Train distance is 1830 km. The speed of the train (in km/h) is one more than twice the total travel time (in hours). If distance = speed * time holds for the same journey, what is the ratio of speed : time that satisfies both conditions?

Introduction / Context:
Many time, speed, and distance questions hide a second relation that links speed and time. Here, the product speed * time equals a fixed distance 1830 km, and an extra linear relation is given between speed and time. We must find a consistent pair and then report the ratio speed : time.

Given Data / Assumptions:

• Distance D = 1830 km.
• Let speed be v km/h and time be t h.
• Additional relation: v = 2t + 1.

Concept / Approach:
Use D = v * t along with v = 2t + 1 to form a quadratic in t. Solve for positive t and then compute v. Finally express v : t in simplest integer terms.

Step-by-Step Solution:

Verification / Alternative check:
Check product: 61 * 30 = 1830 km which matches the distance. The linear relation v = 2t + 1 also holds with t = 30, v = 61.

Why Other Options Are Wrong:
30 : 61 reverses the correct order. 25 : 51 and 51 : 25 do not satisfy v = 2t + 1 nor give v * t = 1830.

Common Pitfalls:
Mixing order of the ratio (speed : time) and discarding the signless positive root only. Some try substituting approximate numbers instead of solving the quadratic cleanly.

Final Answer:
61 : 30