Speed Comparison – Half distance takes four times the time: A certain distance is covered at a speed v1. If half of this distance is covered in four times as long as before, what is the ratio of the two speeds (v1 : v2)?

Introduction / Context:
This tests proportional reasoning. Time is inversely proportional to speed for the same distance. The problem states that for half the original distance, the new time is four times the old time for that half-distance at v1, which fixes the speed ratio directly.

Given Data / Assumptions:

• Original total distance = D and original speed = v1.
• Half distance D/2 at v1 takes time t1 = (D/2)/v1.
• New time for the same half distance is t2 = 4 * t1 at speed v2.

Concept / Approach:
v2 = (D/2)/t2 = (D/2)/(4*(D/2)/v1) = v1/4. Ratio v1 : v2 = 4 : 1.

Step-by-Step Solution:
t1 = (D/2)/v1.t2 = 4t1 ⇒ v2 = (D/2)/t2 = v1/4.Therefore v1 : v2 = 4 : 1.

Verification / Alternative check:
Pick numbers: D = 8, v1 = 8 ⇒ half distance time t1 = 0.5 h. New time t2 = 2 h ⇒ v2 = 2 km/h. Ratio 8 : 2 = 4 : 1.

Why Other Options Are Wrong:
1 : 8, 1 : 4, 8 : 1, and 2 : 1 contradict the direct proportionality established.

Common Pitfalls:
Applying the factor of 4 to distance instead of time or forgetting that the comparison is for the same half distance.

Final Answer:
4 : 1