Two trains running in opposite directions cross a man standing on the platform in 27 seconds and 17 seconds respectively and they cross each other in 23 seconds. The ratio of their speeds is:

Step 1: Let the lengths of the two trains be L₁ and L₂, and their speeds be S₁ and S₂.

• Train 1 crosses a man in 27 seconds ⇒ L₁ = S₁ × 27
• Train 2 crosses a man in 17 seconds ⇒ L₂ = S₂ × 17

Step 2: The two trains cross each other in 23 seconds when moving in opposite directions.

That means:

L₁ + L₂ = (S₁ + S₂) × 23

Step 3: Substitute L₁ and L₂

S₁ × 27 + S₂ × 17 = (S₁ + S₂) × 23

Step 4: Expand and simplify the equation

27S₁ + 17S₂ = 23S₁ + 23S₂
=> 27S₁ - 23S₁ = 23S₂ - 17S₂
=> 4S₁ = 6S₂
=> S₁ / S₂ = 6 / 4 = 3 / 2

Answer: 3 : 2

The ratio of their speeds is 3:2.

This is a classic relative motion problem involving trains. It uses the concept of time = distance/speed, and setting up linear equations from the conditions provided. This type of question is commonly asked in competitive exams like SSC, Railway, and Banking.