Solving Boat and Stream Problems

This problem involves the concept of boat and stream speeds. When a boat travels upstream, its speed is reduced by the speed of the stream. When it travels downstream, its speed is increased by the speed of the stream.

Let:

• Speed of the boat in still water = \(B\) km/min
• Speed of the stream = \(S\) km/min

Therefore:

• Speed upstream = \(B - S\) km/min
• Speed downstream = \(B + S\) km/min

We know that Time = Distance / Speed. We are given two scenarios with distances and times.

Setting Up Equations for Boat and Stream Speed

Scenario 1: 5 km upstream and \(7\frac{1}{2}\) km downstream in 45 minutes.

• \(7\frac{1}{2}\) km is equal to 7.5 km.
• Time taken to go 5 km upstream = \(\frac{5}{B - S}\) minutes
• Time taken to go 7.5 km downstream = \(\frac{7.5}{B + S}\) minutes

The total time for this scenario is 45 minutes. So, the first equation is:

\(\frac{5}{B - S} + \frac{7.5}{B + S} = 45\)  (Equation 1)

Scenario 2: 5 km downstream and 2.5 km upstream in 25 minutes.

• Time taken to go 5 km downstream = \(\frac{5}{B + S}\) minutes
• Time taken to go 2.5 km upstream = \(\frac{2.5}{B - S}\) minutes

The total time for this scenario is 25 minutes. So, the second equation is:

\(\frac{2.5}{B - S} + \frac{5}{B + S} = 25\)   (Equation 2)

Solving the System of Equations

To make these equations easier to solve, let's use substitution. Let:

• \(u = \frac{1}{B - S}\) (which represents the time taken to travel 1 km upstream)
• \(v = \frac{1}{B + S}\) (which represents the time taken to travel 1 km downstream)

Substituting these into our equations, we get:

\(5u + 7.5v = 45\)   (Equation 1)

\(2.5u + 5v = 25\)   (Equation 2)

We can solve this system of linear equations. Multiply Equation 2 by 2 to make the coefficient of \(u\) the same as in Equation 1:

\(2 \times (2.5u + 5v) = 2 \times 25\)

\(5u + 10v = 50\)   (Equation 3)

Now, subtract Equation 1 from Equation 3:

\((5u + 10v) - (5u + 7.5v) = 50 - 45\)

\(2.5v = 5\)

Solving for \(v\):

\(v = \frac{5}{2.5} = 2\)

Now substitute the value of \(v\) into either Equation 1 or Equation 2 to find \(u\). Let's use Equation 2:

\(2.5u + 5(2) = 25\)

\(2.5u + 10 = 25\)

\(2.5u = 25 - 10\)

\(2.5u = 15\)

Solving for \(u\):

\(u = \frac{15}{2.5} = 6\)

Interpreting the Results \(u\) and \(v\)

We found that \(u = 6\) and \(v = 2\).

• \(u = \frac{1}{B - S} = 6\). This means it takes 6 minutes to travel 1 km upstream. The upstream speed is \(\frac{1}{6}\) km/min.
• \(v = \frac{1}{B + S} = 2\). This means it takes 2 minutes to travel 1 km downstream. The downstream speed is \(\frac{1}{2}\) km/min.

Calculating Time for 6 km Downstream

The question asks for the time taken to go 6 km downstream.

The downstream speed is \(\frac{1}{2}\) km/min.

Time = Distance / Speed

Time to go 6 km downstream = \(\frac{6 \text{ km}}{\frac{1}{2} \text{ km/min}}\)

Time = \(6 \times 2\) minutes

Time = 12 minutes

So, it will take 12 minutes to go 6 km downstream.

Revision Table: Boat and Stream Key Concepts

Additional Information: Solving Boat and Stream Problems

Boat and stream problems are a common type in quantitative aptitude tests. They test your understanding of relative speed. Here are some points to remember:

• Always define variables clearly for the speed of the boat in still water and the speed of the stream.
• Remember that downstream speed is \(B+S\) and upstream speed is \(B-S\).
• Be careful with units (km, minutes, hours). Convert them if necessary to maintain consistency throughout the calculation. In this problem, distances are in km and times in minutes, so speeds are in km/min.
• Many problems involve setting up and solving a system of linear equations, often by using substitution for reciprocal speeds (\(\frac{1}{B-S}\) and \(\frac{1}{B+S}\)) as shown in this solution.
• Once you find the speeds, you can calculate time or distance for any given scenario using the basic formula Time = Distance / Speed.