Solving the Puzzle of a Boat's Speed in Still Water

In this article, we will explore a classic problem in physics and mathematics that involves the speed of a boat in still water. This problem is a common challenge in various fields, including competition mathematics and real-world navigation. We will use algebraic methods to solve the problem step by step. Let's dive in!

Problem Statement

A boat takes 5 hours to cover a certain distance going upstream and 3 hours to cover the same distance going downstream. The speed of the current is 2 kilometers per hour. What is the speed of the boat in still water?

Step 1: Set Up the Equations

To solve this problem, we need to use the relationship between speed, time, and distance. Let's denote:

b: Speed of the boat in still water (in km/h) c: Speed of the current 2 km/h d: Distance covered (in km)

When the boat is going downstream, its effective speed is b c. When going upstream, its effective speed is b - c.

Step 2: Apply the Distance Formula

Using the distance formula Distance Speed × Time, we can write:

Downstream: d (b c) × 3 Upstream: d (b - c) × 5

Step 3: Set the Equations Equal to Each Other

Since both expressions equal d, we set them equal to each other:

(b c) × 3 (b - c) × 5

Substitute the speed of the current c 2 km/h:

(b 2) × 3 (b - 2) × 5

Step 4: Expand and Simplify

Expanding both sides gives:

3b 6 5b - 10

Now, rearrange the equation to isolate b:

6 10 5b - 3b

16 2b

b 8

Conclusion

The speed of the boat in still water is 8 kilometers per hour.

Verification

To verify our solution, we can check it with the given conditions:

Upstream: Speed b - 2 8 - 2 6 km/h Time 5 hours Distance Speed × Time 6 × 5 30 km

Downstream: Speed b 2 8 2 10 km/h Time 3 hours Distance Speed × Time 10 × 3 30 km

The calculations confirm that the solution is correct.

Additional Methods

There are other methods to solve this problem, such as:

Algebraic: V d/t, so d Vt d (b 2) × 3 (b - 2) × 5 2b 16 b 8 km/h

Another method using the formula:

Speed of the boat c × (t2 - t1) / (t2 - t1)

Speed of stream 2 km/h Upstream time (t2) 5 hours Downstream time (t1) 3 hours Speed of the boat 2 × (5 - 3) / (5 - 3) 2 × 2 / 2 8 km/h

Conclusion

The speed of the boat in still water is 8 kilometers per hour. This solution confirms that the boat's speed in still water is indeed 8 km/h, considering the effects of the current on its upstream and downstream speeds.