What was the average speed during the journey? Statement I: The first half of…

Concept: Average speed for a journey is total distance divided by total time. When a data-sufficiency question gives average speeds over overlapping portions of the same route, each statement only fixes the average speed over that whole portion, not the time spent in every sub-part of it. If splitting the route around the overlap leaves more unknown segment times than the number of independent equations available, the total time — and hence the overall average speed — stays undetermined even after combining both statements.

Application:

1. Let the total journey distance be D. Split the route into three consecutive parts around the overlap between "first half" and "last two-thirds": Part A = the first one-third of D, Part B = the middle one-sixth of D (the overlap zone, from the one-third mark to the halfway mark), Part C = the last half of D (from the halfway mark to the end).

2. Statement I: the first half (Part A + Part B, distance D/2) was covered at 45 km/hr, so its total time = (D/2) / 45 = D/90.

3. Statement II: the last two-thirds (Part B + Part C, distance 2D/3) was covered at 55 km/hr, so its total time = (2D/3) / 55 = 2D/165.

4. These are only two equations relating three unknown segment times (time on A, time on B, time on C). Even using both statements together, the individual segment times are not pinned down — only their pairwise sums (A+B and B+C) are known.

5. Total journey time = (time on A) + (time on B) + (time on C). This still depends on the unknown time spent on overlapping Part B, so it cannot be evaluated from the two equations alone.

Cross-check (concrete numbers, taking D = 990 km for clean division):

• Split 1: time on A = 6 h, time on B = 5 h, time on C = 7 h. This satisfies both statements (A+B = 11 h, so 495/11 = 45 km/hr; B+C = 12 h, so 660/12 = 55 km/hr), and total time = 18 h, giving an overall average speed of 990/18 = 55 km/hr.

• Split 2: time on A = 3 h, time on B = 8 h, time on C = 4 h. This also satisfies both statements exactly (A+B = 11 h, 45 km/hr; B+C = 12 h, 55 km/hr), but total time = 15 h, giving an overall average speed of 990/15 = 66 km/hr.

• Two different time-splits both honour Statement I and Statement II exactly, yet produce two different overall average speeds — so the two statements together do not fix a unique average speed.

Result: Since combining both statements still leaves the overall average speed undetermined, the data in Statement I and Statement II together is not sufficient to answer the question.