Question

Patna express travels first \(560 \mathrm{~km}\) in 7 hours and rest \(360 \mathrm{~km}\) in 9 hours. What is the average speed of the train? (a) \(39 \mathrm{~km} / \mathrm{h}\) (b) \(43 \mathrm{~km} / \mathrm{h}\) (c) \(63 \mathrm{~km} / \mathrm{h}\) (d) \(57.5 \mathrm{~km} / \mathrm{h}\)

Short answer

a) 50 km/h b) 55 km/h c) 60 km/h d) 57.5 km/h Answer: d) 57.5 km/h

Step-by-step solution

Find the Total Distance traveled

The total distance traveled by Patna express is the sum of the first 560 km and the rest 360 km. So, we need to add 560 km and 360 km. Total Distance = 560 km + 360 km Total Distance = 920 km

Find the Total Time taken

The total time taken by Patna express is the sum of the 7 hours it took for the first 560 km and the 9 hours it took for the remaining 360 km. So, we need to add 7 hours and 9 hours. Total Time = 7 hours + 9 hours Total Time = 16 hours

Calculate Average Speed

Now that we have the total distance and the total time, we can find the average speed of Patna express using the formula: Average Speed = Total Distance / Total Time Average Speed = 920 km / 16 hours Average Speed = 57.5 km/h

Choose the correct answer

From the options given in the exercise, we can see that the correct average speed of the train is: 57.5 km/h (Option (d))

Key concepts

Distance and Time

Understanding distance and time is fundamental to solving average speed problems. Distance refers to how much ground an object has covered during its motion, usually expressed in kilometers or miles.
Time refers to how long it takes to travel that distance, often measured in hours or minutes. To calculate average speed, you need the total distance traveled and the total time taken. In our example, Patna Express traveled a total of 920 km in 16 hours.
Real-life applications of these concepts include planning travel itineraries and estimating arrival times. So a firm grasp of how distance and time relate can help make smart travel decisions.

Train Problems

Train-related problems are a common theme in quantitative aptitude tests. These problems help you practice your ability to deal with moving objects and varying speeds. The classic problem involves calculating the speed of a train based on its traveled distance and time. Trains travel over long distances and can face different speeds due to reasons like stops or varying terrain.
In our problem, Patna express travels two different segments with varying distances and times. By breaking it down into time spent on each segment, we efficiently determine overall performance. Remember, practice with these problems can sharpen your mathematical reasoning and logical thinking.

Speed Formula

The speed formula is a simple yet powerful tool in physics and mathematics. It states that speed is equal to the distance traveled divided by the time taken. Mathematically, this is expressed as:\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]In our exercise, the speed formula helps in calculating the train's average speed over its entire journey. Once we know both the distance and time, determining speed becomes straightforward.
It's essential to ensure that distance and time are in compatible units. Both should be in kilometers and hours, or another consistent unit system, to make your calculations accurate.

Quantitative Aptitude

Quantitative aptitude involves solving numerical problems quickly and correctly, and is a key skill for exams and competitive tests. It covers areas like percentages, ratios, and, importantly, speed, distance, and time problems.
Mastering these can increase scores in tests and improve analytical skills in the workplace. Practicing such problems helps you think logically and solve complex questions under time constraints. By engaging with concepts like those in train problems, students develop not only their calculation abilities but also their understanding of real-world scenarios.
This well-rounded approach can give you a significant advantage in both academic and professional settings.