Efficient Wheat Collection: A Practical Study in Proportional Relationships
In solving real-world problems such as calculating the amount of wheat that can be collected, we often find ourselves utilizing principles from proportional relationships. This article will explore the problem of wheat collection and demonstrate the practical application of these mathematical principles. Specifically, we'll determine how many kilograms of wheat can be collected by 8 workers in 5 days, given the context of 5 workers collecting 120 kilograms of wheat in 3 days. This will involve a detailed analysis using proportional relationships and rate calculations.Understanding the Problem
The problem at hand is straightforward yet illustrative of broader principles. We need to find the relationship between the number of workers, the duration of work, and the quantity of wheat collected. Here are the given parameters: 5 workers can collect 120 kg of wheat. This collection occurs over a period of 3 days. We need to find out how much wheat 8 workers will collect in 5 days.Step-by-Step Solution
To solve this problem, we'll break it down into several steps: Calculate the rate of wheat collection per worker per day. Use this rate to calculate the total collection for 8 workers over 5 days.Step 1: Calculate the Rate of Collection per Worker
The first step involves determining the collection rate per worker. Given the information that 5 workers can collect 120 kg of wheat over 3 days, we can calculate the total collection per day for 5 workers and then determine the rate per worker per day. First, we calculate the total collection per day for 5 workers:[text{Total per day} frac{120 , text{kg}}{3 , text{days}} 40 , text{kg/day}]
Next, we determine the rate at which one worker collects wheat per day. Since 5 workers collectively collect 40 kg of wheat per day, the rate for one worker can be calculated as follows:[text{Rate per worker} frac{40 , text{kg/day}}{5 , text{workers}} 8 , text{kg/worker/day}]
Step 2: Calculate the Total Collection for 8 Workers in 5 Days
With the rate determined, we can now find out how much wheat 8 workers will collect in 5 days. First, we calculate the total collection per day for 8 workers:[text{Total per day for 8 workers} 8 , text{workers} times 8 , text{kg/worker/day} 64 , text{kg/day}]
Next, to find the total collection over 5 days for 8 workers, we multiply the daily collection by the number of days:[text{Total for 5 days} 64 , text{kg/day} times 5 , text{days} 320 , text{kg}]
Therefore, 8 workers will collect 320 kilograms of wheat in 5 days. This solution is based on proportional relationships and rate calculations, demonstrating a practical application of basic mathematical principles in real-world scenarios.Alternative Methods
While the primary method involves step-by-step calculations, there are a few other ways to approach the problem, such as using simple proportion or direct multiplication. Here are two alternative approaches that lead to the same conclusion: Proportional Relationship Method:The number of workers and the days worked are scaled up. We can set up the proportion as follows:
[120 , text{kg} times frac{5 , text{workers}}{3 , text{days}} times frac{8 , text{workers}}{5 , text{workers}} 320 , text{kg}]
This method acknowledges the direct proportionality but simplifies the calculation. Direct Multiplication Method:We can calculate the man-days required, and then scale up:
This method directly scales the given values to obtain the total collection.