The rider traveled half the way at a speed of 10 km / h. Then, for half of the remaining travel time

The rider traveled half the way at a speed of 10 km / h. Then, for half of the remaining travel time, he drove at a speed of 8 km / h, and then walked to the end of the journey at a speed of 4 km / h. Determine the average speed of the rider along the entire path. The answer should be 7.5 km / h.

Let’s denote the entire path that the rider traveled as S, and all the time it took him for this as T. Then the average speed is calculated by the formula:

Vav = S / T.

The condition contains three sections of the path with different data:
S = s1 + s2 + s3; and s1 = S / 2.

T = t1 + t2 + t3; and t2 = t3.

The rider traveled half the way at a speed of 10 km / h:

s1 = S / 2 = 10 * t1;

t1 = S / 20.

For half the remaining time, he drove at a speed of 8 km / h:
s2 = 8 * t2;

t2 = (T – t1) / 2;

s2 = 4 * (T – t1).

The rest of the way he walked on foot at a speed of 4 km / h:

s3 = 4 * t3;

s3 = 4 * t2;

s3 = 4 * (T – t1) / 2;

s3 = 2 * (T – t1).

All the way:

S = s1 + s2 + s3;

S = 10 * t1 + 4 (T – t1) + 2 * (T – t1);

S = 10 * t1 + 6 * (T – t1);

S = 10 * t1 + 6 * T – 6 * t1;

S = 4 * t1 + 6 * T;

S = 4 * S / 20 + 6 * T;

S = S / 5 + 6 * T;

T = (S – S / 5) / 6;

T = 2 * S / 15;

Vav = S / T;

Vav = (S * 15) / (2 * S);

Vav = 7.5 km / h

Answer: The average speed of the rider along the entire route was 7.5 km / h.