Solving Boat and Stream Time Calculation

This problem involves the concepts of boat speed and stream speed, and how they affect travel time upstream and downstream. Let's denote the speed of the boat in still water as \( B \) km/hr and the speed of the stream as \( S \) km/hr.

• When the boat goes upstream, the stream opposes the boat's movement. The effective speed is the difference between the boat's speed and the stream's speed.
• Upstream Speed = \( B - S \) km/hr
• When the boat goes downstream, the stream helps the boat's movement. The effective speed is the sum of the boat's speed and the stream's speed.
• Downstream Speed = \( B + S \) km/hr

The relationship between speed, distance, and time is given by: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).

Setting Up Equations for Boat and Stream Problem

We are given two scenarios with different distances traveled upstream and downstream and the total time taken. We need to convert the times given in minutes to hours because the speeds are in km/hr.

Scenario 1: 3 km upstream and 5 km downstream in 55 minutes.

• 55 minutes = \( \frac{55}{60} \) hours = \( \frac{11}{12} \) hours.
• Time taken for 3 km upstream = \( \frac{3}{B-S} \) hours.
• Time taken for 5 km downstream = \( \frac{5}{B+S} \) hours.
• Equation 1: \( \frac{3}{B-S} + \frac{5}{B+S} = \frac{11}{12} \)

Scenario 2: 4 km upstream and 9 km downstream in 1 hour 25 minutes.

• 1 hour 25 minutes = 60 minutes + 25 minutes = 85 minutes = \( \frac{85}{60} \) hours = \( \frac{17}{12} \) hours.
• Time taken for 4 km upstream = \( \frac{4}{B-S} \) hours.
• Time taken for 9 km downstream = \( \frac{9}{B+S} \) hours.
• Equation 2: \( \frac{4}{B-S} + \frac{9}{B+S} = \frac{17}{12} \)

Solving the System of Equations

To solve these equations, let's use a substitution method. Let \( u = \frac{1}{B-S} \) and \( v = \frac{1}{B+S} \). The equations become:

• Equation 1: \( 3u + 5v = \frac{11}{12} \)
• Equation 2: \( 4u + 9v = \frac{17}{12} \)

Multiply both equations by 12 to eliminate the denominators:

• Equation 1': \( 12 \times (3u + 5v) = 12 \times \frac{11}{12} \Rightarrow 36u + 60v = 11 \)
• Equation 2': \( 12 \times (4u + 9v) = 12 \times \frac{17}{12} \Rightarrow 48u + 108v = 17 \)

Now we have a system of linear equations in terms of \( u \) and \( v \). We can solve this using elimination. Multiply Equation 1' by 4 and Equation 2' by 3 to make the coefficients of \( u \) equal:

• \( 4 \times (36u + 60v) = 4 \times 11 \Rightarrow 144u + 240v = 44 \) (Equation 3)
• \( 3 \times (48u + 108v) = 3 \times 17 \Rightarrow 144u + 324v = 51 \) (Equation 4)

Subtract Equation 3 from Equation 4:

\( (144u + 324v) - (144u + 240v) = 51 - 44 \)

\( 84v = 7 \)

\( v = \frac{7}{84} = \frac{1}{12} \)

Now substitute the value of \( v \) into Equation 1' to find \( u \):

\( 36u + 60(\frac{1}{12}) = 11 \)

\( 36u + 5 = 11 \)

\( 36u = 11 - 5 \)

\( 36u = 6 \)

\( u = \frac{6}{36} = \frac{1}{6} \)

Calculating Upstream and Downstream Speeds

We have \( u = \frac{1}{B-S} = \frac{1}{6} \) and \( v = \frac{1}{B+S} = \frac{1}{12} \).

• From \( u = \frac{1}{B-S} = \frac{1}{6} \), we get \( B-S = 6 \) km/hr (Upstream Speed).
• From \( v = \frac{1}{B+S} = \frac{1}{12} \), we get \( B+S = 12 \) km/hr (Downstream Speed).

Calculating Time to Travel 43.2 km Downstream

We need to find the time it will take to travel 43.2 km downstream. The downstream speed is \( B+S = 12 \) km/hr.

\( \text{Time} = \frac{\text{Distance}}{\text{Downstream Speed}} = \frac{43.2}{12} \) hours.

Let's perform the calculation:

\( \frac{43.2}{12} = \frac{432}{120} \)

Divide both numerator and denominator by common factors. Divide by 12:

\( \frac{432 \div 12}{120 \div 12} = \frac{36}{10} = 3.6 \)

So, the time taken to go 43.2 km downstream is 3.6 hours.

Final Answer Summary

Based on the calculations, the time required to travel 43.2 km downstream at a speed of 12 km/hr is 3.6 hours.

Revision Table: Key Concepts

Additional Information: Boat and Stream Basics

Understanding relative speed is crucial in boat and stream problems. The stream's speed is relative to the ground, while the boat's speed is relative to the water. When moving with the stream (downstream), the speeds add up. When moving against the stream (upstream), the stream's speed is subtracted from the boat's speed.

If you know the upstream speed (U) and downstream speed (D), you can find the boat speed in still water (B) and stream speed (S) using these formulas:

• Boat Speed (in still water): \( B = \frac{D + U}{2} \)
• Stream Speed: \( S = \frac{D - U}{2} \)

In our solved problem, we found Upstream Speed \(B-S = 6\) km/hr and Downstream Speed \(B+S = 12\) km/hr.

• Boat Speed \( B = \frac{12 + 6}{2} = \frac{18}{2} = 9 \) km/hr.
• Stream Speed \( S = \frac{12 - 6}{2} = \frac{6}{2} = 3 \) km/hr.

You can verify these speeds with the original equations if needed. This confirms our calculated speeds are consistent with the problem statements.