Ajay and Vijay undertake to do a piece of work for Rs. 480. Ajay alone can do it in 75 days while Vijay alone can do it in 40 days. With the help of Pradeep, they finish the work in 25 days. How much should Pradeep get for his work?    

Ajay and Vijay undertake to do a piece of work for Rs. 480. Ajay alone can do it in 75 days while Vijay alone can do it in 40 days. With the help of Pradeep, they finish the work in 25 days. How much should Pradeep get for his work?

Pradeep should get Rs. 20 for his work.

Here's how to find Pradeep's share:

1. Calculate individual rates:

   • Ajay's one-day work: 1 work unit / 75 days = 1/75
   • Vijay's one-day work: 1 work unit / 40 days = 1/40

2. Combine Ajay and Vijay's rates:

   • Ajay & Vijay's one-day work together: 1/75 + 1/40 = 7/300

3. Find the total work done in a day by all three:

   • Total one-day work: 1 work unit / 25 days = 1/25

4. Isolate Pradeep's rate:

   • Pradeep's one-day work: 1/25 - 7/300 = 1/75

5. Calculate Pradeep's wages:

   • Pradeep's wages for 25 days: (1/75) * 25 * Rs. 480 = Rs. 20

Therefore, Pradeep should get Rs. 20 for his work.

Time and Work in Mathematics

Time and work problems in mathematics typically involve determining the amount of work done by people working together at different rates or within different time frames. These problems often require you to calculate how long it takes for a certain number of people, working at certain rates, to complete a task together.

Here are some key concepts and steps to solve time and work problems:

1. Rate of Work: The rate of work is usually measured in terms of how much work one person can complete in one unit of time. For example, a person might be able to complete 1/5 of a job in one hour, which means their rate of work is 1/5 of the job per hour.

2. Work Done: Work done is the total amount of work completed by all workers. It's often represented as a fraction or a percentage of the total work.

3. Time: Time is the duration it takes to complete a certain amount of work. It can be in hours, days, or any other unit of time.

4. Basic Formula: The basic formula used in time and work problems is:

   Total Work = Rate of Work × Time

   This can be rearranged to find the time required:

   Time = Total Work / Rate of Work

5. Approach to Solving Problems:

   • Identify the total work to be done.
   • Determine the rates of each person or group involved in the work.
   • Use the formula to find the time required for each person or group to complete the work individually.
   • If people are working together, add their individual times to find the total time required for them to complete the work together.

6. Types of Problems:

   • Problems involving people working together at different rates.
   • Problems involving people joining or leaving the work.
   • Problems involving work being done in stages or in different conditions.

7. Example:

   Suppose it takes 6 hours for person A to complete a task, and it takes 8 hours for person B to complete the same task. How long would it take for both A and B to complete the task if they work together?

   Let's denote the total work as 1 (which means completing the entire task).

   Person A's rate of work = 1 job / 6 hours = 1/6 job per hour.

   Person B's rate of work = 1 job / 8 hours = 1/8 job per hour.

   Working together, their combined rate of work is (1/6 + 1/8) job per hour.

   Total time required = Total work / Combined rate of work = 1 / (1/6 + 1/8) = 1 / (4/24 + 3/24) = 1 / (7/24) = 24 / 7 hours ≈ 3.43 hours

   So, it would take approximately 3.43 hours for both A and B to complete the task working together.