Question

A man rows downstream \(60 \mathrm{~km}\) and upstredit \(36 \mathrm{~km}\), taking 4 hours each time. The speed of the man is : (a) \(15 \mathrm{~km} / \mathrm{h}\) (b) \(16 \mathrm{~km} / \mathrm{h}\) (c) \(8 \mathrm{~km} / \mathrm{h}\) (d) \(12 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: 12 km/h

Step-by-step solution

Introduce the variables

Let's denote the speed of the man in still water as 'x' km/h, and the speed of the stream as 'y' km/h.

Write the equations for downstream and upstream

When the man rows downstream, the actual speed is the sum of the speed of the man and the speed of the stream. So, the equation for downstream will be: (x + y) * time = distance_downstream where time = 4 h, distance_downstream = 60 km. When the man rows upstream, the actual speed is the difference between the speed of the man and the speed of the stream. So, the equation for upstream will be: (x - y) * time = distance_upstream where time = 4 h, distance_upstream = 36 km.

Solve the equations

We now have two equations with two variables 'x' and 'y': 1) (x + y) * 4 = 60 2) (x - y) * 4 = 36 Now, we can simply solve these equations for 'x' and 'y': From equation (1): x + y = 15 From equation (2): x - y = 9 Add both equations: 2x = 24 Therefore, x = 12

Find the speed of the man in still water

Since 'x' represents the speed of the man in still water, we can conclude that the speed of the man is 12 km/h. The correct answer is option (d) 12 km/h.

Key concepts

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality between two expressions. In this problem, algebraic equations are used to represent the relationship between a man's rowing speed, the current of the stream, and the distances traveled. An equation, much like a balance, maintains equality between its two sides, which is key to solving for variables you don't know. For example, consider the two equations given in the solution:

• Downstream: \[ (x + y) \times 4 = 60 \]This equation shows that the combined speed of the man and the stream (x + y) over 4 hours equates to a 60 km journey.
• Upstream:\[ (x - y) \times 4 = 36 \]Here, the difference in speeds (x - y) account for a 36 km journey, again in 4 hours.

These equations help break down the problem by transforming real-world relationships into numeric expressions.To solve these equations, we rearrange them to isolate the variable of interest or perform operations to eliminate one variable, making it easier to solve for the other. This process highlights the usefulness of algebra in problem-solving by allowing us to work step-by-step to find an unknown quantity.

Relative Speed

Relative speed is the speed of one object as observed from another moving object. In this exercise, the relative speed helps us find how the man's speed is affected by the current when moving downstream or upstream.

• **Downstream:** When the current aids the rower, the relative speed is the sum of the man's rowing speed and the stream's speed (x + y).
• **Upstream:** When the rower moves against the current, the relative speed is the man's rowing speed minus the stream's speed (x - y).

Understanding relative speed is crucial in problems involving movement in a medium like water, where the medium affects overall speed. For instance, relative speed helps determine how quickly a boat travels relative to the shore when currents influence its motion. The concept is universally applicable in scenarios, such as walking on a conveyor belt, which adjusts your net speed due to the additional motion, just like in our rowing problem.

Upstream and Downstream Problems

Upstream and downstream problems typically involve determining the speed of a boat or swimmer in a river with a current. These problems capture the essence of movement in a medium where one must account for the medium's effect.

• **Downstream**: When moving downstream, the current assists the vessel, and the effective speed increases due to the current's contribution.
• **Upstream**: Conversely, when moving upstream, the vessel must overcome the current, reducing the effective speed.

Such problems are often solved using two key pieces of data: - The times taken for upstream and downstream journeys. - The distances covered. By establishing equations using these parameters, we can deduce the natural or still water speed of the vessel. These problems improve understanding of how external forces, such as winds or currents, impact travel times and speeds. In our exercise, solving the upstream and downstream movements step by step enabled us to figure out the rower's actual speed in still water, demonstrating the balance of forces and making problems relatable to real-world motion dynamics.