P and S can complete a piece of work in 20 and 15 days respectively. They worked together for 6 days, after which S was replaced by A. If the work would be finished in next 5 days,..

P and S can complete a piece of work in 20 and 15 days respectively. They worked together for 6 days, after which S was replaced by A. If the work would be finished in next 5 days, Find the number of days in which A alone could complete the work.

A alone can complete the work in 40 days.

Let's find the number of days A alone can complete the work.

1. Calculate the combined work rate of P and S:

• P's work rate per day = 1/20 (as P can complete the work in 20 days)
• S's work rate per day = 1/15 (as S can complete the work in 15 days)
• Combined work rate of P and S = P's rate + S's rate = 1/20 + 1/15 = 7/60 (work completed per day by both P and S)

2. Find the work completed by P and S in 6 days:

• Work done by P and S in 6 days = Combined rate * Time = 7/60 * 6 = 7/10 (work completed)

3. Calculate the remaining work:

• Total work = 1 (as the whole work needs to be done)
• Remaining work = Total work - Work done by P and S = 1 - 7/10 = 3/10

4. Find the combined work rate of P and A:

• We know the remaining work (3/10) is completed by P and A in 4 days.
• Let A's work rate per day be x.
• Combined rate of P and A = (Remaining work) / (Time taken) = 3/10 / 4 = 3/40

5. Express the combined rate in terms of P's and A's rates:

• Combined rate of P and A = P's rate + A's rate = 1/20 + x

6. Equate both expressions for the combined rate:

• 1/20 + x = 3/40
• Solve for x (A's rate):
  • x = 3/40 - 1/20
  • x = 1/40

7. Find the number of days A alone can complete the work:

• A's rate per day = 1/40
• Time taken by A to complete the work = 1 / (A's rate) = 1 / (1/40) = 40 days

Therefore, A alone can complete the work in 40 days.

Time and Work in Mathematics

Time and work problems are a common type of mathematical problem that involve determining the amount of time it takes for a certain number of workers, working at a certain rate, to complete a task. These problems often involve concepts of rates, proportions, and basic arithmetic.

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

1. Understand the problem: Read the problem carefully and identify what needs to be solved. Understand the given information, such as the number of workers, their individual rates of work, and the task to be completed.

2. Identify the variables: Typically, you'll have variables such as the number of workers, the rate at which they work, and the time taken to complete the task. Assign variables to these quantities.

3. Formulate the equation: Use the concept that work done equals the product of the rate of work and the time taken. The basic equation is:

   Total work = (Rate of work of one worker) × (Number of workers) × (Time taken)

   Alternatively, you can also use the formula:

   Total work = (1/Time taken) × (Number of workers)

4. Solve the equation: Once you have the equation, solve it to find the unknown quantity, which could be the time taken, the rate of work, or the number of workers.

5. Check your answer: Ensure that your solution makes sense in the context of the problem. Sometimes, it's helpful to double-check your calculations to avoid errors.

Let's consider an example:

Example: If it takes 6 hours for 8 workers to complete a task, how long will it take for 12 workers to complete the same task?

Solution:

1. Understand the problem: We want to find out how long it takes for 12 workers to complete a task if it took 8 workers 6 hours.

2. Identify the variables: Let T be the time taken for 12 workers to complete the task.

3. Formulate the equation:

   • For 8 workers: 6 = (1/T) × 8
   • For 12 workers: Total work = (1/T) × 12

4. Solve the equation:

   • From the equation for 8 workers: T = 8/6 = 4/3 hours.
   • Substitute T = 4/3 into the equation for 12 workers: Total work = 1 / (4/3) × 12 = 9 hours.

5. Check your answer: It makes sense that it takes less time with more workers, so 9 hours seems reasonable.