Problem 16 - Basic Engineering Mathematics

Question

If 3 people can complete a task in 4 hours, find how long it will take 5 people to complete the same task, assuming the rate of work remains constant.

Step-by-Step Solution

Answer

5 people will take 2.4 hours to complete the task.

Step 1: Determine the work rate per person

First, calculate the total work done by 3 people in 4 hours. Since it takes 3 people 4 hours to complete the task, the total work can be expressed as the product of the number of workers and time: \[ W = 3 \text{ people} \times 4 \text{ hours} = 12 \text{ person-hours} \] This means the total task requires 12 person-hours to complete.

Step 2: Establish the work rate for one person

Now that we know the total task requires 12 person-hours, we find the rate at which one person completes the work. If 3 people complete the task in 4 hours, one person would take 12 hours to finish the task alone. The formula for the rate of one person is: \[ R_{1} = \frac{1 \text{ task}}{12 \text{ hours}} = \frac{1}{12} \text{ task/hour} \] This means one person does 1/12 of the task per hour.

Step 3: Calculate the work rate for 5 people

Next, calculate the rate at which 5 people work together. Given one person’s rate is \( \frac{1}{12} \) task/hour, 5 people would have a combined rate: \[ R_{5} = 5 \times \frac{1}{12} = \frac{5}{12} \text{ task/hour} \] So, 5 people together can complete \( \frac{5}{12} \) of the task in one hour.

Step 4: Find the time it takes for 5 people to complete the task

Finally, determine the time it takes for 5 people to complete 1 whole task at the rate of \( \frac{5}{12} \) task/hour. Use the formula for time: \[ T = \frac{\text{Total Work}}{\text{Rate}} = \frac{1 \text{ task}}{\frac{5}{12} \text{ task/hour}} \] This simplifies to: \[ T = \frac{12}{5} \text{ hours} \] Therefore, it will take 5 people 2.4 hours to complete the task.

Key Concepts

Rate of Work

In solving work rate problems, the core concept revolves around understanding how much work can be accomplished over time. The "Rate of Work" is foundational as it quantifies the amount of a task completed per unit of time. Imagine if you were solving a puzzle. If you know how many pieces you can place in an hour, you'll know your rate of work. Here's how it applies to our problem:

When multiple workers come together, their rates combine to increase efficiency. Thus, knowing how much one person alone can do in an hour (defined as rate per person) helps scale this to the entire group. For instance:

• A single person's work rate would be \(\frac{1}{12}\) task/hour, meaning they complete one-twelfth of the task in one hour.
• When 5 people work together, we say their combined rate is 5 times each individual’s rate, therefore \(R_5 = 5 \times \frac{1}{12} = \frac{5}{12}\) task/hour.

Understanding this helps conceptualize distribution of workload among workers.

Person-Hours Calculation

Calculating person-hours is critical; it's a unit to represent work done by several people over time. It's like multiplying the driving speed by the duration to know how far you've traveled.

For the given task, knowing it takes 3 people 4 hours to complete the work means the task totals 12 person-hours. In essence:

• 3 people working for 4 hours each contributes individual hours of work, amounting collectively to 12 person-hours.

Essentially, person-hours quantify the labor needed to complete work. It's a useful metric to estimate or allocate sufficient manpower to finish a task efficiently.

Task Completion Time

Understanding how long it takes a group to finish a task helps with proper planning and management. To calculate this time, we divide the total work (as represented by a full task which equals 1) by the group’s work rate.

Given 5 people working at a combined rate of \(\frac{5}{12}\) task per hour, the formula for task completion time becomes:

• \(T = \frac{1 \text{ task}}{\frac{5}{12} \text{ task/hour}}\), which simplifies to \(T = \frac{12}{5} \text{ hours}\).
• This means the task will be completed in 2.4 hours.

By understanding rates and person-hours, it's easier to predict task completion time, optimizing schedules and resource allocation.