Opposite Travel – Ratio of speeds from post-meeting completion times: Two cyclists start simultaneously from A to B and from B to A. After meeting, they take 16 h and 25 h, respectively, to complete the remaining distances. What is the ratio of the first cyclist’s speed to the second’s?

Introduction / Context:
When two travelers start toward each other, the times they take to finish after meeting encode the ratio of their speeds. A standard identity says v1/v2 = sqrt(t2/t1), where t1 and t2 are the post-meeting finish times of travelers 1 and 2.

Given Data / Assumptions:

• After meeting, first finishes in t1 = 16 h.
• After meeting, second finishes in t2 = 25 h.
• They started simultaneously and travel on the same straight route.

Concept / Approach:
Let initial distance be D and meeting time be T. Then t1 = (v2*T)/v1 and t2 = (v1*T)/v2, giving t1/t2 = (v2/v1)^2 and thus v1/v2 = sqrt(t2/t1).

Step-by-Step Solution:
v1/v2 = sqrt(25/16) = 5/4.

Verification / Alternative check:
Square both sides: (v1/v2)^2 = 25/16 = t2/t1, which matches the identity derived from meeting-time relations.

Why Other Options Are Wrong:
Ratios other than 5 : 4 conflict with the square-root relationship forced by the 16 h and 25 h data.

Common Pitfalls:
Reversing t1 and t2, which would invert the ratio, or directly dividing times rather than using the square-root relation.

Final Answer:
5 : 4