Understanding Rowing Speeds

This problem involves calculating the distance a man travels on a river, considering his rowing speed and the river's current. We need to understand how the current affects his speed when moving downstream (with the current) and upstream (against the current).

• Speed in Still Water: The speed the man can row if the water were not moving. Let's denote this as $S_w$. Given $S_w = 9$ km/h.
• Speed of the River Current: The speed at which the water is flowing. Let's denote this as $S_c$. Given $S_c = 4$ km/h.

Calculating Rowing Speeds

The direction of rowing relative to the current changes the effective speed:

• Downstream Speed ($S_d$): When rowing downstream, the man's speed is added to the current's speed. $$ S_d = S_w + S_c $$ $$ S_d = 9 \text{ km/h} + 4 \text{ km/h} = 13 \text{ km/h} $$
• Upstream Speed ($S_u$): When rowing upstream, the current's speed subtracts from the man's speed. $$ S_u = S_w - S_c $$ $$ S_u = 9 \text{ km/h} - 4 \text{ km/h} = 5 \text{ km/h} $$

Relating Time and Distance

The problem states that the time taken for the upstream journey is 8 hours longer than the time taken for the downstream journey for the same distance. Let the distance be $D$ km.

• Time taken downstream, $T_d = \frac{\text{Distance}}{\text{Downstream Speed}} = \frac{D}{S_d} = \frac{D}{13}$ hours.
• Time taken upstream, $T_u = \frac{\text{Distance}}{\text{Upstream Speed}} = \frac{D}{S_u} = \frac{D}{5}$ hours.
• We are given the relationship: $T_u = T_d + 8$ hours.

Step-by-Step Distance Calculation

Now, we can set up an equation using the time difference:

1. Substitute the expressions for $T_u$ and $T_d$: $$ \frac{D}{5} = \frac{D}{13} + 8 $$
2. Rearrange the equation to solve for $D$. Move the terms with $D$ to one side: $$ \frac{D}{5} - \frac{D}{13} = 8 $$
3. Find a common denominator for the fractions (which is $5 \times 13 = 65$): $$ \frac{13D}{65} - \frac{5D}{65} = 8 $$
4. Combine the fractions: $$ \frac{13D - 5D}{65} = 8 $$ $$ \frac{8D}{65} = 8 $$
5. Solve for $D$. Multiply both sides by 65: $$ 8D = 8 \times 65 $$
6. Divide both sides by 8: $$ D = \frac{8 \times 65}{8} $$ $$ D = 65 $$

Therefore, the distance to the place is 65 km.

Final Answer Summary

The distance calculated based on the given rowing speeds and the time difference between upstream and downstream journeys is 65 km.