10. A boat takes 6 hours to travel 8 km upstream and 32 km downstream, and it takes 7 hours to travel 20 km upstream and 16 km downstream. Find the speed of the boat in still water and the speed of the stream.

Solution. Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr.

∴  Speed of the boat upstream = (x – y) km/hr

and speed of the boat downstream = (x + y) km/hr

We know that, Time = Distance ÷ Speed

As per the first condition,

8/(x – y) + 32/(x + y) = 6 ....... eq. no. (1)

As per the second condition,

20/(x – y) + 16/(x + y) = 7 ....... eq. no. (2)

Let 1/(x – y) =  m and 1/(x + y) = n

∴ Equation No. (1) will become,

8m + 32n = 6 ...... eq. no. (3)

and Equation Number (2) will become,

20m + 16n = 7 ....... eq. no. (4)

Multiplying equation no. (4) by 2, we get

40m  + 32n = 14 ...... eq. no. (5)

Subtracting equation (3) from equation (5)

40m + 32n = 14

8m + 32n = 6

(-)    (-)      (-)

32m        =  8

∴ m = 8/32

∴ m = ¼

Substituting  m = ¼ in equation number (3)

∴ 8m + 32n = 6

∴ 8(¼) + 32n = 6

∴ 2 + 32n = 6

∴ 32n = 6 – 2

∴ 32n = 4

∴ n = 4/32

∴ n = 1/8

Resubstituting the values of m and n we get,

m = 1/(x – y) ∴ ¼ = 1/(x – y) ∴ x – y = 4...... eq. no. (A)

n = 1/(x + y) ∴ 1/8 = 1/(x + y) ∴ x + y = 8 ....... eq. no. (B)

Adding equations (A) and (B) ,

x – y = 4

x + y = 8

2x     = 12

∴ x = 12/2

∴ x = 6

Substituting x = 6 in equation (B),

∴ x + y = 8

∴ 6 + y = 8

∴ y = 8 – 6

∴ y = 2

∴ The speed of boat in still water is 6 km/hr and speed of stream is  2 km/ hr.