Given Information:
- Length of road to be constructed = 6 km
- Total time given to complete the road = 200 days
- Number of persons employed initially = 140
- Length of road completed in 60 days = 1.5 km

Approach:
To find the number of additional people needed to finish the work on time, we need to calculate the work done by the current workforce and then determine the additional workforce required to complete the remaining work within the given time frame.

Calculating Work Done:
- The total work to be done is constructing a 6 km road.
- In the first 60 days, only 1.5 km of road has been completed.
- Therefore, the work done in 60 days = 1.5 km.

Calculating Work Remaining:
- The remaining work can be calculated as the difference between the total work and the work done in 60 days.
- Remaining work = Total work - Work done in 60 days
- Remaining work = 6 km - 1.5 km = 4.5 km.

Calculating Work Done by Each Person:
- To calculate the number of additional people needed, we need to determine the work done by each person in the first 60 days.
- Work done by each person in 60 days = Work done / Number of persons employed initially
- Work done by each person in 60 days = 1.5 km / 140 persons.

Calculating Work Done by Each Person per Day:
- Since the work done by each person is calculated for 60 days, we need to find the work done by each person per day.
- Work done by each person per day = Work done by each person in 60 days / 60 days.
- Work done by each person per day = (1.5 km / 140 persons) / 60 days.

Calculating Total Work Done:
- Now, we need to calculate the total work done by all the people for the remaining 140 days.
- Total work done = Work done by each person per day * Number of persons * Remaining days
- Total work done = (1.5 km / 140 persons) / 60 days * 140 persons * 140 days.

Calculating Additional Workforce:
- To find the number of additional people needed to finish the work on time, we need to calculate the work remaining after 140 days and divide it by the work done by each person per day.
- Additional workforce needed = Remaining work / (Work done by each person per day * Remaining days)
- Additional workforce needed = 4.5 km / ((1.5 km / 140 persons) / 60 days * 140 persons * 140 days).

Simplifying the Equation:
- Simplifying the above equation will give us the number of additional people needed.
- Additional workforce needed = 40.

Therefore, the contractor would need to employ an additional 40 people to finish the work exactly on time.