Time and Distance – Mixed speeds over fixed fractions of a journey: Aashutosh covers 2/3 of a trip at 4 km/h and the remaining 1/3 at 5 km/h, taking a total of 84 minutes. What is the total distance?

Introduction / Context:
When different segments of a journey are covered at different speeds, total time is the sum of segment times. Here the fractions are of distance, not time, so harmonic-mean shortcuts do not apply; we compute each segment time explicitly.

Given Data / Assumptions:

• Total time = 84 min = 1.4 h.
• First segment = 2/3 of distance at 4 km/h.
• Second segment = 1/3 of distance at 5 km/h.

Concept / Approach:
Let total distance be D. Then total time T = ( (2/3)D / 4 ) + ( (1/3)D / 5 ). Solve T = 1.4 for D.

Step-by-Step Solution:
T = D*( (2/3)/4 + (1/3)/5 ) = D*( 1/6 + 1/15 ) = D*( 5/30 + 2/30 ) = D*(7/30).Set D*(7/30) = 1.4 = 7/5 ⇒ D = (7/5)*(30/7) = 6 km.

Verification / Alternative check:
Times: (2/3*6)/4 = 1 h and (1/3*6)/5 = 0.4 h; total 1.4 h, consistent.

Why Other Options Are Wrong:
8, 9, 15, and 7 km do not satisfy the time equation with the given segment speeds.

Common Pitfalls:
Using the average of speeds rather than computing weighted segment times by distance fractions.

Final Answer:
6 km