Two pipes A and B can fill a tank is 8 minutes and 14 minutes respectively. If both the pipes are opened simultaneously, and the pipe A is closed after 3 minutes.

Two pipes A and B can fill a tank is 8 minutes and 14 minutes respectively. If both the pipes are opened simultaneously, and the pipe A is closed after 3 minutes, Then how much more time will it take to fill the tank by pipe B?

It will take approximately 5.75 minutes for pipe B to fill the remaining portion of the tank.

To solve this problem, let's first find the rate at which each pipe fills the tank.

Pipe A fills the tank in 8 minutes, so its rate of filling is 1/8 of the tank per minute.

Pipe B fills the tank in 14 minutes, so its rate of filling is 1/14 of the tank per minute.

When both pipes A and B are opened simultaneously, they fill the tank at a combined rate.

Combined rate = Rate of A + Rate of B

Combined rate = 1/8 + 1/14

Combined rate = (7/56 + 4/56) = 11/56

So, together, pipes A and B fill 11/56 of the tank per minute.

Now, let's calculate how much of the tank is filled after 3 minutes when both pipes are open.

Tank filled in 3 minutes = Combined rate × Time

Tank filled in 3 minutes = (11/56) × 3

Tank filled in 3 minutes = 33/56 of the tank

After 3 minutes, pipe A is closed, and only pipe B continues to fill the tank.

So, we need to find out how much more time it will take for pipe B to fill the remaining portion of the tank.

Remaining portion of the tank = 1 - 33/56

Remaining portion of the tank = 56/56 - 33/56

Remaining portion of the tank = 23/56 of the tank

Now, we need to calculate how much time it will take for pipe B to fill 23/56 of the tank.

Time taken by B = (Remaining portion of the tank) ÷ (Rate of B)

Time taken by B = (23/56) ÷ (1/14)

Time taken by B = (23/56) × (14/1)

Time taken by B = 322/56

Time taken by B ≈ 5.75 minutes

So, after pipe A is closed, it will take approximately 5.75 minutes for pipe B to fill the remaining portion of the tank.

Time and Work in Mathematics

Time and work problems in mathematics typically involve calculating the amount of work done by a group of people or machines working together over a period of time. These problems often require understanding how the rate of work of each individual or machine contributes to the total work completed.

Here are some key concepts and strategies for solving time and work problems:

1. Rate of Work: The rate of work of a person or machine is the amount of work they can complete in a unit of time. It's usually expressed as "work done per unit time," such as tasks per hour or items produced per day.

2. Time and Work Relationship: The fundamental relationship between time, work, and rate of work is expressed as:

   Work = Rate × Time

   This formula states that the amount of work completed is equal to the rate of work multiplied by the time spent working.

3. Inverse Relationship: There's an inverse relationship between the rate of work and the time taken to complete a task. If someone's rate of work increases, the time taken to complete the task decreases, and vice versa.

4. Common Types of Problems:

   a. Single Person or Machine: Calculate how long it takes for a single person or machine to complete a task.

   b. Multiple Persons or Machines Working Together: Determine how long it takes for a group of people or machines to complete a task when working together.

   c. Work Stopped or Started: Adjustments are sometimes made to the time or work done when work is started or stopped at various points.

5. Fractional Work: Sometimes, problems involve fractions of work. For example, if a person completes half of a task before another person takes over, you need to account for this fractional work.

6. Efficiency and Work Distribution: In problems involving multiple workers, it's important to consider the efficiency of each worker and how the work is distributed among them.

7. Common Strategies:

   a. Using Rates: Express the rates of work of each person or machine involved, and then use them to calculate the total work completed.

   b. Setting Up Equations: Formulate equations based on the given information and solve them to find the unknowns.

   c. Fractional Parts: If someone completes only a part of the work, express it as a fraction of the total work.

   d. Work Done in 1 Unit of Time: Sometimes, it's easier to consider how much work is done by all workers in 1 unit of time, then scale it up to find the total time required for the entire task.

8. Practice: Like any mathematical concept, practice is essential for mastering time and work problems. Work through a variety of problems to become comfortable with different scenarios and approaches.

By understanding these principles and practicing regularly, you can effectively solve time and work problems in mathematics.