Introduction to Work-Duration Problems

When dealing with work-duration problems, we often find ourselves calculating how long it takes for one or more individuals to complete a specific task based on their individual or combined efficiencies. This article aims to solve a specific example and provide a detailed breakdown of the steps and approaches used to find the most efficient solutions.

Solving the Given Problem

The problem at hand is as follows: B can work twice as much as A can do in a day. If they together can do 4/15 parts of the work in 10 days, how many days will B finish the work alone?

Let's start by defining variables:

Let A take x days to complete the work alone.

Then B will take 3x days to complete the work alone, based on the given condition that B works twice as much as A.

The part of the work A can do in one day is 1/x, and B can do 1/3x of the work in one day.

Together, they complete 1/x 1/3x 1/x (1 1/3) 1/x (4/3) 4/3x of the work in one day.

Applying the Given Condition

They can complete 4/15 of the work in 10 days. So, together they can do:

4/3x * 10 4/15

Simplifying, we have:

40/3x 4/15

Multiplying both sides by 3x, we get:

40 4x/5

Multiplying both sides by 5 and dividing by 40, we get:

x 25

So, A can do the work alone in 25 days, and B can do it alone in 75 days (3x).

Conclusion

Therefore, B, working alone, can complete the work in 75 days. This method can be generalized to other similar problems where the efficiency of workers is involved.

Key Concepts and Methods

Defining Variables: Assigning a variable to the time taken by each worker to complete the work alone.

Setting Up Equations: Relating the work done in one day to the total work done over a given period.

Multiplying and Simplifying: Utilizing mathematical operations to solve for the unknown variable.

Reciprocals: Understanding how the reciprocal of the time taken is the rate at which each worker completes the work.

These steps not only provide a clear solution to the given problem but also offer a systematic approach to solving similar work-duration problems.

Additional Tips

Check for Consistency: Ensure that the relationships and calculations are consistent and lead to a logical and reasonable solution.

Practice: Regular practice will help in understanding and applying the concepts more effectively.

Vary the Conditions: Try to solve similar problems with different conditions to enhance your understanding.

By mastering these concepts and techniques, you can efficiently solve complex work-duration problems and improve your problem-solving skills.

Keywords

Keywords: work efficiency, algebraic problem solving, time and work problems