Problem Setup and Initial Conditions

Let the initial distance be $D_1 = 1400$ km, initial time duration be $d_1 = 80$ days, and initial rest time per day be $T_{rest1} = 10$ hours.

The new distance is $D_2 = 1000$ km, the new rest time per day is $T_{rest2} = 16$ hours, and the new speed is $S_2 = 2.5 \times S_1$, where $S_1$ is the initial speed.

We need to find the new number of days, $d_2$.

Initial Walking Time Calculation

A day has 24 hours. The initial time available for walking per day is:

Walking hours per day (initial) $= 24 - T_{rest1} = 24 - 10 = 14$ hours/day.

Total initial walking time $= d_1 \times (\text{Walking hours per day (initial)}) = 80 \times 14 = 1120$ hours.

The initial speed $S_1$ is given by:

$S_1 = \frac{D_1}{\text{Total initial walking time}} = \frac{1400 \text{ km}}{1120 \text{ hours}} = 1.25$ km/hour.

New Conditions and Speed Calculation

The new time available for walking per day is:

Walking hours per day (new) $= 24 - T_{rest2} = 24 - 16 = 8$ hours/day.

The new speed $S_2$ is 2.5 times the initial speed:

$S_2 = 2.5 \times S_1 = 2.5 \times 1.25 = 3.125$ km/hour.

Final Calculation for New Duration

The total walking time required to cover the new distance $D_2$ at speed $S_2$ is:

Total required walking time $= \frac{D_2}{S_2} = \frac{1000 \text{ km}}{3.125 \text{ km/hour}} = 320$ hours.

The new number of days $d_2$ can be calculated using the new walking hours per day:

$d_2 = \frac{\text{Total required walking time}}{\text{Walking hours per day (new)}} = \frac{320 \text{ hours}}{8 \text{ hours/day}} = 40$ days.

Alternative Ratio Method

The number of days ($d$) is directly proportional to the distance ($D$) and inversely proportional to the walking hours per day ($24 - T_{rest}$) and the speed ($S$).

So, $d \propto \frac{D}{(24 - T_{rest}) \times S}$.

Therefore, $\frac{d_2}{d_1} = \frac{D_2}{D_1} \times \frac{(24 - T_{rest1})}{(24 - T_{rest2})} \times \frac{S_1}{S_2}$.

Substituting the known values:

• $D_1 = 1400$, $d_1 = 80$, $T_{rest1} = 10 \implies (24 - 10) = 14$ hours/day
• $D_2 = 1000$, $d_2 = ?$, $T_{rest2} = 16 \implies (24 - 16) = 8$ hours/day
• $S_2 = 2.5 \times S_1 \implies \frac{S_1}{S_2} = \frac{1}{2.5}$

Plugging these into the formula:

$\frac{d_2}{80} = \frac{1000}{1400} \times \frac{14}{8} \times \frac{1}{2.5}$

$\frac{d_2}{80} = \frac{10}{14} \times \frac{14}{8} \times \frac{1}{2.5}$

$\frac{d_2}{80} = \frac{5}{7} \times \frac{14}{8} \times \frac{1}{2.5}$

$\frac{d_2}{80} = \frac{5 \times 14}{7 \times 8 \times 2.5}$

$\frac{d_2}{80} = \frac{70}{140} = \frac{1}{2}$

This calculation appears incorrect. Let's re-evaluate the ratio.

Correct ratio setup:

Speed $S = \frac{\text{Distance}}{\text{Total Walking Hours}} = \frac{D}{d \times (24 - T_{rest})}$.

So, $S \times d \times (24 - T_{rest}) = D$.

$\frac{S_2 \times d_2 \times (24 - T_{rest2})}{S_1 \times d_1 \times (24 - T_{rest1})} = \frac{D_2}{D_1}$.

Rearranging for $d_2$:

$d_2 = d_1 \times \frac{D_2}{D_1} \times \frac{S_1}{S_2} \times \frac{(24 - T_{rest1})}{(24 - T_{rest2})}$.

