Question Details

Two automated cargo capsules, capsule X and capsule Y, are traveling along parallel tracks. The length of capsule X is 25% greater than that of capsule Y. When moving in opposite directions, they completely pass each other in 12 seconds. When moving in the same direction, capsule X passes capsule Y in 60 seconds. What is the ratio of the speed of capsule X to the speed of capsule Y?

Options

5: 7

4: 5

3: 5

3: 2

3: 4

Show Answer

Correct Answer :

Option D

3: 2

Solution :

The correct option is 3: 2.

To find the ratio of the speed of capsule X to the speed of capsule Y, we need to determine the total distance covered when the capsules pass each other and set up relative speed equations for both opposite-direction and same-direction movements.

Step 1: Express the lengths of capsule X and capsule Y
Let the length of capsule Y be:
$L_{Y}$
Since the length of capsule X is 25% greater than that of capsule Y, we express the length of capsule X ( $L_{X}$ ) as:
$L_{X} = L_{Y} + 0.25 L_{Y} = 1.25 L_{Y} = \frac{5}{4} L_{Y}$
When two moving capsules pass each other completely, the total distance covered relative to each other is equal to the sum of their lengths:
$Total Distance = L_{X} + L_{Y} = 1.25 L_{Y} + L_{Y} = 2.25 L_{Y} = \frac{9}{4} L_{Y}$

Step 2: Formulate relative speed equations
Let $S_{X}$ be the speed of capsule X and $S_{Y}$ be the speed of capsule Y.

Case 1: Moving in opposite directions
When moving in opposite directions, the relative speed is the sum of their individual speeds:
$Relative Speed (Opposite) = S_{X} + S_{Y}$
They completely pass each other in 12 seconds:
$S_{X} + S_{Y} = \frac{Total Distance}{Time} = \frac{2.25 L_{Y}}{12} = \frac{9 L_{Y}}{48} = \frac{3 L_{Y}}{16}$
(Equation 1)

Case 2: Moving in the same direction
When moving in the same direction, capsule X passes capsule Y, so the relative speed is the difference of their speeds:
$Relative Speed (Same) = S_{X} - S_{Y}$
They pass each other in 60 seconds:
$S_{X} - S_{Y} = \frac{Total Distance}{Time} = \frac{2.25 L_{Y}}{60} = \frac{9 L_{Y}}{240} = \frac{3 L_{Y}}{80}$
(Equation 2)

Step 3: Calculate individual speeds and find the ratio
Add Equation 1 and Equation 2:
$2 S_{X} = \frac{3 L_{Y}}{16} + \frac{3 L_{Y}}{80} = \frac{15 L_{Y} + 3 L_{Y}}{80} = \frac{18 L_{Y}}{80}$
Dividing both sides by 2:
$S_{X} = \frac{9 L_{Y}}{80}$

Subtract Equation 2 from Equation 1:
$2 S_{Y} = \frac{3 L_{Y}}{16} - \frac{3 L_{Y}}{80} = \frac{15 L_{Y} - 3 L_{Y}}{80} = \frac{12 L_{Y}}{80}$
Dividing both sides by 2:
$S_{Y} = \frac{6 L_{Y}}{80}$

Now, determine the ratio of $S_{X}$ to $S_{Y}$ :
$\frac{S_{X}}{S_{Y}} = \frac{9 L_{Y} / 80}{6 L_{Y} / 80} = \frac{9}{6} = \frac{3}{2}$