Bridge Building: Understanding Work and Time Problems

This problem involves a classic concept in quantitative aptitude: work and time. To solve this, we need to determine the work rate of men and women relative to each other and then calculate the total work required to build the bridge. Finally, we can find out how many men are needed to complete this total work in one week.

Problem Analysis: Bridge Construction Work

Let's define the work rate of one man as 'M' units of work per week, and the work rate of one woman as 'W' units of work per week. The total work required to build the bridge remains constant.

• Scenario 1: 1200 men and 500 women build the bridge in 2 weeks.
• Scenario 2: 900 men and 250 women build the bridge in 3 weeks.

Step-by-Step Solution: Calculating Men Needed

1. Establish Equations for Total Work

The total work done is the product of the number of workers, their individual rates, and the time taken.

From Scenario 1:

Total Work $= (1200 \times \text{M} + 500 \times \text{W}) \times 2$ weeks

Total Work $= 2400 \text{M} + 1000 \text{W} \quad \text{(Equation 1)}$

From Scenario 2:

Total Work $= (900 \times \text{M} + 250 \times \text{W}) \times 3$ weeks

Total Work $= 2700 \text{M} + 750 \text{W} \quad \text{(Equation 2)}$

2. Find the Relationship Between Men's and Women's Work Rates

Since the total work is the same in both scenarios, we can equate Equation 1 and Equation 2:

$2400 \text{M} + 1000 \text{W} = 2700 \text{M} + 750 \text{W}$

Rearrange the terms to group M and W:

$1000 \text{W} - 750 \text{W} = 2700 \text{M} - 2400 \text{M}$

$250 \text{W} = 300 \text{M}$

Divide both sides by 50 to simplify:

$5 \text{W} = 6 \text{M}$

This means that the work done by 5 women is equivalent to the work done by 6 men. We can express the work rate of one woman in terms of a man's work rate:

$\text{W} = \frac{6}{5} \text{M}$

3. Calculate Total Work in Terms of Men's Work Units

Now, substitute the value of 'W' from the relationship $W = \frac{6}{5} M$ into either Equation 1 or Equation 2 to find the total work in terms of men's work units. Let's use Equation 1:

Total Work $= 2400 \text{M} + 1000 \text{W}$

Total Work $= 2400 \text{M} + 1000 \left(\frac{6}{5} \text{M}\right)$

Total Work $= 2400 \text{M} + (200 \times 6) \text{M}$

Total Work $= 2400 \text{M} + 1200 \text{M}$

Total Work $= 3600 \text{M}$

This means the entire bridge building project requires 3600 "man-weeks" of work.

4. Determine Men Needed to Build the Bridge in One Week

We need to find out how many men ('X') are required to build the bridge in 1 week. The total work done by X men in 1 week must be equal to the total work required for the bridge.

Work done by X men in 1 week $= \text{X} \times \text{M} \times 1$ week $= \text{X M}$

Equate this to the total work required:

$\text{X M} = 3600 \text{M}$

Divide both sides by M:

$\text{X} = 3600$

Therefore, 3600 men will be needed to build the bridge in one week.

The total work required is equivalent to 3600 units of a man's work for one week. To complete this in exactly one week, 3600 men would be needed.