Mathematical Expressions for Calculating Work Done

Calculating the fraction of work done by a person in a given time can be done using straightforward formulas based on the rate of work and the time worked. In this article, we will derive mathematical expressions for determining the fraction of work done by a man who can finish a job in 3 hours and 20 minutes.

Understanding the Basic Formula

The core of this calculation is the formula for the fraction of work done:

Fraction of work done Time elapsed / Total time to complete the job

Total Time to Complete the Job

In the scenario, the man can complete the job in 3 hours and 20 minutes. Converting this into hours:

Conversion to Hours

3 hours 20 minutes 3 (20/60) hours 3.33 hours

Calculating the Fraction of Work Done

The fraction of work done can then be expressed as:

Fraction of work done Time elapsed / 3.33 hours

This value will range from 0 to 1, where 0 indicates no work done and 1 indicates the entire job completed.

Additional Formulas

Sometimes it helps to break down the problem using additional formulas. For example, if we have the formula:

W R x T

where:

W Work R Rate (units of work per unit of time) T Time (in hours)

The rate can be calculated as:

R W / T

Rate Calculation

If the man can complete the job in 3 hours and 20 minutes (3.33 hours), the rate is:

R 1 / 3.33 hours 1 / 10 / 3 3/10 units of work per hour

Thus, the total work done (W) can be expressed as:

W (3/10) x T where T is expressed in hours worked

Example Calculations

Let's demonstrate with a couple of examples:

In 1 hour: Work done (3/10) x 1 3/10 of the job. In 10 hours: Work done (3/10) x 10 3 (the whole job).

More Detailed Calculation

Using a more detailed approach, we can express the fraction of work done as:

Fraction of work done t / 3.33 hours where t is the time elapsed in hours.

General Case for t Hours

For a general case where the man has been working for t hours (0 le; t le; 3.33), the fraction of work done is:

Fraction of work done t / 3.33 3t / 10

Thus, if t is 1 hour, the fraction of work done is 3/10, and if t is 3.33 hours, the fraction is 1 (the whole job).

These calculations provide a clear and concise way to express the fraction of work done based on the time elapsed and the total time to complete the job.