Solving Boat and Stream Time Problems

This problem involves calculating the time a boat takes to travel specific distances both upstream and downstream. The key to solving such boat and stream problems is understanding how the speed of the stream affects the boat's speed in different directions.

Let's define our variables:

• Let the speed of the boat in still water be \(B\) km/h.
• Let the speed of the stream be \(S\) km/h.

When the boat travels upstream, it goes against the current, so its effective speed is reduced. Upstream speed = \(B - S\) km/h.

When the boat travels downstream, it goes with the current, so its effective speed is increased. Downstream speed = \(B + S\) km/h.

We are given two scenarios relating distance, speed, and time. The formula relating these is Time = Distance / Speed.

Setting Up the Equations for Boat Travel Time

Based on the information given, we can set up a system of equations:

Scenario 1: 27 km upstream and 33 km downstream in 6 hours.

Time upstream + Time downstream = Total Time

\(\frac{\text{Distance Upstream}}{\text{Upstream Speed}} + \frac{\text{Distance Downstream}}{\text{Downstream Speed}} = \text{Total Time}\)

\(\frac{27}{B - S} + \frac{33}{B + S} = 6\) (Equation 1)

Scenario 2: 36 km upstream and 22 km downstream in the same time (6 hours).

Time upstream + Time downstream = Total Time

\(\frac{36}{B - S} + \frac{22}{B + S} = 6\) (Equation 2)

Solving the System of Equations

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

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

Now, the equations become linear:

\(27u + 33d = 6\) (Equation 1 simplified)

\(36u + 22d = 6\) (Equation 2 simplified)

We can solve this system for \(u\) and \(d\). Let's multiply Equation 1 by 4 and Equation 2 by 3 to eliminate \(u\):

\(4 \times (27u + 33d) = 4 \times 6 \implies 108u + 132d = 24\) (Equation 3)

\(3 \times (36u + 22d) = 3 \times 6 \implies 108u + 66d = 18\) (Equation 4)

Subtract Equation 4 from Equation 3:

\((108u + 132d) - (108u + 66d) = 24 - 18\)

\(108u - 108u + 132d - 66d = 6\)

\(66d = 6\)

\(d = \frac{6}{66} = \frac{1}{11}\)

Now substitute the value of \(d\) back into Equation 1 (simplified):

\(27u + 33\left(\frac{1}{11}\right) = 6\)

\(27u + 3 = 6\)

\(27u = 6 - 3\)

\(27u = 3\)

\(u = \frac{3}{27} = \frac{1}{9}\)

So, we found that \(u = \frac{1}{9}\) hours per km upstream and \(d = \frac{1}{11}\) hours per km downstream.

Calculating Time for the Final Scenario

The question asks for the time it will take to go 36 km upstream and 44 km downstream.

Time = Distance \(\times\) Time per km

Time for 36 km upstream = \(36 \times u = 36 \times \frac{1}{9}\) hours

Time for 36 km upstream = 4 hours

Time for 44 km downstream = \(44 \times d = 44 \times \frac{1}{11}\) hours

Time for 44 km downstream = 4 hours

Total time for the final scenario = Time upstream + Time downstream

Total time = 4 hours + 4 hours = 8 hours.

Therefore, it will take 8 hours to go 36 km upstream and 44 km downstream.

Revision Table: Key Values

Additional Information: Boat and Stream Concepts

Boat and stream problems are a common type in quantitative aptitude tests. They test your understanding of relative speed.

• Speed of boat in still water (B): This is the inherent speed of the boat without any influence from the stream.
• Speed of stream or current (S): This is the speed of the flowing water.
• Upstream: When the boat moves against the direction of the stream. The net speed is \(B - S\). This speed is always less than the speed of the boat in still water. For the boat to move upstream, \(B\) must be greater than \(S\).
• Downstream: When the boat moves in the same direction as the stream. The net speed is \(B + S\). This speed is always greater than the speed of the boat in still water.
• Relationship between speeds: If you know the upstream speed \(U\) and downstream speed \(D\), you can find the speed of the boat in still water and the speed of the stream using these formulas:
  • Boat Speed (\(B\)) = \(\frac{D + U}{2}\)
  • Stream Speed (\(S\)) = \(\frac{D - U}{2}\)
  In our problem, Upstream Speed = \(1/u = 9\) km/h and Downstream Speed = \(1/d = 11\) km/h. Using the formulas: Boat Speed (\(B\)) = \(\frac{11 + 9}{2} = \frac{20}{2} = 10\) km/h. Stream Speed (\(S\)) = \(\frac{11 - 9}{2} = \frac{2}{2} = 1\) km/h. We can verify this: \(B - S = 10 - 1 = 9\) (Upstream Speed) and \(B + S = 10 + 1 = 11\) (Downstream Speed). These match the speeds calculated from \(u\) and \(d\).
• Time, Distance, Speed formula: Time = Distance / Speed, Distance = Speed \(\times\) Time, Speed = Distance / Time. These fundamental formulas are crucial for solving boat and stream problems.