Understanding Work and Time Problems

This question deals with the concept of work and time, specifically how the number of workers, the time spent working (days and hours per day), and the amount of work are related. A common principle in these problems is that the total amount of 'work' done is proportional to the number of workers, the number of days they work, and the hours they work each day, assuming constant efficiency.

The Man-Days-Hours Formula

For a fixed amount of work, the product of the number of men ($M$), the number of days ($D$), and the number of hours per day ($H$) is constant. We can write this as:

$\text{Total Work} = M \times D \times H$

If the work is the same in two different scenarios, we can equate the products:

$M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2$

Where the subscript '1' refers to the first scenario and '2' refers to the second scenario.

Applying the Formula to the Problem

In the given problem, we have two scenarios:

• Scenario 1: 18 men complete the work in 15 days working 8 hours a day.
• Scenario 2: A certain number of men ($M_2$) complete the same work in 12 days working 9 hours a day.

Let's list the values:

• $M_1 = 18$ men
• $D_1 = 15$ days
• $H_1 = 8$ hours/day
• $M_2 = ?$ (unknown number of men)
• $D_2 = 12$ days
• $H_2 = 9$ hours/day

Step-by-Step Calculation

Using the formula $M_1 \times D_1 \times H_1 = M_2 \times D_2 \times H_2$, we substitute the known values:

$18 \times 15 \times 8 = M_2 \times 12 \times 9$

Now, we calculate the product on the left side:

$18 \times 15 = 270$

$270 \times 8 = 2160$

So, the equation becomes:

$2160 = M_2 \times 12 \times 9$

Next, calculate the product of days and hours in the second scenario:

$12 \times 9 = 108$

The equation is now:

$2160 = M_2 \times 108$

To find $M_2$, we need to isolate it by dividing both sides of the equation by 108:

$M_2 = \frac{2160}{108}$

Performing the division:

$M_2 = 20$

So, 20 men can complete the same work in 12 days working 9 hours a day.

Conclusion

The number of men required in the second scenario is 20. This corresponds to option 3.

Equating the M × D × H products for both scenarios:

$2160 = M_2 \times 108$

$M_2 = \frac{2160}{108} = 20$

Revision Table: Work and Time Concepts

Additional Information: Inverse Proportionality in Work Problems

In work and time problems, variables are often inversely proportional when the total work is fixed. For example:

• If the number of men increases, the number of days required to complete the same work decreases (inverse proportionality).
• If the hours worked per day increases, the number of days required decreases (inverse proportionality).
• If the number of men decreases, the number of hours per day they need to work might increase to finish in the same number of days (inverse proportionality).

The formula $M \times D \times H = \text{Constant Work}$ shows this relationship clearly. If Work is constant, any increase in one variable (like M) must be balanced by a decrease in one or both of the other variables (D or H).

In this problem, both the number of days (from 15 to 12) and the hours per day (from 8 to 9) changed, requiring a calculation involving all three factors.