Workers Paving A Track: A Math Problem

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Workers Paving a Track: A Math Problem

Let's dive into a classic math problem involving workers, paving, and time. This type of problem often appears in aptitude tests and helps illustrate the concepts of direct and inverse variation. We'll break down the problem step by step to make sure you understand not just the solution, but also the logic behind it.

Understanding the Problem

The problem states: If 15 workers pave 500m of track in 30 days, how many days will it take 10 workers to pave 300 meters of track if the yields are the same?

To solve this, we need to figure out how the number of workers, the length of the track, and the number of days are related. More workers should mean less time to pave the same length of track (inverse variation). A longer track should mean more time to pave it (direct variation).

Breaking Down the Variables

  • Workers: The number of people doing the work.
  • Track Length: The amount of track to be paved (in meters).
  • Days: The time it takes to complete the work.

We're assuming that each worker works at the same rate, which simplifies the problem significantly. This means the 'yields are the same' for everyone.

Setting Up the Proportion

This problem can be solved using the concept of combined variation. We can set up a proportion that relates the variables:

(Workers1 * Days1) / Work1 = (Workers2 * Days2) / Work2

Where:

  • Workers1 = 15
  • Days1 = 30
  • Work1 = 500 (meters)
  • Workers2 = 10
  • Days2 = ? (This is what we want to find)
  • Work2 = 300 (meters)

Plugging in the Values

Now, let's plug in the known values into the equation:

(15 * 30) / 500 = (10 * Days2) / 300

Solving for Days2

First, simplify both sides of the equation:

450 / 500 = (10 * Days2) / 300

  1. 9 = (10 * Days2) / 300

Now, multiply both sides by 300 to isolate the term with Days2:

  1. 9 * 300 = 10 * Days2

  2. 270 = 10 * Days2

Finally, divide both sides by 10 to solve for Days2:

Days2 = 270 / 10

Days2 = 27

The Answer

It will take 10 workers 27 days to pave 300 meters of track.

Alternative Approach: Finding the Work Rate

Another way to solve this is by finding the work rate of a single worker. Here’s how:

  1. Total work done in the first scenario: 500 meters
  2. Total worker-days: 15 workers * 30 days = 450 worker-days
  3. Work rate per worker-day: 500 meters / 450 worker-days = 10/9 meters per worker-day (approximately 1.11 meters per worker per day).

Now, let's apply this work rate to the second scenario:

  1. Total work to be done: 300 meters
  2. Number of workers: 10
  3. Combined work rate: 10 workers * (10/9) meters/worker-day = 100/9 meters per day
  4. Days required: 300 meters / (100/9 meters per day) = 300 * (9/100) = 27 days

So, using this method, we arrive at the same answer: it will take 10 workers 27 days to pave 300 meters of track.

Key Takeaways

  • Understanding Variation: This problem illustrates both direct and inverse variation. The length of the track varies directly with the number of days, while the number of workers varies inversely with the number of days.
  • Setting Up Proportions: Setting up the correct proportion is crucial. Make sure you understand how each variable affects the outcome.
  • Units: Always pay attention to the units. In this case, we're dealing with workers, meters, and days. Keeping track of the units helps ensure that your calculations are correct.

Why is this important?

These types of problems are more than just abstract math exercises. They help develop your problem-solving skills, which are useful in many real-world situations. For example, project management often involves similar calculations to estimate how long a task will take based on the number of people working on it.

Real-World Applications

  • Construction: Estimating the time required to complete a building project based on the number of workers and the size of the project.
  • Manufacturing: Calculating the production rate of a factory based on the number of machines and workers.
  • Software Development: Predicting how long it will take to develop a software application based on the number of developers and the complexity of the project.

Common Mistakes to Avoid

  • Incorrectly Identifying Variation: Confusing direct and inverse variation can lead to incorrect proportions.
  • Not Keeping Track of Units: Failing to keep track of the units can result in errors in your calculations.
  • Assuming Constant Work Rate: This problem assumes that all workers work at the same rate. In real-world scenarios, this may not be the case.

Let's Summarize!

So, to recap, the answer to the question is 27 days. Remember to break down the problem, identify the relationships between the variables, and set up the proportion carefully. Guys, mastering these types of problems not only boosts your math skills but also enhances your ability to tackle real-world challenges! Happy problem-solving!