Understanding the Boat and Stream Problem

This problem involves the concepts of boat speed in still water, stream speed, and how they affect the speed of the boat when moving downstream (with the current) and upstream (against the current).

Given Information

• Ratio of boat speed in still water to stream speed: \(v_b : v_s = 3 : 1\)
• Speed of the stream (\(v_s\)): 4 km/hr
• Total time for the round trip journey: 45 minutes

Calculating Boat Speed in Still Water

We are given the ratio of the boat's speed in still water (\(v_b\)) to the stream's speed (\(v_s\)) as 3:1. This can be written as:

\[ \frac{v_b}{v_s} = \frac{3}{1} \]

Since \(v_s = 4\) km/hr, we can find \(v_b\):

\[ v_b = 3 \times v_s = 3 \times 4 \text{ km/hr} = 12 \text{ km/hr} \]

So, the speed of the boat in still water is 12 km/hr.

Calculating Downstream and Upstream Speeds

When the boat moves downstream, the speed of the stream adds to the boat's speed in still water. The downstream speed (\(v_d\)) is:

\[ v_d = v_b + v_s = 12 \text{ km/hr} + 4 \text{ km/hr} = 16 \text{ km/hr} \]

When the boat moves upstream, the speed of the stream opposes the boat's speed in still water. The upstream speed (\(v_u\)) is:

\[ v_u = v_b - v_s = 12 \text{ km/hr} - 4 \text{ km/hr} = 8 \text{ km/hr} \]

Converting Total Time to Hours

The total time for the round trip is given as 45 minutes. To use this with speeds in km/hr, we convert minutes to hours:

\[ 45 \text{ minutes} = \frac{45}{60} \text{ hours} = \frac{3}{4} \text{ hours} \]

Setting up the Equation for Distance

Let \(D\) be the distance of the one-way trip (from start to the turn-around point). The time taken to travel downstream is \(t_d = \frac{D}{v_d}\), and the time taken to travel upstream is \(t_u = \frac{D}{v_u}\).

The total time for the round trip is the sum of the downstream and upstream times:

\[ \text{Total Time} = t_d + t_u \] \[ \frac{3}{4} = \frac{D}{16} + \frac{D}{8} \]

Solving for the One-Way Distance (D)

To solve for \(D\), we find a common denominator for the fractions on the right side, which is 16:

\[ \frac{3}{4} = \frac{D}{16} + \frac{2D}{16} \] \[ \frac{3}{4} = \frac{D + 2D}{16} \] \[ \frac{3}{4} = \frac{3D}{16} \]

Now, we can solve for \(D\). Multiply both sides by 16:

\[ \frac{3}{4} \times 16 = 3D \] \[ 3 \times 4 = 3D \] \[ 12 = 3D \]

Divide by 3:

\[ D = \frac{12}{3} = 4 \text{ km} \]

So, the distance of the one-way trip is 4 km.

Finding the Total Distance of the Whole Trip

The question asks for the distance of the whole trip, which is a round trip. This means the total distance is twice the one-way distance.

\[ \text{Total Distance} = 2 \times D \] \[ \text{Total Distance} = 2 \times 4 \text{ km} = 8 \text{ km} \]

The distance of the whole trip is 8 km.

Revision Table: Boat and Stream Key Concepts

Additional Information on Boat and Stream Problems

Boat and stream problems are common in competitive exams. They test your understanding of relative speed. The key idea is that the stream's speed either helps (downstream) or hinders (upstream) the boat's movement. Always remember to use consistent units for speed (km/hr or m/s) and time (hours or seconds) before performing calculations.

In round trip problems, the distance traveled downstream is equal to the distance traveled upstream. The total time is the sum of the time taken for each leg of the journey.