Solving a Speed, Distance, and Time Problem

Understanding the relationship between speed, distance, and time is fundamental in many real-life scenarios. This article will guide you through solving a problem that highlights how changes in speed can affect the distance traveled in the same time frame. Let’s break down the problem and explore the concepts involved:

Problem Statement

A person walks at 12 km/hr instead of 10 km/hr and covers 1 kilometer more in the same time. The task is to find the actual distance traveled by him at 10 km/hr in the same time. We’ll solve this step-by-step.

Step-by-Step Solution

1. Define Variables:

Let the time taken to walk be ( t ) hours and the distance traveled at 10 km/hr be ( d ) kilometers.

When walking at 10 km/hr, the distance can be expressed as:

( d 10t )

When walking at 12 km/hr, the distance traveled would be:

( d 1 12t )

2. Set Up the Equation:

Now, we can set up the equation based on the two expressions for ( d ):

( 10t 1 12t )

3. Solve for ( t ):

Rearranging the equation gives:

( 1 12t - 10t )

( 1 2t )

( t frac{1}{2} ) hours

4. Find the Distance ( d ):

Substituting ( t ) back into the equation for distance ( d ):

( d 10t 10 times frac{1}{2} 5 ) km

Thus, the actual distance traveled by him at 10 km/hr in the same time is 5 kilometers.

Basic Principle of Speed, Time, and Distance

The relationship between speed, time, and distance is foundational. Here are some key principles to keep in mind:

1. Direct Proportion:

Distance is directly proportional to speed when time is constant. If the speed increases by a certain factor, the distance covered also increases by the same factor.

2. Relative Increase in Distance:

Given that the speed increases from 10 km/hr to 12 km/hr, the speed increases by (frac{12 - 10}{10} frac{1}{5}).

Since increasing the speed by (frac{1}{5}) causes an increase of 1 km in distance, the original distance must be 5 km.

3. Simplified Calculation:

Another way to solve the problem is by directly calculating the time taken at the increased speed and then determining the distance covered at the original speed. If the increased distance is 1 km, the time taken difference is (frac{1}{2}) hour. Hence, the distance at 10 km/hr is (10 times frac{1}{2} 5) km.

In conclusion, the problem highlights the importance of understanding the foundational principles of speed, time, and distance. By applying these basic principles, we can effectively solve real-world problems and make informed calculations.