Solving Shyam's Boat Travel Time Problem

This problem involves calculating the speed of a boat in still water and the speed of the stream to determine the time taken for an upstream journey. We need to find out how long Shyam would take to travel 36 km upstream.

Step 1: Calculate Speed in Still Water

Shyam rowed a distance of 24 km in 3 hours in still water. The formula for speed is distance divided by time.

Speed in still water ($S_s$) = Distance / Time

Using LaTeX notation:

$$ S_s = \frac{\text{Distance}}{\text{Time}} = \frac{24 \text{ km}}{3 \text{ hours}} $$

$$ S_s = 8 \text{ km/hr} $$

So, Shyam's speed in still water is 8 km/hr.

Step 2: Calculate Speed Downstream

Shyam rowed 20 km in 2 hours downstream. We can calculate his downstream speed using the same formula.

Speed downstream ($S_d$) = Distance / Time

Using LaTeX notation:

$$ S_d = \frac{\text{Distance}}{\text{Time}} = \frac{20 \text{ km}}{2 \text{ hours}} $$

$$ S_d = 10 \text{ km/hr} $$

Shyam's speed when rowing downstream is 10 km/hr.

Step 3: Calculate Speed of the Stream

The speed downstream is the sum of the speed in still water and the speed of the stream ($S_w$).

Formula: $$ S_d = S_s + S_w $$

We know $S_d = 10$ km/hr and $S_s = 8$ km/hr. We can find $S_w$:

$$ 10 \text{ km/hr} = 8 \text{ km/hr} + S_w $$

Rearranging the formula to solve for $S_w$:

$$ S_w = S_d - S_s $$

$$ S_w = 10 \text{ km/hr} - 8 \text{ km/hr} $$

$$ S_w = 2 \text{ km/hr} $$

The speed of the stream is 2 km/hr.

Step 4: Calculate Speed Upstream

The speed upstream is the speed in still water minus the speed of the stream.

Formula: $$ S_u = S_s - S_w $$

Using the values we found:

$$ S_u = 8 \text{ km/hr} - 2 \text{ km/hr} $$

$$ S_u = 6 \text{ km/hr} $$

Shyam's speed when rowing upstream is 6 km/hr.

Step 5: Calculate Time to Travel Upstream

We need to find the time it would take Shyam to travel 36 km upstream. We use the formula: Time = Distance / Speed.

Time upstream = Distance upstream / Speed upstream

Using LaTeX notation:

$$ \text{Time Upstream} = \frac{36 \text{ km}}{S_u} $$

$$ \text{Time Upstream} = \frac{36 \text{ km}}{6 \text{ km/hr}} $$

$$ \text{Time Upstream} = 6 \text{ hours} $$

Therefore, Shyam would take 6 hours to travel 36 km upstream.