Solving the Speed of a Boat in Still Water with Various Stream Speeds

Understanding the speed of a boat in still water, relative to the speed of a stream, is a classic application of algebra in real-world scenarios. This article will explore different methods to solve the problem of finding the speed of a boat in still water given the stream's speed and the distances traveled upstream and downstream.

Method 1: Algebraic Equation

Let the speed of the boat in still water be x km/h. The speed of the stream is 7 km/h. Therefore, the speed of the boat upstream is x - 7 km/h, and the speed of the boat downstream is x 7 km/h. If the boat travels 6 miles upstream in the same time it takes to travel 14 miles downstream, we can set up the following equation:

{eq}6 / (x - 7) 14 / (x 7)

After solving this equation, we get:

{eq}x 15.57

Hence, the speed of the boat in still water is approximately 15.57 km/h.

Method 2: Simplified Equations with Variables

Let the speed of the boat in still water be x km/h. The speed of the stream is 5 km/h. The speed of the boat upstream is x - 5 km/h, and the speed of the boat downstream is x 5 km/h. If the boat travels 6 miles upstream and downstream, we set up the following equation:

{eq}6 / (x - 5) 6 / (x 5)

After solving this equation, we get:

{eq}x 6.10

Hence, the speed of the boat in still water is approximately 6.10 km/h.

Method 3: Cross-Multiplication Technique

Let the speed of the boat in still water be x kph. Assume the speed of the stream is 3 km/h. The upstream speed is x - 3 km/h, and downstream speed is x 3 km/h. If the distance traveled upstream is 4 km in 12 minutes and downstream is 12 km in 6 minutes, the equation becomes:

{eq}4 / (x - 3) 12 / (x 3)

Cross-multiplying, we get:

{eq}12x - 36 4x 12

After simplifying, we get:

{eq}8x 48

{eq}x 6

Hence, the speed of the boat in still water is 6 kph.

Method 4: Stream Speeds and Distance

Let the speed of the boat in still water be x kph. The speed of the stream is 4 km/h. The upstream speed is x - 4 km/h, and downstream speed is x 4 km/h. If the boat travels 6 miles upstream in 42 minutes and 10 miles downstream in 60 minutes, the equation becomes:

{eq}6 / (x - 4) 10 / (x 4)

Cross-multiplying, we get:

{eq}6x 24 1 - 40

After simplifying, we get:

{eq}4x 64

{eq}x 16

Hence, the speed of the boat in still water is 16 kph.

Conclusion

Through various methods, we have solved the problem of finding the speed of a boat in still water given the stream's speed and the distances traveled upstream and downstream. Each method illustrates the application of algebra and logic in real-world scenarios. Understanding these solutions can provide valuable insights into solving similar mathematical problems.