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?
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A30 : 61
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B61 : 30
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C25 : 51
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D51 : 25
Answer
Correct Answer: 61 : 30
Explanation
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:
D = v * t = 1830.Given v = 2t + 1.So t * (2t + 1) = 1830.2t^2 + t - 1830 = 0.Discriminant = 1 + 4 * 2 * 1830 = 14641 = 121^2.t = ( -1 + 121 ) / ( 2 * 2 ) = 120 / 4 = 30 h (positive root).v = 2 * 30 + 1 = 61 km/h.Thus speed : time = 61 : 30.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