Train \(X\) crosses a man standing on the platform in 24 seconds and train \(Y\) crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of \(X\) to speed of \(Y\)?
This problem involves calculating the ratio of the speeds of two trains, Train X and Train Y, based on how long they take to cross a stationary man and how long they take to cross each other.
Key concepts used here are:
Let:
We are given the following information:
Train X crossing a man:
Using the speed-distance-time formula, where distance is the length of the train:
\(v_X = L_X / 24\)
From this, we can express the length of Train X in terms of its speed:
\(L_X = 24 * v_X\)
Train Y crossing a man:
Similarly, for Train Y:
\(v_Y = L_Y / 18\)
Expressing the length of Train Y in terms of its speed:
\(L_Y = 18 * v_Y\)
Trains X and Y crossing each other:
Since they are moving in opposite directions, their relative speed is \(v_X + v_Y\).
The total distance to cover is the sum of their lengths, \(L_X + L_Y\).
The time taken is 20 seconds.
Therefore, the equation is:
\(v_X + v_Y = (L_X + L_Y) / 20\)
Substituting lengths into the crossing equation:
Now, substitute the expressions for \(L_X\) and \(L_Y\) from steps 1 and 2 into the equation from step 3:
\(v_X + v_Y = ( (24 * v_X) + (18 * v_Y) ) / 20\)
Solving for the ratio of speeds:
Multiply both sides by 20:
\(20 * (v_X + v_Y) = 24 * v_X + 18 * v_Y\)
Distribute the 20:
\(20 * v_X + 20 * v_Y = 24 * v_X + 18 * v_Y\)
Rearrange the terms to group speeds of X on one side and speeds of Y on the other:
\(20 * v_Y - 18 * v_Y = 24 * v_X - 20 * v_X\)
Simplify the terms:
\(2 * v_Y = 4 * v_X\)
Now, find the ratio of the speed of X to the speed of Y (\(v_X / v_Y\)):
\(v_X / v_Y = 2 / 4\)
Simplify the fraction:
\(v_X / v_Y = 1 / 2\)
The ratio of the speed of Train X to the speed of Train Y is 1:2.
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