Question

a boat goes 30km upstream and 44km down stream in 10 hours in 13 hours it can go 40km upstream and 55km down stream determine the speed of the stream and that of the boat in still water​

Answer 1

• Speed of boat in still water = 8 km/h
• Speed of Stream = 3 km/h

Step-by-step explanation:

$\bf\large {\pink{Given :-}}$

• boat goes 30km upstream and 44km down stream in 10 hours .
• hours it can go 40km upstream and 55km downstream in 13 hrs.

$\bf\large {\pink{To \:find :-}}$

• The speed of stream and that of boat in still water .

$\bf\large {\pink{Formula \:used :-} }$

Time = distance/speed

$\bf\large {\pink{Solution :- }}$

• Let speed of boat in still water = x km/h
• Speed of stream = y km/h

Therefore,

• speed of upstream = ( x – y ) km/h

and,

• speed of downstream = ( x + y ) km/h

A.T.Q.

$\\\sf \dfrac{30}{x - y} + \dfrac{44}{x + y} = 10\: .............eq.(1) \\ \\\sf \dfrac{40}{x - y} + \dfrac{55}{x + y} = 13\:........eq.(2)\\ \\\sf let \: \: \: \: \: \dfrac{1}{x - y} = a \: \\ \\\sf \dfrac{1}{x + y} = b$

Therefore,

$\sf 30a + 44b = 10\:.........eq.(3) \\ \\\sf 40a + 55 a = 13\:.........eq.(4)$

multiply by 4 in eq.(3)and by 3 in eq.(4)

$\\ \sf 120a +176 = 40\:...........eq.(5) \\ \\ \sf 120a +165b = 39\:........eq.(6)$

Subtract eq.(6) from eq.(5)

$\\ \sf 120a + 165 b= 39 \\ \sf 120a + 176b =40 \\ \sf\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\\sf - 11b = - 1 \\ \\\sf 11 b = 1 \\ \\\sf b \: = \dfrac{1}{11}$

Put the value of b = 1/11 in eq.(3)

$\\ \sf 30a + 44 \times \dfrac{1}{11} = 10 \\ \\\sf 30a + 4 = 10 \\ \\\sf 30a = 6 \\ \\\sf a = \dfrac{1}{5}$

Now,

$\\ \sf a = \dfrac{1}{x - y} = \dfrac{1}{5} \\ \\\sf x - y = 5\: ........eq.(8) \\ \\\sf b = \dfrac{1}{x + y} = \frac{1}{11} \\ \\ \sf x + y = 11\:.......eq.(9)$

Add eq.(8) and (9)

$\\ \sf x - y = 5 \\ \sf x + y = 11 \\ \underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \sf 2x = 16 \\ \\ \sf x = \dfrac{16}{2} = 8 \\ \\\sf \boxed{\sf x = 8}$

Put the value of x = 8 in eq.(9)

$\\ \sf 8 + y = 11 \\ \\ \sf y = 11 - 8 \\ \\\sf\boxed{ \sf y = 3}$

Hence, speed of boat in still water = x = 8km/h

speed of stream = y = 3km/h