A contractor agreed to construct a 6 km road in 200 days. He employed 140 persons for the work. After 60 days, he realized that only 1.5 km road has been completed. How many additional people would he need to employ in order to finish the work exactly on time?

The correct answer is 40.

This is a classic Work and Labour problem. We use the fundamental relationship that connects the number of workers, days, and the amount of work done:

$\frac{M_1\times D_1}{W_1}=\frac{M_2\times D_2}{W_2}$

where M = number of men, D = number of days, and W = amount of work (km of road).

Step 1: Identify what has been done and what remains.

• Total road to be built = 6 km
• Total time allowed = 200 days
• Workers initially employed = 140
• After 60 days, road completed = 1.5 km

So the remaining work and time are:
• Remaining road = 6 − 1.5 = 4.5 km
• Remaining days = 200 − 60 = 140 days

Step 2: Apply the Man-Days-Work formula.

From the first phase (already completed), we know the rate of work:

$\frac{M_1\times D_1}{W_1}=\frac{M_2\times D_2}{W_2}$

Substituting the known values:

$\frac{140\times 60}{1.5}=\frac{M_2\times 140}{4.5}$

Step 3: Solve the left-hand side.

$\frac{140\times 60}{1.5}=\frac{8400}{1.5}=5600$

Step 4: Solve for M₂.

$5600=\frac{M_2\times 140}{4.5}$

Multiply both sides by 4.5:

$M_2\times 140=5600\times 4.5=25200$

Divide both sides by 140:

$M_2=\frac{25200}{140}=180$

Step 5: Find the additional people needed.

Additional workers = M₂ − Original workers = 180 − 140 = 40

Why does this make sense intuitively?
In 60 days with 140 workers, only 1.5 km (one-quarter of 6 km) was completed. But only 140 out of 200 days remain — that's 70% of the total time left for 75% of the work. The workers are behind schedule, so more people are needed. Exactly 40 additional workers (bringing the total to 180) will ensure the remaining 4.5 km is completed in the remaining 140 days at the same rate of work per person per day.