A company contracts to paint 3 houses. Mr. Brown can paint a house in 6 days while Mr. Black would take 8 days and Mr. Blue 12 days. After 8 days Mr. Brown goes on vacation.

A company contracts to paint 3 houses. Mr.Brown can paint a house in 6 days while Mr.Black would take 8 days and Mr.Blue 12 days. After 8 days Mr.Brown goes on vacation and Mr. Black begins to work for a period of 6 days. How many days will it take Mr.blue to complete the contract?

It will take Mr. Blue 11 days to complete the contract.

Here's how you can solve this problem:

1. Work done by Mr. Brown:

• Mr. Brown can paint 1 house in 6 days, so in 8 days he paints 8/6 = 1.33 houses.

2. Work remaining after Mr. Brown leaves:

• Since they contracted to paint 3 houses, there are 3 - 1.33 = 1.67 houses remaining.

3. Work done by Mr. Black:

• Mr. Black can paint 1 house in 8 days, so in 6 days he paints 6/8 = 0.75 houses.

4. Work remaining for Mr. Blue:

• After Mr. Black's work, there are 1.67 - 0.75 = 0.92 houses left to paint.

5. Time taken by Mr. Blue:

• Mr. Blue can paint 1 house in 12 days, so to paint 0.92 houses he will take 0.92 * 12 = 11 days (approximately).

Therefore, it will take Mr. Blue 11 days to complete the contract.

Time and Work in Mathematics

Time and work problems in mathematics deal with calculating the time taken to complete a certain task when different individuals or machines work together at different rates. These types of problems often involve understanding the concept of work rate or efficiency and using that information to find the time it takes to complete a task.

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

1. Work Rate or Efficiency: The work rate or efficiency of a person or a machine is the amount of work they can complete in a unit of time. It's usually expressed in terms of work per hour or work per day.

2. Inverse Proportionality: Time and work are inversely proportional. This means that if the number of workers increases, the time required to complete a task decreases, and vice versa. Mathematically, if W represents the total work to be done, R represents the combined work rate of all workers, and T represents the time taken to complete the work, then W = R × T.

3. Fractional Work: Sometimes, workers might not work for the entire duration. In such cases, you need to calculate the fraction of the work they complete in a given time period.

4. Simultaneous Equations: Time and work problems often involve setting up simultaneous equations to represent the relationship between the number of workers, their individual work rates, and the total time taken to complete the task.

5. Units Conversion: Ensure that all units are consistent. If work rates are given in terms of days, hours, or minutes, make sure to convert them into the same unit before performing calculations.

6. Practice: Time and work problems can be solved efficiently with practice. Familiarize yourself with different problem-solving techniques and strategies, and work on a variety of problems to build your skills.

Here's a simple example to illustrate:

Example: If it takes 6 hours for 8 workers to complete a job, how long would it take for 12 workers to complete the same job, assuming they work at the same rate?

Solution: Let T be the time required for 12 workers to complete the job. Using the inverse proportionality concept: 8 × 6 = 12 × T T = (8 × 6) / 12 = 4 hours

So, it would take 12 workers 4 hours to complete the job.

Remember, time and work problems come in various forms, so it's essential to understand the underlying concepts and adapt your problem-solving approach accordingly.