Understanding Boat and Stream Speed Calculation

This question involves calculating the speed of the current given the speed of a boat moving downstream (in the direction of the stream) and upstream (against the direction of the stream). We need to find the speed of the current using the provided information.

Formulas for Boat and Stream Problems

Let $S_b$ represent the speed of the boat in still water (km/hr).
Let $S_c$ represent the speed of the current (km/hr).

• When the boat moves downstream, the speed of the current adds to the boat's speed. So, the downstream speed ($S_d$) is:

  $$ S_d = S_b + S_c $$

• When the boat moves upstream, the speed of the current subtracts from the boat's speed. So, the upstream speed ($S_u$) is:

  $$ S_u = S_b - S_c $$

Applying Formulas to the Given Data

From the question, we have:

• Speed downstream ($S_d$) = 46 km/hr
• Speed upstream ($S_u$) = 36 km/hr

We can set up two equations based on the formulas:

1. Equation 1: $$ S_b + S_c = 46 $$
2. Equation 2: $$ S_b - S_c = 36 $$

Step-by-Step Calculation for Current Speed

Our goal is to find the speed of the current ($S_c$). We can solve these two equations simultaneously.

Method 1: Subtracting the equations

Subtract Equation 2 from Equation 1:

$$ (S_b + S_c) - (S_b - S_c) = 46 - 36 $$

Simplify the equation:

$$ S_b + S_c - S_b + S_c = 10 $$

Combine like terms:

$$ 2S_c = 10 $$

Solve for $S_c$:

$$ S_c = \frac{10}{2} $$

$$ S_c = 5 $$

So, the speed of the current is 5 km/hr.

Method 2: Finding boat speed first

Add Equation 1 and Equation 2:

$$ (S_b + S_c) + (S_b - S_c) = 46 + 36 $$

Simplify the equation:

$$ 2S_b = 82 $$

Solve for $S_b$ (speed of the boat in still water):

$$ S_b = \frac{82}{2} $$

$$ S_b = 41 $$

Now, substitute the value of $S_b$ (41 km/hr) into Equation 1:

$$ 41 + S_c = 46 $$

Solve for $S_c$:

$$ S_c = 46 - 41 $$

$$ S_c = 5 $$

The speed of the current is 5 km/hr.

Conclusion

Both methods confirm that the speed of the current is 5 km/hr. This matches one of the options provided.