A train starts from Q with speed of 30 km per hour towards P which is at a distance of 240 km from Q. An insect starts flying with speed of 45 km per hour at the same time from P towards Q and meets the train at some point. It turns back immediately and flies towards P. When it reaches P, it turns back and flies towards the train. It continues to do so until the train reaches P. What is the sum of all time intervals when its is moving in the same direction as the train during the journey. It is given that the speeds of both the train and the insect remain constant?a)3 hoursb)4 hoursc)2 hoursd)6 hoursCorrect answer is option 'B'. Can you explain this answer?

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A train starts from Q with speed of 30 km per hour towards P which is at a distance of 240 km from Q. An insect starts flying with speed of 45 km per hour at the same time from P towards Q and meets the train at some point. It turns back immediately and flies towards P. When it reaches P, it turns back and flies towards the train. It continues to do so until the train reaches P. What is the sum of all time intervals when its is moving in the same direction as the train during the journey. It is given that the speeds of both the train and the insect remain constant?

a)
3 hours
b)
4 hours
c)
2 hours
d)
6 hours

Correct answer is option 'B'. Can you explain this answer?

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A train starts from Q with speed of 30 km per hour towards P which is ...

To solve this problem, we need to find the time intervals when the insect is moving in the same direction as the train during the journey from Q to P. Let's break down the problem step by step:

1. Finding the time taken by the train to reach P:
The speed of the train is given as 30 km/h, and the distance from Q to P is 240 km. We can use the formula Time = Distance / Speed to find the time taken by the train to reach P.
Time taken by the train = 240 km / 30 km/h = 8 hours

2. Finding the time taken by the insect to meet the train:
The insect is flying towards the train with a speed of 45 km/h. Since the insect and the train are moving towards each other, their effective speed is the sum of their individual speeds.
Effective speed = Speed of the insect + Speed of the train
Effective speed = 45 km/h + 30 km/h = 75 km/h
Using the formula Time = Distance / Speed, we can find the time taken by the insect to meet the train.
Time taken by the insect = 240 km / 75 km/h = 3.2 hours

3. Finding the time intervals when the insect is moving in the same direction as the train:
Since the insect and the train start at the same time, the insect starts moving towards P as soon as the train starts from Q. When the insect meets the train, it immediately turns back and starts flying towards P. This process continues until the train reaches P.
During this journey, the insect is moving in the same direction as the train for a certain time interval. We need to find the sum of all these time intervals.
From step 2, we know that the insect takes 3.2 hours to meet the train. Since the total journey time of the train is 8 hours (step 1), the insect will again meet the train after 8 - 3.2 = 4.8 hours.
This pattern will continue until the train reaches P. We can see that the time intervals between each meeting of the insect and the train form an arithmetic progression with a common difference of 4.8 hours.
To find the sum of these time intervals, we can use the formula for the sum of an arithmetic progression:
Sum = (n/2)(2a + (n-1)d)
where n is the number of terms, a is the first term, and d is the common difference.

In this case, the first term is 3.2 hours, the common difference is 4.8 hours, and the number of terms is (8 / 4.8) + 1 = 2 + 1 = 3 (since the insect meets the train three times).
Substituting these values into the formula, we get:
Sum = (3/2)(2 * 3.2 + (3-1) * 4.8)
Sum = (3/2)(6.4 + 2 * 4.8)
Sum = (3/2)(6.4 + 9.6)
Sum = (3/2)(16)
Sum = 24/2
Sum = 12 hours

Therefore, the sum of all time intervals when the insect is moving in the same direction as the train during the journey is 12 hours, which corresponds to option

Answered by Wizius Careers · View profile

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A train starts from Q with speed of 30 km per hour towards P which is ...

The insect and the train will repeat the same process after the insect reaches its initial point.
Consider the distance between the insect and the train when the insect starts flying from P be d.
Distance between point where both meet and

Then both will move in the same direction.
Time taken by the insect to reach P again=

(which is the required time interval)
Distance traveled by the train when the insect reaches

Distance between the train and the insect when the insect reaches P = 3d/5-2d/5 = d/5
The same process will be repeated again with distance

and so on.
Then time intervals will be d₁/75,d₂/75,d₃/75 and so on.
Sum of the time intervals = (d/75)(1+1/5+1/25+1/125.........)
Now the initial distance,d =240 km
Sum of the time intervals =

B is the answer.
Alternate Solution:
Total time taken by the train to reach Q=240/30=8 hours
The insect will move towards the train meet at some place and return back. Since the speed of insect is constant, no matter what time the train takes, the time taken by insect towards the train and and away from the train will remain same each time because the insect travels same distance both towards the train and away from the train.
Total time taken in the direction of the train=Total time taken against the direction of the train=8/2=4 hours

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A train starts from Q with speed of 30 km per hour towards P which is at a distance of 240 km from Q. An insect starts flying with speed of 45 km per hour at the same time from P towards Q and meets the train at some point. It turns back immediately and flies towards P. When it reaches P, it turns back and flies towards the train. It continues to do so until the train reaches P. What is the sum of all time intervals when its is moving in the same direction as the train during the journey. It is given that the speeds of both the train and the insect remain constant?a)3 hoursb)4 hoursc)2 hoursd)6 hoursCorrect answer is option 'B'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A train starts from Q with speed of 30 km per hour towards P which is at a distance of 240 km from Q. An insect starts flying with speed of 45 km per hour at the same time from P towards Q and meets the train at some point. It turns back immediately and flies towards P. When it reaches P, it turns back and flies towards the train. It continues to do so until the train reaches P. What is the sum of all time intervals when its is moving in the same direction as the train during the journey. It is given that the speeds of both the train and the insect remain constant?a)3 hoursb)4 hoursc)2 hoursd)6 hoursCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A train starts from Q with speed of 30 km per hour towards P which is at a distance of 240 km from Q. An insect starts flying with speed of 45 km per hour at the same time from P towards Q and meets the train at some point. It turns back immediately and flies towards P. When it reaches P, it turns back and flies towards the train. It continues to do so until the train reaches P. What is the sum of all time intervals when its is moving in the same direction as the train during the journey. It is given that the speeds of both the train and the insect remain constant?a)3 hoursb)4 hoursc)2 hoursd)6 hoursCorrect answer is option 'B'. Can you explain this answer?.

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