Understanding Work, Workers, and Time Problems

This question involves a common type of problem where the amount of work done is related to the number of workers, the hours they work per day, and the number of days they take to complete the work. The core idea is that the total amount of work required to demolish the building is constant.

Formulating the Work Relationship

We can think of the total work as the product of the number of workers, the number of days, and the number of hours worked per day, assuming each worker works at the same rate. This relationship can be expressed as:

\( \text{Work} = \text{Number of Workers} \times \text{Number of Days} \times \text{Hours per Day} \)

Since the work (demolishing the same building) is the same in both scenarios given in the question, we can set up an equation:

\( (\text{Workers}_1 \times \text{Days}_1 \times \text{Hours}_1) = (\text{Workers}_2 \times \text{Days}_2 \times \text{Hours}_2) \)

Applying the Given Information

From the question, we have the following information:

• Scenario 1: 14 workers, 16 hours per day, 81 days.
• Scenario 2: 18 workers, 24 hours per day, unknown number of days.

Let's plug these values into our equation:

\( 14 \times 81 \times 16 = 18 \times \text{Days}_2 \times 24 \)

Solving for the Unknown Number of Days

Our goal is to find \(\text{Days}_2\). We can rearrange the equation to solve for \(\text{Days}_2\):

\( \text{Days}_2 = \frac{14 \times 81 \times 16}{18 \times 24} \)

Now, let's calculate the value:

• First, multiply the numbers on the left side: \(14 \times 81 \times 16 = 1134 \times 16 = 18144\)
• Next, multiply the known numbers on the right side: \(18 \times 24 = 432\)
• So, the equation becomes: \(18144 = 432 \times \text{Days}_2\)
• Now, divide 18144 by 432 to find \(\text{Days}_2\):

\( \text{Days}_2 = \frac{18144}{432} \)

Let's perform the division:

\( \frac{18144}{432} = 42 \)

Alternatively, we can simplify the fraction directly:

\( \text{Days}_2 = \frac{14 \times 81 \times 16}{18 \times 24} \)

• Divide 81 by 9 and 18 by 9: \( \frac{14 \times 9 \times 16}{2 \times 24} \)
• Divide 16 by 8 and 24 by 8: \( \frac{14 \times 9 \times 2}{2 \times 3} \)
• Cancel out the 2 in the numerator and denominator: \( \frac{14 \times 9}{3} \)
• Divide 9 by 3: \( 14 \times 3 \)
• \(14 \times 3 = 42\)

Both methods give the same result.

Conclusion

Therefore, 18 workers working 24 hours per day can demolish the same building in 42 days.

Revision Table: Work, Workers, Hours, Days

Additional Information on Work Rate

In problems like these, the 'work rate' of a single worker is often assumed to be constant. The total work rate of a group of workers is the sum of their individual rates. When workers work more hours per day, their effective contribution per day increases.

The formula \(M \times D \times H = \text{Constant Work}\) is based on the idea that the total 'worker-hours' (or 'worker-day-hours') needed for a specific task is fixed. As you increase any factor on the left side (Workers, Days, Hours), you generally need to decrease another factor to keep the total constant, assuming the total work remains the same.

Key concepts to remember:

• If the number of workers increases, the number of days needed decreases (assuming constant hours/day and work rate).
• If the hours worked per day increase, the number of days needed decreases (assuming constant workers and work rate).
• This type of problem is an example of inverse proportionality combined with direct proportionality in a multi-variable relationship.