Substitute values:

$d_2 = 80 \times \frac{1000}{1400} \times \frac{1}{2.5} \times \frac{(24 - 10)}{(24 - 16)}$

$d_2 = 80 \times \frac{10}{14} \times \frac{1}{2.5} \times \frac{14}{8}$

$d_2 = 80 \times \frac{5}{7} \times \frac{1}{2.5} \times \frac{7}{4}$

$d_2 = 80 \times \frac{5 \times 1 \times 7}{7 \times 2.5 \times 4}$

$d_2 = 80 \times \frac{35}{70}$

$d_2 = 80 \times \frac{1}{2}$

Wait, the ratio method result $d_2=40$ conflicts with the first method. Let's recheck the first method carefully.

Initial Speed $S_1 = 1.25$ km/hr. Correct.

New Speed $S_2 = 2.5 \times 1.25 = 3.125$ km/hr. Correct.

Total walking time needed for 1000 km = $1000 / 3.125 = 320$ hours. Correct.

New walking time per day = $24 - 16 = 8$ hours/day. Correct.

Number of days $d_2 = 320 / 8 = 40$ days. Correct.

Let's recheck the ratio formula application.

$d_2 = 80 \times \frac{1000}{1400} \times \frac{1}{2.5} \times \frac{14}{8}$

$d_2 = 80 \times (\frac{1000}{1400}) \times (\frac{1}{2.5}) \times (\frac{14}{8})$

$d_2 = 80 \times (\frac{5}{7}) \times (\frac{2}{5}) \times (\frac{7}{8})$

$d_2 = 80 \times \frac{5 \times 2 \times 7}{7 \times 5 \times 8}$

$d_2 = 80 \times \frac{70}{280}$

$d_2 = 80 \times \frac{1}{4}$

$d_2 = 20$ days.

There seems to be a discrepancy between the two methods or my application of the ratio method. Let's rethink the relationship.

Speed is distance covered per unit *actual walking time*. Total walking time depends on days and hours per day.

Let $W_1$ be the total walking hours in the first scenario. $W_1 = d_1 \times (24 - T_{rest1}) = 80 \times (24 - 10) = 80 \times 14 = 1120$ hours.

Speed $S_1 = D_1 / W_1 = 1400 / 1120 = 1.25$ km/h.

Let $W_2$ be the total walking hours in the second scenario.

New speed $S_2 = 2.5 \times S_1 = 2.5 \times 1.25 = 3.125$ km/h.

New distance $D_2 = 1000$ km.

The total walking time required for the second scenario is $W_2 = D_2 / S_2 = 1000 / 3.125 = 320$ hours.

In the second scenario, the man walks for $24 - T_{rest2} = 24 - 16 = 8$ hours per day.

The number of days required is $d_2 = W_2 / (\text{hours walked per day}) = 320 / 8 = 40$ days.

The first method (step-by-step calculation) appears robust. Let's re-examine the ratio setup for potential errors.

Relationship: $S = D / (d \times (24 - T_{rest}))$

$\implies S \times d \times (24 - T_{rest}) = D$

Consider ratios:

$\frac{S_2 \times d_2 \times (24 - T_{rest2})}{S_1 \times d_1 \times (24 - T_{rest1})} = \frac{D_2}{D_1}$

Substitute $S_2 = 2.5 S_1$: $\frac{2.5 S_1 \times d_2 \times (24 - T_{rest2})}{S_1 \times d_1 \times (24 - T_{rest1})} = \frac{D_2}{D_1}$

Cancel $S_1$: $\frac{2.5 \times d_2 \times (24 - T_{rest2})}{d_1 \times (24 - T_{rest1})} = \frac{D_2}{D_1}$

Solve for $d_2$: $d_2 = d_1 \times \frac{D_2}{D_1} \times \frac{(24 - T_{rest1})}{(24 - T_{rest2})} \times \frac{1}{2.5}$

Plug in values:

$d_2 = 80 \times \frac{1000}{1400} \times \frac{(24 - 10)}{(24 - 16)} \times \frac{1}{2.5}$

$d_2 = 80 \times \frac{10}{14} \times \frac{14}{8} \times \frac{1}{2.5}$

$d_2 = 80 \times \frac{10}{14} \times \frac{14}{8 \times 2.5}$

$d_2 = 80 \times \frac{10}{14} \times \frac{14}{20}$

$d_2 = 80 \times \frac{10 \times 14}{14 \times 20}$

$d_2 = 80 \times \frac{140}{280}$

$d_2 = 80 \times \frac{1}{2}$

$d_2 = 40$ days.

The ratio method now yields 40 days, consistent with the first method. The error was in the simplification step during the previous attempt.

Final Answer is 40 days.