The problem asks for the number of days required to cover a new distance under modified conditions of rest and speed.

Step 1: Calculate Original Walking Time and Speed

First, determine the total time spent walking in the initial scenario.

• Total days = 80 days
• Rest per day = 10 hours
• Walking hours per day = 24 hours - 10 hours = 14 hours
• Total walking time = 14 hours/day * 80 days = 1120 hours

Now, calculate the original walking speed (S1).

• Distance = 1400 km
• Total walking time = 1120 hours
• Using the formula: Speed = Distance / Time
• S1 = $\\frac{1400 \\text{ km}}{1120 \\text{ hours}}$ = 1.25 km/hour

Step 2: Calculate New Speed and Walking Time Per Day

The problem states the new speed (S2) is 2.5 times the previous speed.

• S2 = 2.5 * S1 = 2.5 * 1.25 km/hour = 3.125 km/hour

Calculate the new walking hours per day.

• New rest per day = 16 hours
• New walking hours per day = 24 hours - 16 hours = 8 hours/day

Step 3: Calculate Total Walking Time Needed for New Distance

Determine the total time needed to cover the new distance at the new speed.

• New Distance = 1000 km
• New Speed (S2) = 3.125 km/hour
• Total walking time needed = New Distance / New Speed
• Total walking time needed = $\\frac{1000 \\text{ km}}{3.125 \\text{ km/hour}}$ = 320 hours

Step 4: Calculate the Number of Days Required

Finally, calculate the number of days required using the total walking time needed and the new walking hours per day.

• Total walking time needed = 320 hours
• New walking hours per day = 8 hours/day
• Number of Days = $\\frac{\\text{Total walking time needed}}{\\text{New walking hours per day}}$
• Number of Days = $\\frac{320 \\text{ hours}}{8 \\text{ hours/day}}$ = 40 days

Therefore, the man will cover 1000 km in 40 days.