January Production

Alright, first things first: In January, the factory crushed it, hitting 105% of its monthly production plan. This means they produced more than what was initially planned. The key here is understanding that 100% represents the target, and anything over that is a bonus! Think of it like getting extra credit on a test. The factory's overachievement in January sets the stage for our calculations. We've got our baseline, and that's important for later.

Let's imagine the monthly plan as 'X'. In January, the factory produced 1.05X (105% of X). To get this, we convert the percentage to a decimal (105 / 100 = 1.05) and multiply it by the planned production, which we're calling 'X.' The 'X' is a variable, representing the initial target. The great thing is that we don't need to know the exact amount, but we can still solve the problem. This is a common strategy in percentage problems – focusing on the relationship between the planned and actual output. Using 'X' allows us to compare the production of the factory.

So, January's production is 1.05X. Remember this figure, it is important for the next step. Understanding this initial overperformance is the foundation for calculating the overall overfulfillment across both months. If January had been the only month to consider, we'd already have our answer. But now, we move on to February. It will be even more interesting!

Figuring out February's Production Boost

Now, let's zoom into February. The problem tells us that the factory produced 4% more than it did in January. This is another percentage increase, but this time, it's relative to January's output. To calculate February's production, we need to add 4% of January's production to January's production itself. Again, the 'X' is important here.

We know January's production was 1.05X. To find 4% of that, we multiply 1.05X by 0.04 (4 / 100 = 0.04). This gives us 0.042X. This is the extra production achieved in February compared to January. Now, we add that to the January production to find February's total production: 1.05X + 0.042X = 1.092X. This result shows us how much the factory produced in February.

To recap: In February, the factory produced 1.092X. This is a very important figure. Comparing February to January, we see that February's output is also above the planned target. Both months had some success in production. This shows that the factory has been doing very well!

The Two-Month Production Plan: A Combined Analysis

Next step: We must combine the production figures for January and February to see how the factory did overall. We need to find the total production for the two months and compare it to the total planned production for the two months. This is how we'll find the percentage of overfulfillment.

In January, the factory produced 1.05X, and in February, it produced 1.092X. So, the total production for the two months is 1.05X + 1.092X = 2.142X. This is the actual amount of the production for the two months.

Now, let's consider the plan. If the monthly plan is X, then the two-month plan is 2X (X for January + X for February). This is our target! Now, we have all the numbers, it's time to put it all together. What is the difference between the target and the actual performance? Let's go!

Calculating the Overfulfillment Percentage

Okay, guys, here comes the final stretch! We've got the total actual production (2.142X) and the total planned production (2X). To find the overfulfillment percentage, we need to figure out how much the actual production exceeded the plan, and then express that as a percentage of the plan. This is the core of our solution. We take our time here, so it's super easy to follow!

First, find the difference between actual production and the planned production: 2.142X - 2X = 0.142X. This represents the extra production the factory achieved over the two months. To get the percentage, we divide the extra production by the total planned production (2X) and multiply by 100: (0.142X / 2X) * 100. The 'X's cancel out, which is good! It simplifies the calculation and we get (0.142 / 2) * 100 = 7.1%. So, the factory overfulfilled the two-month plan by 7.1%.

And there you have it! The factory exceeded its two-month production plan by 7.1%. This calculation shows how to solve this specific problem. You can definitely rock this type of problem now. The key is to break it down, focus on understanding each step, and then put everything together to get the final answer. Amazing!

Final Answer and Takeaways

So, the correct answer is 2) 7.1. Congratulations! You've successfully navigated this percentage problem. Let's recap the key takeaways:

  • Understand Percentages: Always convert percentages to decimals when doing calculations. Remember, 100% means the whole target, and anything more means overachievement.
  • Break it Down: Separate the problem into manageable steps: January, February, combined production, and the final percentage calculation.
  • Use Variables: Using 'X' (or any variable) to represent the plan allows you to work with the relationships between the planned and actual production.

Keep practicing, and you'll become a percentage problem-solving pro in no time! Keep going, and you'll find similar problems much easier. You have all the tools you need to do great in math! Great job, everyone!