Finding Boat Speed in Still Water

This problem involves calculating the speed of a boat in still water and the speed of the stream using information about upstream and downstream journeys.

Let's define the variables:

• Let the speed of the boat in still water be $B$ km/hr.
• Let the speed of the stream be $S$ km/hr.

Based on these variables, we can define the upstream and downstream speeds:

• Upstream speed = Speed of boat against the stream = $(B - S)$ km/hr.
• Downstream speed = Speed of boat with the stream = $(B + S)$ km/hr.

We are given information about the time taken for two different journeys. Remember that Time = Distance / Speed.

Setting Up Equations for Boat Speed

Case 1: A boat covers 2 km upstream and 4 km downstream in 120 minutes.

First, convert the time to hours: 120 minutes = $\frac{120}{60}$ hours = 2 hours.

Time taken for 2 km upstream = $\frac{2}{B - S}$ hours.

Time taken for 4 km downstream = $\frac{4}{B + S}$ hours.

The total time for Case 1 is 2 hours. So, we get the equation:

$\frac{2}{B - S} + \frac{4}{B + S} = 2$ (Equation 1)

Case 2: The boat covers 4 km upstream and 6 km downstream in 210 minutes.

Convert the time to hours: 210 minutes = $\frac{210}{60}$ hours = $\frac{21}{6}$ hours = $\frac{7}{2}$ hours.

Time taken for 4 km upstream = $\frac{4}{B - S}$ hours.

Time taken for 6 km downstream = $\frac{6}{B + S}$ hours.

The total time for Case 2 is $\frac{7}{2}$ hours. So, we get the equation:

$\frac{4}{B - S} + \frac{6}{B + S} = \frac{7}{2}$ (Equation 2)

Solving for Upstream and Downstream Speeds

We now have a system of two equations with two variables, $\frac{1}{B - S}$ and $\frac{1}{B + S}$. To make solving easier, let's use substitution:

• Let $u = \frac{1}{B - S}$
• Let $v = \frac{1}{B + S}$

Substituting $u$ and $v$ into Equation 1 and Equation 2:

Equation 1 becomes: $2u + 4v = 2$

Dividing by 2, we get: $u + 2v = 1$ (Equation 3)

Equation 2 becomes: $4u + 6v = \frac{7}{2}$ (Equation 4)

Now we can solve the system of linear equations (Equation 3 and Equation 4) for $u$ and $v$. Let's multiply Equation 3 by 4:

$4 \times (u + 2v) = 4 \times 1$

$4u + 8v = 4$ (Equation 5)

Subtract Equation 4 from Equation 5:

$(4u + 8v) - (4u + 6v) = 4 - \frac{7}{2}$

$2v = \frac{8}{2} - \frac{7}{2}$

$2v = \frac{1}{2}$

$v = \frac{1}{4}$

Now substitute the value of $v = \frac{1}{4}$ back into Equation 3:

$u + 2 \left(\frac{1}{4}\right) = 1$

$u + \frac{1}{2} = 1$

$u = 1 - \frac{1}{2}$

$u = \frac{1}{2}$

Calculating Boat Speed and Stream Speed

We found $u = \frac{1}{B - S} = \frac{1}{2}$ and $v = \frac{1}{B + S} = \frac{1}{4}$.

From $u = \frac{1}{B - S} = \frac{1}{2}$, we get:

$B - S = 2$ (Equation 6)

From $v = \frac{1}{B + S} = \frac{1}{4}$, we get:

$B + S = 4$ (Equation 7)

Now we have a simple system of equations for $B$ and $S$. Add Equation 6 and Equation 7:

$(B - S) + (B + S) = 2 + 4$

$2B = 6$

$B = \frac{6}{2}$

$B = 3$

The speed of the boat in still water is 3 km/hr.

We can also find the speed of the stream by substituting $B=3$ into Equation 7:

$3 + S = 4$

$S = 4 - 3$

$S = 1$ km/hr.

Summary of Speeds

Final Answer

The speed of the boat in still water is 3 km/hr.