Let the speed of boat in still water = x km/h

Let the speed of stream = y km/h

Obviously, x>y. (Otherwise the question cannot be solved)

**Upstream**** ** **Downstream**

** **

velocity, v = x - y km/h v = x + y km/h

distance, d = 30 km d = 44 km

velocity = distance/time v = d/t

∴ v = d/t ∴x + y = 44/t

∴ x - y = 30/t ∴t = 44 / (x+y) hours

∴ time t = 30 / (x-y) hours

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Now, the boat takes 10 hours to travel 30 km upstream and 44 km downstream.

∴ 30 / (x-y) + 44 / (x+y) = 10 ------------------- (1)

Similarly the boat takes 13 hours to travel 40 km upstream and 55 km downstream

∴ 40 / (x-y) + 55 / (x+y) = 13 ---------------- (2)

Let 1/(x-y) = a and 1/(x+y) = b

So, for equation (1)

30a + 44b = 10

∴2 (15a + 22b) = 10

∴15a + 22b = 5 ---------------(3)

For equation (2)

40a + 55b = 13 ---------------(4)

Solving equations (3) and (4) by Elimination method.

15a + 22b = 5 Equation (3) * 8

40a + 55b = 13 Equation (4) * -3

∴120a + 176b = 40

-120a - 165b = -39 Adding both equations

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∴ 11b = 1

∴ **b = 1/11**

Putting b = 1/11 in equation (3)

∴ 15a + 22 (1/11) = 5

∴ 15a + 2 = 5

∴15a = 5 - 2

∴15a = 3

∴a = 3/15

∴**a = 1/5**

Now,

a = 1/5 and b = 1/11

∴1/(x-y) = 1/5 ∴1/(x+y) = 1/11

∴x - y = 5 ---------(5) ∴x + y = 11 ------------(6)

Solving equations (5) and (6) by Elimination Method,

x - y = 5

x + y = 11 Adding (5) and (6)

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∴2x = 16

∴**x = 8**

Putting x = 8 in equation (6)

∴ 8 + y = 11

∴y = 11 - 8

∴**y = 3**

Thus,

Speed of boat in still water = x = **8 km/h**

Speed of stream = y = **3 km/h**