A boatman can row 2 km against the stream in 20 minutes and return in 10 minutes. Find the rate of flow of the current. 

A boatman can row 2 km against the stream in 20 minutes and return in 10 minutes. Find the rate of flow of the current.

The rate of flow of the current is 3 km/hr.

Let's denote the speed of the boatman in still water as 'b' km/hr and the speed of the stream as 'c' km/hr.

When the boatman is rowing against the stream, the effective speed is (b - c) km/hr. Similarly, when rowing with the stream, the effective speed is (b + c) km/hr.

Given that the boatman can row 2 km against the stream in 20 minutes, we have:

(b - c) * (20/60) = 2

Simplify this to:

(1/3)(b - c) = 2

b - c = 6

And when the boatman returns in 10 minutes, we have:

(b + c) * (10/60) = 2

Simplify this to:

(1/6)(b + c) = 2

b + c = 12

Now, we can solve these two equations simultaneously to find the values of 'b' and 'c'.

Adding the two equations:

(b - c) + (b + c) = 6 + 12

2b = 18

b = 9

Now, substituting b = 9 into one of the equations to find 'c':

9 - c = 6

c = 9 - 6

c = 3

So, the rate of flow of the current is 3 km/hr.

Speed, Distance, and Time in Mathematics

In mathematics, speed, distance, and time are fundamental concepts related to motion. They are interconnected through a set of formulas that allow you to calculate one of them if you know the other two.

Here's a breakdown of each concept and their relationship:

1. Speed:

• Definition: Speed refers to the rate at which an object travels a certain distance. It signifies how fast or slow an object is moving.
• Units: Speed is typically measured in units like kilometers per hour (km/h), meters per second (m/s), miles per hour (mph), etc.
• Formula: Speed (S) is calculated by dividing the distance (D) traveled by the time (T) taken:
  • S = D / T

2. Distance:

• Definition: Distance refers to the total length or gap covered by an object during its movement from one point to another.
• Units: Distance is measured in units like kilometers (km), meters (m), miles (mi), etc.
• Formula: Distance (D) is calculated by multiplying the speed (S) by the time (T) taken:
  • D = S * T

3. Time:

• Definition: Time refers to the duration or interval during which the movement occurs.
• Units: Time is typically measured in units like hours (h), minutes (min), seconds (s), etc.
• Formula: Time (T) is calculated by dividing the distance (D) traveled by the speed (S):
  • T = D / S

These formulas form a proportional relationship, meaning:

• Speed is directly proportional to distance: If the distance traveled increases while the time remains constant, the speed also increases (and vice versa).
• Speed is inversely proportional to time: If the time taken increases while the distance traveled remains constant, the speed decreases (and vice versa).

These concepts are frequently used in various mathematical applications, including:

• Solving word problems: These problems often involve scenarios involving moving objects, and you need to use the speed, distance, and time formulas to find the missing information.
• Calculating rates: Rates, such as interest rates or work rates, can be seen as a form of speed, relating change in a quantity over time.
• Understanding motion in physics: Kinematics, a branch of physics, heavily relies on these concepts to analyze the motion of objects.

By understanding the relationships between speed, distance, and time, you can solve various problems and gain a deeper understanding of motion in both mathematics and the real world.