Solving Work and Time Problems: X Boys and 2X Boys

This question involves a classic scenario in work and time problems where the number of workers and the amount of work change, and we need to find the new time taken. The core principle is that the total work done is proportional to the number of workers and the time they spend working, assuming their efficiency is constant.

Understanding the Relationship between Men, Work, and Time

In problems like this, we often use the formula based on the idea that Men $\times$ Days $\times$ Efficiency = Work. If efficiency is constant (or the same for all workers), we can simplify this. For two different scenarios doing the same work, we have $M_1 \times D_1 = M_2 \times D_2$. If the work done is different, the relationship becomes $M_1 \times D_1 \times W_2 = M_2 \times D_2 \times W_1$, where $W_1$ and $W_2$ represent the amounts of work in scenario 1 and scenario 2, respectively.

Setting Up the Problem

Let's break down the given information into two scenarios:

Scenario 1: The Initial Condition

• Number of boys ($M_1$): X
• Time taken ($D_1$): 80 days
• Amount of work done ($W_1$): Let's consider this as 1 unit of work (the full work).

Scenario 2: The New Condition

• Number of boys ($M_2$): 2X
• Time taken ($D_2$): This is what we need to find. Let's call it T days.
• Amount of work done ($W_2$): Half of the same work, which is 0.5 units (or $\frac{1}{2}$).

Applying the Work-Time Formula

We will use the formula that relates men, days, and work done for two scenarios:

\( M_1 \times D_1 \times W_2 = M_2 \times D_2 \times W_1 \)

Step-by-Step Calculation

Now, let's substitute the values from our scenarios into the formula:

\( X \times 80 \times 0.5 = 2X \times T \times 1 \)

Simplify both sides of the equation:

Left side: \( X \times 80 \times 0.5 = 80X \times 0.5 = 40X \)

Right side: \( 2X \times T \times 1 = 2XT \)

So the equation becomes:

\( 40X = 2XT \)

We need to solve for T. We can divide both sides by 2X (assuming X is not zero, which it must be for there to be boys working):

\( \frac{40X}{2X} = T \)

\( \frac{40}{2} = T \)

\( 20 = T \)

So, the time taken by 2X boys to do half of the work is 20 days.

Conclusion

Using the relationship between the number of workers, time, and the amount of work, we found that 2X boys will take 20 days to complete half of the work that X boys completed in 80 days.

Revision Table: Work and Time Concepts

Additional Information: Efficiency in Work Problems

Sometimes, work and time problems introduce the concept of efficiency. If the efficiency of workers is different, the formula expands to include efficiency (E). The total work done would be proportional to Men $\times$ Days $\times$ Efficiency. For two scenarios with potentially different efficiencies:

\( M_1 \times D_1 \times E_1 \times W_2 = M_2 \times D_2 \times E_2 \times W_1 \)

In this particular problem, since all workers are referred to simply as 'boys' without any mention of different capabilities, we assume their efficiency is the same ($E_1 = E_2$), which is why the efficiency term cancels out and we can use the simplified formula \( M_1 \times D_1 \times W_2 = M_2 \times D_2 \times W_1 \).

Always carefully read the question to identify if efficiency is a factor to consider. If not mentioned, assume identical efficiency for all workers of the same type.