A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can
go 40 km upstream and 55 km downstream. Determine the speed of the stream and
that of the boat in still water.
Answers
Step-by-step explanation:
Let the speed of boat in still water=x=x km\hr and The speed of stream=y=y km\hr
Speed of boat at downstream
\Rightarrow \left (x+y \right )km/hr⇒(x+y )km/hr
Speed of boat at upstream
\Rightarrow \left (x-y \right )km/ hr⇒(x−y )km/hr
\because time =\cfrac{distance}{speed}∵time=
speed
distance
Time taken to cover 30 km upstream \Rightarrow \cfrac{30}{x-y}⇒
x−y
30
Time taken to cover 44 km downstream\Rightarrow \cfrac{44}{x+y}⇒
x+y
44
According to the first condition,
\Rightarrow \dfrac{30}{x-y}=\dfrac{44}{x+y}=10⇒
x−y
30
=
x+y
44
=10
Time taken to cover 40 km upstream \Rightarrow \cfrac{40}{x-y}⇒
x−y
40
Time taken to cover 55 km downstream \Rightarrow \cfrac{55}{x+y}⇒
x+y
55
According to the second condition,
\Rightarrow \dfrac{40}{x-y}=\dfrac{55}{x+y}=13⇒
x−y
40
=
x+y
55
=13
Let \dfrac{1}{x-y}=u\quad and \quad\dfrac{1}{x+y}=v
x−y
1
=uand
x+y
1
=v
\Rightarrow 30u+44v=10.....eq1⇒30u+44v=10.....eq1
\Rightarrow 40u+55v=13.....eq2⇒40u+55v=13.....eq2
Multiplying eq1 by 3 and eq2 by 5 and subtract both
\Rightarrow \left (150u+220v=50 \right )-\left (160u+220v=52 \right )⇒ (150u+220v=50 )−(160u+220v=52 )
\Rightarrow -10u=-2\Rightarrow u=\dfrac{1}{5}⇒−10u=−2⇒u=
5
1
put u=\dfrac{1}{5}u=
5
1
in eq1
\Rightarrow 30\times\dfrac{1}{5}+44v=10\Rightarrow 44v=4\Rightarrow v=\dfrac{1}{4}⇒30×
5
1
+44v=10⇒44v=4⇒v=
4
1
\Rightarrow u=\dfrac{1}{x-y}=\dfrac{1}{5}\Rightarrow x-y=5...eq3⇒u=
x−y
1
=
5
1
⇒x−y=5...eq3
\Rightarrow v=\dfrac{1}{x+y}=\dfrac{1}{11}\Rightarrow x+y=11...eq4⇒v=
x+y
1
=
11
1
⇒x+y=11...eq4
Subtracting eq3 and eq4, we get
\Rightarrow x=8⇒x=8
Put x=8x=8 in eq3
\Rightarrow y=3⇒y=3
Hence, the speed of the boat in still water=8=8km\hr
The speed of stream=3=3km\hr
Answer:
Speed of stream = 3 km / hr.
Speed of boat in still water = 8 km / hr.
Step-by-step explanation:
Let the speed of the boat in still water be a km / hr and stream be b km / hr
For upstream = a - b
For downstream = a + b
We know :
Speed = Distance / Time
Case 1 .
10 = 30 / a - b + 44 / a + b
Let 1 / a - b = x and 1 / a + b = y
30 x + 44 y = 10 ... ( i )
Case 2 .
13 = 40 / a - b + 55 / a + b
40 x + 55 y = 13 ... ( i )
Multiply by 4 in ( i ) and by 3 in ( ii )
120 x + 176 y = 40
120 x = 40 - 176 y ... ( iii )
120 x + 165 y = 39
120 = 39 - 165 y ... ( iv )
From ( iii ) and ( iv )
40 - 176 y = 39 - 165 y
11 y = 1
y = 1 / 11
120 x = 40 - 176 y
120 x = 40 - 176 / 11
x = 1 / 5
Now :
1 / a - b = 1 / 5
a - b = 5
a = 5 + b ... ( v )
1 / a + b = 1 / 11
a + b = 11
a = 11 - b ... ( vi )
From ( v ) and ( vi )
11 - b = 5 + b
2 b = 6
b = 3
a = 5 + b
a = 5 + 3
a = 8
Hence we get answer.