Introduction

The question of how long it takes for two individuals to complete a job when working together is a common problem in the realm of work rate and time management. This article explores the process of determining the time it takes for Anu and Banu to complete a job when working together, using the concept of work rate. We will delve into the mathematics behind this problem and apply it to find a solution.

Understanding Work Rate

The work rate of an individual is the fraction of the work they can complete in a day. If Anu can do a job in 10 days, her work rate is ( frac{1}{10} ) of the job per day. Similarly, if Banu can do the same job in 5 days, his work rate is ( frac{1}{5} ) of the job per day.

Combining Work Rates

To determine the combined work rate of Anu and Banu, we need to add their individual work rates. The combined work rate is the amount of work they can complete in a single day when working together.

Method 1: Direct Work Rates Addition

The combined work rate can be calculated as follows:

A’s 1 day work ( frac{1}{10} )

B’s 1 day work ( frac{1}{5} )

Combined work rate ( frac{1}{10} frac{1}{5} frac{1}{10} frac{2}{10} frac{3}{10} )

This means that together, Anu and Banu can complete ( frac{3}{10} ) of the job in one day. To find the total time required to complete the job, we need to find the reciprocal of the combined work rate.

Method 2: LCM and Efficiency Approach

Using the least common multiple (LCM) approach, we set the total work to 30 units (the LCM of 5 and 10).

Efficiency of A ( frac{30}{5} 6 ) units per day

Efficiency of B ( frac{30}{10} 3 ) units per day

Total efficiency 6 3 9 units per day

Time to complete the work ( frac{30}{9} 3 frac{1}{3} ) days

This confirms that working together, Anu and Banu can complete the job in ( frac{10}{3} ) days or 3.33 days.

Conclusion

Thus, the time taken by Anu and Banu to complete the job together is ( frac{10}{3} ) days, approximately 3.33 days or 3 days and 8 hours. This method of calculating work rates and combining them is a fundamental concept in managing resources and understanding the efficiency of collaborative work.