Two cyclists P and Q cycle towards each other along a straight road at 20 km/h and 25 km/h respectively. If they meet after 56 minutes, what was the distance (in km) between them when they started?

Introduction / Context:
This is a direct application of relative speed when two objects move towards each other. The distance between them decreases at the sum of their speeds. By combining their speeds and using the time until they meet, we can easily compute the initial separation between the two cyclists.

Given Data / Assumptions:

• Speed of cyclist P = 20 km/h.
• Speed of cyclist Q = 25 km/h.
• They ride towards each other.
• They meet after 56 minutes.
• Speeds are constant and along a straight line.

Concept / Approach:
When two bodies move towards each other, their relative speed is the sum of their individual speeds. The distance between them at the start equals relative speed multiplied by the time until they meet. Time must be converted into hours to match the speed units of km/h.

Step-by-Step Solution:
Speed of P = 20 km/h. Speed of Q = 25 km/h. Relative speed = 20 + 25 = 45 km/h. Time until they meet = 56 minutes = 56 / 60 hour = 14 / 15 hour. Distance between them = relative speed * time. Distance = 45 * (14 / 15) = 3 * 14 = 42 km.

Verification / Alternative check:
You can compute how far each cyclist travels: P travels 20 * (14 / 15) = 280 / 15 km, and Q travels 25 * (14 / 15) = 350 / 15 km. The sum of these is (280 + 350) / 15 = 630 / 15 = 42 km, matching the computed distance.

Why Other Options Are Wrong:

• 36 km, 39 km and 45 km do not match the product of 45 km/h and 14 / 15 hour. If used, they produce a meeting time differing from 56 minutes.

Common Pitfalls:
Some students mistakenly subtract speeds when objects move towards each other, which is incorrect. Others forget to convert 56 minutes into hours, leading to unit mismatch. Keeping units consistent and remembering that for opposite directions we add speeds is key.

Final Answer:
The initial distance between the two cyclists was 42 km.