The question is from the topic Relative Speed. This is a classic CAT level question testing your understanding in the concept of relative speed from Time Speed and Distance. Time Speed and Distance is a favorite in CAT Exam and appears more often than expected in the CAT Quantitative Aptitude section in the CAT Exam

Question 27 : Two trains left from two stations P and Q towards station Q and station P respectively. 3 hours after they met, they were 675 Km apart. First train arrived at its destination 16 hours after their meeting and the second train arrived at its destination 25 hours after their meeting. How long did it take for the first train to make the whole trip?

- 18h
- 36h
- 25h
- 48h

36h

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Total distance travelled by both the trains before meeting = D. This distance will be covered in the proportion of their speeds. Clarify with the diagram.

3 hours after meeting distance travelled by A = S_{A} * 3 and by B = S_{B} * 3

Therefore, 3 (S_{A} + S_{B}) = 675 => S_{A} + S_{B} = 225.

Now the remaining distance to be covered by first train is DS_{B}/(S_{A} + S_{B})

Therefore, time taken = \\frac{DS_{B}}{(S_{A} + S_{B})S_{A}}\\) = 16 --------- (1)

Similarly, \\frac{DS_{A}}{(S_{A} + S_{B})S_{A}}\\) = 25 --------- (2)

Dividing equation 1 by 2 -

\\frac{S_A^2}{S_B^2}\\) = \\frac{25}{16}\\) => \\frac{S_A}{S_B}\\) = \\frac{5}{4}\\).

Therefore, S_{A} + \\frac{4}{5}\\) * S_{A} = 225

=> S_{A} = 125 Km/hr and S_{B} = 100 Km/hr

From equation 2, \\frac{D}{S_A}\\) = 16 * \\frac{225}{100}\\) = 36h,

which is the time taken for the first train to complete the journey.

The question is ** "How long did it take for the first train to make the whole trip?"**

Choice B is the correct answer.

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