A boat goes downstream 60 km in 4 hrs, upstream 40 km in 5 hrs. Find speed of boat in still water.

Unraveling Boat Speed: A Step-by-Step Guide to Solving Downstream and Upstream Problems

Boats on rivers create interesting math puzzles! Ever wondered how to calculate a boat's speed when the current is helping or hindering it? This post will show you how to solve boat speed problems, specifically focusing on downstream and upstream scenarios.

The Problem: A boat travels downstream 60 km in 4 hours. Going upstream, the same boat covers 40 km in 5 hours. The question is: What is the boat's speed in still water?

This guide will walk you through solving this step-by-step. Let's get started!

I. Understanding the Basics: Speed, Distance, and Time

Before diving in, let's refresh some key concepts. We'll be using these a lot.

• Speed = Distance / Time: This is the fundamental formula we'll use throughout.
• Still Water Speed: This is how fast the boat goes without any current affecting it. It's the boat's "true" speed.
• Downstream Speed: This is the boat's speed plus the speed of the current. The current helps the boat move faster.
• Upstream Speed: This is the boat's speed minus the speed of the current. The current slows the boat down.

II. Setting Up the Equations: Downstream and Upstream

Now, let's use the information from our problem to set up our equations.

Downstream:

• Distance: 60 km
• Time: 4 hours
• Downstream Speed: 60 km / 4 hours = 15 km/hr
• Representation: Downstream speed can be written as: boat's speed (b) + current's speed (c) or b + c

Upstream:

• Distance: 40 km
• Time: 5 hours
• Upstream Speed: 40 km / 5 hours = 8 km/hr
• Representation: Upstream speed can be written as: boat's speed (b) - current's speed (c) or b - c

III. Formulating Two Equations

Based on our calculations, we now have two crucial equations:

• Equation 1: Downstream Speed = b + c = 15 km/hr
• Equation 2: Upstream Speed = b - c = 8 km/hr

IV. Solving the Equations: Finding the Boat's Speed

We can solve these equations using a method called elimination. Here’s how it works:

Step 1: Notice that we have +c and -c in our equations. If we add the equations together, the 'c' values will cancel each other out.

Step 2: Add the equations:

(b + c) + (b - c) = 15 + 8

2b = 23

Step 3: Simplify and isolate 'b' (boat's speed):

b = 23 / 2

b = 11.5 km/hr

Step 4: The boat's speed in still water is 11.5 km/hr!

V. Verifying the Solution

Always check your answer! Let's see if our boat's speed makes sense.

Let’s plug our answer (b = 11.5 km/hr) back into the original equations:

Downstream: 11.5 + c = 15 => c = 3.5 km/hr (Current's speed is 3.5 km/hr)

Upstream: 11.5 - c = 8 => c = 3.5 km/hr (Same current speed!)

Since the current speed is the same in both cases, our answer is correct.

VI. Conclusion

You did it! We successfully solved the boat speed problem.

Here's a recap:

• We understood the concepts of speed, distance, time, downstream, and upstream.
• We calculated downstream and upstream speeds.
• We set up two equations.
• We used elimination to find the boat's speed in still water (11.5 km/hr).
• We verified our answer.

Key Takeaways:

• Downstream problems involve adding the current's speed.
• Upstream problems involve subtracting the current's speed.
• Setting up equations is key to solving these problems.

Want to try another problem? Practice makes perfect! Feel free to leave a comment if you have any questions.