The boat travels 96 km downstream from A to B and back in 14 hours. A raft departed from A at the same time

The boat travels 96 km downstream from A to B and back in 14 hours. A raft departed from A at the same time. On the way back, the boat met a raft at a distance of 24 km from A. Determine the speed of the boat in still water and the speed of the current.

Let x be the boat’s own speed and the river speed (and the speed of the raft).

(x – a) is the speed of the boat upstream, (x + a) is the speed of the boat downstream.

Let us express the time of the boat on the way from A to B: 96 / (x + a).

Let us express the time of the boat on the way from B to the place of meeting with the raft: the boat passed from B 96 – 24 = 72 km, the time is equal to 72 / (x – a).

Let us express the time of the raft on the way: 24 / a.

Let’s make the equation:

24 / a = 96 / (x + a) + 72 / (x – a).

Divide the equation by 24:

1 / a = 4 / (x + a) + 3 / (x – a).

1 / a = (4x – 4a + 3x + 3a) / (x + a) (x – a).

1 / a = (7x – a) / (x² – a²).

a (7x – a) = x² – a².

7ax – a² + a² = x².

7ax = x², divide by x: x = 7a.

The time of movement of the boat from A to B is equal to 96 / (x + a), the travel time back is equal to 96 / (x – a), only 14 hours, we make the equation:

96 / (x + a) + 96 / (x – a) = 14, divide by 2:

48 / (x + a) + 48 / (x – a) = 7.

(48x – 48a + 48x + 48a) / (x + a) (x – a) = 7.

96x / (x² – a²) = 7.

7x² – 7a² = 96x.

x = 7a (see above).

7 * 49a² – 7a² = 96 * 7a.

336а² = 672а, divide by 336а:

a = 2 (km / h) – river flow speed.

x = 7a = 7 * 2 = 14 (km / h) – own speed of the boat.