Two pipes A and B can fill a cistern in 30 minutes and 40 minutes respectively. Both the pipes are opened. Find when the second pipe B must be turned of so the cistern may just be full in 10 minutes. 

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Efficient cistern filling strategy: Calculate the ideal shutdown time for pipe B to achieve a complete fill within 10 minutes, alongside pipe A, operating with a 30-minute fill time.

Two pipes A and B can fill a cistern in 30 minutes and 40 minutes respectively. Both the pipes are opened. Find when the second pipe B must be turned of so the cistern may just be full in 10 minutes.

Pipe B must be turned off after 26.67 minutes to ensure the cistern is full in exactly 10 minutes.

Let’s find the time when pipe B needs to be turned off to fill the cistern in 10 minutes.

1. Find the portion filled by pipe A in 10 minutes:

• Pipe A fills the cistern in 30 minutes.
• In 1 minute, pipe A fills 1/30th of the cistern.
• In 10 minutes, pipe A fills (1/30) * 10 = 1/3rd of the cistern.

2. Calculate the remaining portion to be filled by pipe B:

• The total cistern capacity is 1.
• Pipe A fills 1/3rd, so the remaining portion for pipe B is 1 – 1/3 = 2/3rd of the cistern.

3. Determine the time needed for pipe B to fill the remaining portion:

• Pipe B fills the entire cistern in 40 minutes.
• To fill 2/3rd of the cistern, the time taken by pipe B is (2/3) * 40 = 26.67 minutes (rounded to two decimal places).

Therefore, pipe B must be turned off after 26.67 minutes to ensure the cistern is full in exactly 10 minutes.

Time and Work in Mathematics

Time and work problems in mathematics involve determining the amount of work done by individuals or groups working together over a certain period of time. These problems typically require understanding the rate at which each individual or group can complete a task and then using that information to find out how long it would take them to complete the task when working together.

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Here’s a general approach to solving time and work problems:

1. Understand the task: Read the problem carefully to understand what work needs to be done and how many people or groups are involved.

2. Identify the rates: Determine the rate at which each person or group can complete the work. This could be given in terms of work per unit time (e.g., tasks completed per hour).

3. Use the formula: The basic formula used in time and work problems is: Work = Rate×Time If there are multiple individuals or groups involved, you can add their rates to find the combined rate.

4. Solve for the unknown: Once you have the combined rate, use it to find the time it would take to complete the work.

5. Check your answer: Make sure your solution makes sense in the context of the problem. For example, ensure the time is reasonable, and the units are correct.

Let’s consider an example:

Example: If A can complete a task in 6 hours and B can complete the same task in 8 hours, how long would it take for both A and B to complete the task working together?

Solution:

1. Rate of A = 1/6 tasks per hour Rate of B = 1/8 tasks per hour
2. Combined rate = 1/6 + 1/8 = 7/24 tasks per hour
3. Let t be the time taken for both A and B to complete the task together.
4. Work = Rate × Time 1 = (7/24) × t Solving for t: t = 24/7 hours
5. The time taken for both A and B to complete the task together is approximately 3.43 hours.

That’s the basic process for solving time and work problems in mathematics. Remember to practice with various examples to become proficient in solving them.

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