Solving the River Speed Riddle: Speed of the Boat in Still Water and the Current

Imagine a motorboat navigating the currents of a river with ease, yet sometimes faced with the challenge of going upstream against the flow. To truly understand its performance, we need to uncover the rate of the boat in still water and the speed of the current. This article provides a detailed step-by-step guide on how to solve these problems mathematically using the given data.

Understanding the Equations and Formulas

Let's consider the following problem:

A motorboat travels 150 miles in 6 hours going upstream. It travels 222 miles going downstream in the same amount of time. What is the rate of the boat in still water and what is the rates of the current?

To solve this, we can use the basic formula for speed, which is:

SPEED DISTANCE / TIME

We can break down the scenario into two parts: traveling upstream and traveling downstream.

Step-by-Step Solution

Let:

Speed of the boat in still water be b miles per hour. Speed of the current be c miles per hour.

We are given:

Upstream distance 150 miles, time 6 hours. Downstream distance 222 miles, time 6 hours.

Building the Equations

When traveling upstream, the speed of the boat is reduced by the current. Thus, the effective speed is:

Speed Uphill b - c (miles per hour)

Given that the boat travels 150 miles in 6 hours upstream, we can write the equation as:

frac16; * (b - c) 150

This simplifies to:

b - c 25 (1)

Similarly, when traveling downstream, the speed of the boat is increased by the current. Thus, the effective speed is:

Speed Downhill b c (miles per hour)

Given that the boat travels 222 miles in 6 hours downstream, we can write the equation as:

frac16; * (b c) 222

This simplifies to:

b c 37 (2)

Solving the System of Equations

We now have a system of two linear equations:

b - c 25 b c 37

Adding these two equations will eliminate the variable c and help us find b:

(b - c) (b c) 25 37

2b 62

b 31 (3)

Substitute b 31 back into one of the equations to find c. Using Equation 1:

31 - c 25

c 31 - 25

c 6 (4)

Hence, we have:

The speed of the boat in still water is 31 miles per hour (mph). The speed of the current is 6 miles per hour (mph).

Bringing it to Life with Additional Examples

Let's look at a couple of additional examples to solidify our understanding of this concept:

Example 1

A boat travels 162/3 miles in 3 hours going upstream and 132/3 miles in the same time going downstream.

Using the formula:

Speed Distance / Time

We calculate:

Upstream:

(162/3) / 3 54 mph - 44/2 5 miles/hour (rate of the current)

Downstream:

(132/3) / 3 44 miles/hour (rate of the boat in still water)

Final Results:

The speed of the boat in still water is 44 miles/hour. The rate of the current is 5 miles/hour.

Example 2

A boat travels 152 kilometers in 4 hours going upstream and 224 kilometers in the same time going downstream.

Using the formula:

Speed Distance / Time

We calculate:

Upstream:

(152) / 4 38

Downstream:

(224) / 4 56

Add:

2x 94

x 47 km/h (speed in still water)

y 9 km/h (rate of current)

Final Results:

The speed of the boat in still water is 47 km/h. The speed of the current is 9 km/h.

Example 3

A boat travels 52 kilometers in 2 hours going upstream and 80 kilometers in the same time going downstream.

Using the formula:

Speed Distance / Time

We calculate:

Upstream:

(52) / 2 26

Downstream:

(80) / 2 40

Add:

2x 66

x 33 km/h (speed in still water)

y 14 km/h (rate of current)

Final Results:

The speed of the boat in still water is 33 km/h. The speed of the current is 14 km/h.

Example 4

A boat travels 104 kilometers in 2 hours going upstream and 160 kilometers in the same time going downstream.

Using the formula:

Speed Distance / Time

We calculate:

Upstream:

(104) / 2 52

Downstream:

(160) / 2 80

Add:

2x 132

x 66 km/h (speed in still water)

y 14 km/h (speed of stream)

Final Results:

The speed of the boat in still water is 66 km/h. The speed of the current is 14 km/h.

Understanding the river speed, the rate of the boat, and the current is essential for planning and executing successful navigational routes. By mastering these calculations, you can optimize your boat's performance and ensure a safer and more efficient journey.