Problem 34 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

An automobile driver travels from plane to a hill station \(120 \mathrm{~km}\) distant at an average speed of \(30 \mathrm{~km}\) per hour. He then makes the return trip at an average speed of \(25 \mathrm{~km}\) per hour. He covers another \(120 \mathrm{~km}\) distance on plane at an average speed of \(50 \mathrm{~km}\) per hour. His average speed over the entire distance of \(360 \mathrm{~km}\) will be (A) \(\frac{30+25+50}{3} \mathrm{~km} / \mathrm{h}\) (B) \((30 \cdot 25 \cdot 50)^{\frac{1}{3}}\) (C) \(\frac{3}{\frac{1}{30}+\frac{1}{25}+\frac{1}{50}} \mathrm{~km} / \mathrm{h}\) (D) None of these

Step-by-Step Solution

Answer

The average speed is approximately 32.14 km/h, so the answer is D.

Step 1: Understand the Problem

We have three journeys: the first from plane to hill at 30 km/h, the return trip at 25 km/h, and another on plane at 50 km/h, each covering 120 km. We need to find the average speed for the entire 360 km trip.

Step 2: Calculate Time for Each Segment

For the first segment (120 km at 30 km/h), time taken is \(\frac{120}{30} = 4\) hours. For the return trip (120 km at 25 km/h), time taken is \(\frac{120}{25} = 4.8\) hours. For the last segment (120 km at 50 km/h), time taken is \(\frac{120}{50} = 2.4\) hours.

Step 3: Calculate Total Time for the Entire Trip

Add the time for each segment: \(4 + 4.8 + 2.4 = 11.2\) hours in total.

Step 4: Calculate Average Speed

Average speed is total distance divided by total time. Therefore, \(\text{Average speed} = \frac{360}{11.2} \approx 32.14\) km/h.

Step 5: Determine the Correct Option

Compare \(32.14\) km/h with the options given. Since neither option A nor B nor C matches \(32.14\), the correct answer is D, None of these.

Key Concepts

Time and Distance

Understanding the relationship between time and distance is crucial when solving problems related to average speed. To calculate time, you use the formula:

• Time = Distance ÷ Speed

In the given problem, the driver covers three distinct segments of his journey, each 120 km long, with different speeds. Calculating the time taken for each segment allows us to determine how long the entire journey will take. For the first segment, it takes 4 hours (120 km at 30 km/h). For the second, it requires 4.8 hours (120 km at 25 km/h), and for the third, just 2.4 hours (120 km at 50 km/h). These calculations are straightforward and require basic division, which is foundational in time and distance problems.
Maintaining consistency in units (like kilometers and hours) is important to ensure accuracy in calculations. Once you have the time for each journey part, you can easily add them to find the total travel time.

Physics of Motion

The physics of motion help explain how different speeds affect travel time over a given distance. Average speed is a useful concept because it provides a simple measure to understand overall motion, especially when speeds vary throughout a journey.

• Average Speed = Total Distance ÷ Total Time

In our example, although the driver has different speeds for each section, the overall average speed reflects his complete journey over 360 km. Average speed differs from instantaneous speed, which would be the speed at a particular moment. Calculating total average speed involves considering every segment of the journey.
In physics, the change in speed or the idea of different average speeds can also be tied to concepts like acceleration, but in this scenario, we're focused on discrete segments of travel at constant speeds, which simplifies the calculations.

Problem Solving in Mathematics

Problem solving in mathematics often involves breaking down a complex scenario into manageable steps. For our problem, we begin by understanding the situation: three journeys each with known distance and speed.
Then, using the formula \[\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}\]we calculate the time for each segment. By adding these times, we get the total time for all the journeys. With the total distance (360 km) and total time (11.2 hours) figured out, finding the average speed becomes a simple division exercise.

• Step 1: Calculate each segment's duration
• Step 2: Sum these times for total journey time
• Step 3: Divide the total distance by total time for average speed

These kinds of problems enhance mathematical reasoning, helping students develop strategies to tackle various types of questions. Understanding each part, such as why the mathematical method we chose is correct, is just as important as getting the right answer. This encourages deeper comprehension, allowing students to apply similar techniques to related problems.