Short Answer:

Group work problems are solved step by step by first finding the work done by each worker in one day. This is called one day work or efficiency. After that, the efficiencies of all workers are added to find their combined work.

In simple words, we calculate individual work, add it for the group, and then find the total time needed to complete the task. This method makes group work problems easy and clear.

Detailed Explanation:

Group work problem steps

Group work problems are an important part of Time and Work in mathematics. These problems explain how two or more people complete a task together. Since each worker may have a different speed, we need a proper step-by-step method to solve such questions correctly.

The main idea in group work problems is efficiency. Efficiency means the amount of work done in one unit of time, usually one day. When many workers work together, their efficiencies are added. This gives the combined efficiency of the group. Once we know the combined efficiency, we can find how much time the group will take to complete the whole work.

Step 1: Read the problem carefully

The first step is to read the question carefully and understand what is given. Usually, the problem gives the time taken by each worker to complete the same work alone. For example, one worker may complete the work in 6 days and another in 12 days. We must identify these values clearly before starting the calculation.

This step is important because wrong reading can lead to wrong answers. We should also check whether all workers work together from the beginning or whether someone joins or leaves later. If the problem has such conditions, we must solve it in parts.

Step 2: Take total work as one unit

The next step is to take the total work as one complete unit. This means the whole work is assumed to be 1. This makes the calculation simple because we can use fractions to represent the work done by each worker in one day.

For example, if a worker completes the whole work in 5 days, then the work done in one day is 1/5. If another worker completes the same work in 10 days, then their one day work is 1/10.

This method is very useful because it helps us compare the work speed of different workers easily.

Step 3: Find one day work of each worker

After taking total work as 1, we find the one day work of each worker. One day work is found by dividing the total work by the number of days taken by the worker.

For example, if Worker A completes a task in 8 days, then A’s one day work is 1/8. If Worker B completes the same task in 12 days, then B’s one day work is 1/12.

This one day work is also called the efficiency of the worker. A bigger one day work means higher efficiency, and a smaller one day work means lower efficiency.

Step 4: Add the efficiencies

When workers work together, we add their one day work. This gives the combined efficiency of the group. Combined efficiency tells us how much work the group completes in one day.

For example, if A’s one day work is 1/8 and B’s one day work is 1/12, then their combined one day work is 1/8 + 1/12. After adding, we get the amount of work done by both workers together in one day.

This step is the most important part of solving group work problems because it shows the total speed of the group.

Step 5: Find total time taken

After finding the combined efficiency, we can find the total time needed to complete the work. Since total work is taken as 1, the time taken is found by dividing total work by combined efficiency.

The formula is: Time = Total Work ÷ Combined Efficiency.

For example, if the combined efficiency is 5/24, then time taken will be 1 ÷ 5/24, which is 24/5 days. This means the group will complete the work in 24/5 days.

This final step gives the answer to most group work problems.

Step 6: Use LCM method when needed

Sometimes, fractions may look difficult. In such cases, the LCM method can be used. In this method, we take total work as the LCM of the given days. This avoids fractions and makes the solution easier.

For example, if one worker takes 6 days and another takes 8 days, then the LCM of 6 and 8 is 24. So, total work is taken as 24 units. The first worker does 4 units per day, and the second worker does 3 units per day. Together they do 7 units per day.

Then total time is found by dividing total work by combined work per day. This method is especially useful in exams and quick calculations.

Special group work cases

Some group work problems are simple, but some have extra conditions. In such cases, we must solve them carefully and step by step.

Workers joining or leaving

Sometimes, one worker may leave after a few days, or another worker may join later. In these problems, we calculate the work done in different parts. First, we find the work done by the workers who started together. Then we calculate the remaining work and solve it according to the new group.

This method helps us avoid confusion and gives the correct answer.

Equal workers

If all workers have the same efficiency, group work becomes easier. For example, if one worker completes a task in 10 days, then two equal workers will complete it in 5 days. This is because the efficiency becomes double.

When workers are equal, more workers reduce the time in the same proportion.

Unequal workers

If workers have different efficiencies, we cannot simply divide time equally. We must find each worker’s one day work separately and then add them.

This is important because a faster worker contributes more, while a slower worker contributes less.

Conclusion:

Group work problems are solved by finding each worker’s one day work, adding their efficiencies, and then calculating the total time required. The method becomes easy when we follow the steps carefully. For difficult problems, the LCM method and part-wise calculation help in getting the correct answer.