Toy Car Assembly: Men Vs. Output In 5 Days
Hey guys! Let's dive into a classic problem of proportionality and figure out how many toy cars a team of workers can assemble. This is a common type of question you might see in math problems, especially when dealing with concepts like direct variation and ratios. We're going to break down this toy car assembly problem step by step, making sure you understand not just the answer, but the why behind it. So, buckle up and let's get started!
Understanding the Problem
In this toy car assembly scenario, we're given that a team of 8 men can assemble 20 toy cars in 5 days. The core of the problem lies in understanding the relationship between the number of men, the number of toy cars assembled, and the time taken. Here’s the key question we need to answer: If the team increases to 12 men, how many toy cars can they assemble in the same 5-day period?
To tackle this, we'll assume that each man works at the same rate. This is a crucial assumption because it allows us to directly compare the output of different team sizes. We're also keeping the time constant at 5 days, which simplifies our calculations and lets us focus solely on the impact of the increased workforce. The goal is to find out how the increased manpower translates into an increased number of assembled toy cars. We will delve into the concept of direct proportionality, where an increase in the number of men directly leads to an increase in the number of toy cars assembled, provided the time remains constant. Understanding this relationship is crucial for solving the problem accurately and efficiently. So, let’s put on our thinking caps and get ready to solve this fun little puzzle!
Setting Up the Proportion
To solve this, we can use the concept of direct proportion. Direct proportion means that if one quantity increases, the other quantity increases at the same rate. In our case, the number of men and the number of toy cars assembled are directly proportional, assuming the time is constant. Let's set up a proportion to represent this relationship. We can express the initial scenario as a ratio: 8 men / 20 toy cars. This ratio tells us the number of men required to assemble a certain number of toy cars.
Now, let's represent the unknown number of toy cars that 12 men can assemble as 'x'. We can set up another ratio: 12 men / x toy cars. Since the number of men and the number of toy cars are directly proportional, we can equate these two ratios: 8/20 = 12/x. This equation forms the foundation for solving the problem. We now have a simple proportion that we can solve for 'x'. Solving proportions involves cross-multiplication, which is a straightforward algebraic technique. By setting up the proportion correctly, we've transformed the word problem into a mathematical equation that we can easily solve. So, let's move on to the next step and find out how many toy cars those 12 men can assemble!
Solving the Proportion
Now that we've set up our proportion (8/20 = 12/x), it's time to solve for 'x', which represents the number of toy cars 12 men can assemble in 5 days. The standard method for solving proportions is cross-multiplication. This involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the two products equal to each other. So, let's apply cross-multiplication to our proportion: 8 * x = 20 * 12. This simplifies to 8x = 240. Now, we need to isolate 'x' to find its value. To do this, we divide both sides of the equation by 8: x = 240 / 8.
Performing the division, we get x = 30. So, what does this '30' mean? It means that 12 men can assemble 30 toy cars in 5 days, given the same working conditions and assuming each man works at the same rate. This solution makes intuitive sense. We increased the workforce, and as a result, we expect the output to increase as well. This step-by-step solution demonstrates how to use proportions to solve real-world problems involving direct variation. So, with this value in hand, we're ready to confidently state the answer to our problem. Let's wrap it up in the final section!
The Answer and Explanation
Alright, guys, we've crunched the numbers and found our answer! Based on our calculations, 12 men can assemble 30 toy cars in 5 days. This corresponds to option (b) in the original question. But let's not just stop at the answer; let's quickly recap why this is the correct answer. We started with the information that 8 men assemble 20 toy cars in 5 days. We then used the concept of direct proportion to figure out how many toy cars 12 men could assemble in the same amount of time.
By setting up the proportion 8/20 = 12/x and solving for 'x', we found that x = 30. This demonstrates a direct relationship between the number of workers and the output, assuming the time and individual work rates remain constant. In simpler terms, if you increase the number of workers, you'll increase the number of toy cars assembled. This type of problem is a great example of how mathematical concepts can be applied to real-world scenarios. Understanding proportions and direct variation is super helpful in many situations, from manufacturing and production to cooking and scaling recipes. So, next time you encounter a similar problem, remember the steps we've covered, and you'll be able to solve it like a pro!