Boat and Stream Problem Solving

This problem involves calculating the speed of a man in still water based on his speeds rowing downstream and upstream. We are given the distance and time taken for the downstream journey, and a relationship between the downstream and upstream times. We need to find the distance covered in one hour in still water.

Understanding Boat and Stream Concepts

In boat and stream problems, we use the following concepts:

• Speed Downstream: Speed of the boat/man + Speed of the stream. The current helps the movement.
• Speed Upstream: Speed of the boat/man - Speed of the stream. The current opposes the movement.
• Speed in Still Water: The inherent speed of the boat/man without the effect of the stream.

Let:

• \(V_m\) = Speed of the man in still water (km/h)
• \(V_s\) = Speed of the stream (km/h)
• Speed Downstream = \(V_d = V_m + V_s\) (km/h)
• Speed Upstream = \(V_u = V_m - V_s\) (km/h)

Calculating Downstream Speed

The man takes 15 minutes to row 16 km downstream.

• Distance downstream = 16 km
• Time downstream = 15 minutes

First, convert the time to hours:

\(\text{Time downstream} = 15 \text{ minutes} = \frac{15}{60} \text{ hours} = \frac{1}{4} \text{ hours}\)

Now, calculate the speed downstream:

\(\text{Speed Downstream} = \frac{\text{Distance}}{\text{Time}} = \frac{16 \text{ km}}{\frac{1}{4} \text{ hours}} = 16 \times 4 \text{ km/h} = 64 \text{ km/h}\)

So, we have our first equation: \(\boldsymbol{V_m + V_s = 64}\)

Calculating Upstream Time and Speed

The question states that the time taken to row 16 km downstream is 25% less than the time taken to row the same distance upstream.

Let \(T_u\) be the time taken for the upstream journey in minutes.

Time downstream = 15 minutes.

15 minutes is 25% less than \(T_u\). This means 15 minutes is \(100\% - 25\% = 75\%\) of \(T_u\).

So, \(15 = 0.75 \times T_u\)

To find \(T_u\):

\(T_u = \frac{15}{0.75} = \frac{15}{\frac{3}{4}} = 15 \times \frac{4}{3} = 5 \times 4 = 20 \text{ minutes}\)

The time taken for the upstream journey is 20 minutes.

Convert the upstream time to hours:

\(\text{Time upstream} = 20 \text{ minutes} = \frac{20}{60} \text{ hours} = \frac{1}{3} \text{ hours}\)

Distance upstream = 16 km.

Now, calculate the speed upstream:

\(\text{Speed Upstream} = \frac{\text{Distance}}{\text{Time}} = \frac{16 \text{ km}}{\frac{1}{3} \text{ hours}} = 16 \times 3 \text{ km/h} = 48 \text{ km/h}\)

So, we have our second equation: \(\boldsymbol{V_m - V_s = 48}\)

Finding Speed in Still Water

We have a system of two linear equations:

1. \(V_m + V_s = 64\)
2. \(V_m - V_s = 48\)

To find \(V_m\) (speed in still water), we can add the two equations:

\((V_m + V_s) + (V_m - V_s) = 64 + 48\)

\(2V_m = 112\)

\(V_m = \frac{112}{2}\)

\(V_m = 56 \text{ km/h}\)

The speed of the man in still water is 56 km/h.

Calculating Distance in One Hour in Still Water

The question asks how many kilometres the man can row in an hour in still water. This is simply the speed of the man in still water multiplied by 1 hour.

\(\text{Distance in 1 hour} = \text{Speed in still water} \times 1 \text{ hour}\)

\(\text{Distance} = 56 \text{ km/h} \times 1 \text{ hour} = 56 \text{ km}\)

Rounding to the Nearest Whole Number

The calculated distance is 56 km, which is already a whole number. So, the rounded answer is 56.

Summary of Calculations

The final answer is 56 km.

Revision Table: Boat and Stream Formulas

Additional Information: Percentage Relationships in Time

Understanding percentage increase/decrease is crucial for solving time-based problems.

• If A is P% less than B, it means A is \((100-P)\%\) of B. So, \(A = \frac{100-P}{100} \times B\). In our problem, downstream time (15 min) is 25% less than upstream time (\(T_u\)), so \(15 = \frac{100-25}{100} \times T_u = \frac{75}{100} \times T_u = 0.75 \times T_u\).
• If A is P% more than B, it means A is \((100+P)\%\) of B. So, \(A = \frac{100+P}{100} \times B\). For example, if the upstream time was 20 minutes, the downstream time (15 minutes) is 5 minutes less. The percentage less is \(\frac{5}{20} \times 100\% = 25\%\). This confirms our calculation that 15 minutes is 25% less than 20 minutes.