Speed: distance, time, and why average speed is not the average of two speeds

Speed arrives late in P6 and carries a lot of marks. The formula itself is easy. What makes these questions hard is that they usually hide one quantity that stays the same across two journeys — and finding it is the whole problem.

The relationships

Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed

Keep units consistent. If speed is in km/h, time must be in hours — 45 minutes is 0.75 h or ¾ h, never 45. Mixed units are the most common careless error here.

The average-speed trap

A car travels from Town A to Town B at 60 km/h and returns along the same road at 40 km/h. The whole trip takes 5 hours. What is the distance between A and B?

Before solving: the average speed of this trip is not 50 km/h. That’s the trap, and it catches almost everyone. The car spends longer at the slower speed, so the slow leg pulls the average down.

Average speed is always:

Average speed = Total distance ÷ Total time

Now solve. The distance is the same both ways — that’s the constant.

Let the distance be d km.

Time there  = d ÷ 60
Time back   = d ÷ 40
Total time  = 5 hours

d/60 + d/40 = 5

Use a common denominator of 120:

2d/120 + 3d/120 = 5
        5d/120  = 5
          d/24  = 5
             d  = 120

The distance is 120 km.

Check: 120 ÷ 60 = 2 h, 120 ÷ 40 = 3 h, total 5 h ✓

And the average speed? Total distance 240 km ÷ 5 h = 48 km/h — not 50. Worth seeing once so you never assume otherwise.

Same distance vs same time

Almost every hard speed question is one of these two.

Same distance — a return journey, or two people covering the same route. Distance is your constant; write both times in terms of it.

Same time — two objects travelling towards each other, or starting together. Time is your constant; write both distances in terms of it.

For two objects moving towards each other, they close the gap at the sum of their speeds. For one chasing another, the gap closes at the difference.

Two cyclists start 60 km apart and ride towards each other at 12 km/h and 8 km/h. How long until they meet?

They close at 12 + 8 = 20 km/h.

Time = 60 ÷ 20 = 3 hours

Check: in 3 h one covers 36 km, the other 24 km, totalling 60 km ✓

Common mistakes

• Averaging the two speeds. Only valid when the times are equal, which is almost never the case in these questions.
• Minutes left as minutes. 1 h 30 min is 1.5 h.
• Adding speeds when one object chases another. Chasing means subtract.
• Forgetting a stop. If the question mentions resting, that time counts in the total but covers no distance.

Quick check

Meera walks to school at 4 km/h and cycles home along the same route at 12 km/h. The round trip takes 1 hour. How far is the school from her home?

   d/4 + d/12 = 1

Common denominator 12:
 3d/12 + d/12 = 1
        4d/12 = 1
          d/3 = 1
            d = 3