Question

'A' goes 10 km distance with average speed of \(6 \mathrm{~km} / \mathrm{h}\) while rest \(20 \mathrm{~km}\) he travels with an average speed of \(15 \mathrm{~km} / \mathrm{h}\). What is the average speed of ' \(A\) ' during the whole journey? (a) \(10 \mathrm{~km} / \mathrm{h}\) (b) \(12 \mathrm{~km} / \mathrm{h}\) (c) \(13 \mathrm{~km} / \mathrm{h}\) (d) \(14.5 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: 10 km/h

Step-by-step solution

Calculate time taken for each part of the journey

To calculate the time taken for each part of the journey, divide the distance by the average speed. Time taken for the first 10 km: 10 km / 6 km/h = 5/3 h Time taken for the remaining 20 km: 20 km / 15 km/h = 4/3 h

Calculate total time and distance of the journey

Add up the time taken for each part and the distances covered to get the total time and distance. Total time: (5/3 h) + (4/3 h) = 3 h Total distance: 10 km + 20 km = 30 km

Calculate the average speed of the entire journey

To find the average speed, divide the total distance by the total time. Average speed = Total distance / Total time Average speed = 30 km / 3 h Average speed = 10 km/h So, the correct answer is (a) \(10 \mathrm{~km} / \mathrm{h}\).

Key concepts

Time and Distance Problems

When we encounter time and distance problems in mathematics, it is essential to understand the relation between speed, time, and distance. The primary formula to remember is Distance = Speed × Time. This fundamental relationship allows us to solve for any one of the variables if the other two are known.

In practical problems, finding the average speed over an entire journey can be trickier when the speed varies in different segments of the trip. As seen in the exercise, 'A' travels different distances at different speeds. The key to solving such problems is to break down the journey into segments where the speed is constant, calculate the time for each segment, and then combine these to determine the total time and distance.

Breaking Down the Journey

You must calculate the time for the first part of the journey, where 'A' travels at 6 km/h, and then do the same for the second part, where the speed is 15 km/h. You would then add these times to find the total time it took for the entire journey. This step-by-step approach is vital as it provides a clear and organized path to the solution.

Quantitative Aptitude

Quantitative aptitude consists of various topics and skills, including numerical reasoning, problem-solving, and data interpretation. It's not just about calculating numbers; it's about understanding the logical foundation behind the numbers. In our exercise, quantitative aptitude is represented by calculating the average speed, which demands attentiveness to detail.

To enhance quantitative aptitude, especially in time and distance problems, practice is crucial. Begin with understanding the formulas and how they are derived. Then, attempt various problems with increasing complexity. Keep an eye out for tricks such as changes in speed, as seen in this problem.

Frequent Practice and Conceptual Understanding

Regular practice with different types of problems can improve your ability to quickly identify the most efficient method to find a solution. Building a strong foundation in the concepts will make it easier to apply them to a range of real-world scenarios.

Speed, Time, and Distance

Understanding the relationship between speed, time, and distance is critical for solving transportation and travel related problems, as well as for many standardized tests. For constant speed, the formula Distance = Speed × Time works perfectly. But when speed changes, the average speed for the journey is not just the average of the speeds but is instead defined by the total distance divided by the total time.

The average speed is a concept that is often misunderstood. It is not the mean of the speeds but rather a weighted average based on the time spent at each speed.

The Importance of Considering Time

This means that for each segment of the trip, you must consider the time spent traveling at each speed. Once you have the total time and the total distance, you can accurately calculate the average speed. As demonstrated in the exercise, despite intuitively thinking the average speed might be between 6 and 15 km/h, the calculation shows that the correct answer is 10 km/h, based on the total distance of 30 km traveled in 3 hours.