A boat covers 32km upstream and 36km downstream in 7 hours. it covers 40 km upstream and 48 km downstream in 9 hours. find the speed in still water and in stream.
Answers
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Answer :
Speed in still water = 10 km / hr
Speed in stream = 2 km / hr
Step-by-step explanation :
Let the speed of the boat in still water = x km/hr
Speed of stream = y km/hr
First Condition :
1) Speed of the boat in downstream = (x + y) km/hr
Distance travelled = d₁ = 36
Speed = Distance / Time
⇒ t₁ ( time ) = 36 / ( x + y )
2) Speed of the upstream = ( x - y )km/hr
Distance travelled = d₂ = 32
Speed = Distance / Time
⇒ t₂ ( time ) = 32 / ( x - y )
Total time = 7 hour
∴ t₁ + t₂ = 7
⇒ { 36 / (x + y) } + { 32 / (x - y) } = 7 ...…(i)
Second Condition :
1) Speed of the boat in downstream = (x + y) km/hr
Distance travelled = d₃ = 48
Speed = Distance / Time
⇒ t₃ ( Time ) = 48 / ( x + y )
2) Speed of the boat in upstream = (x - y) km/hr
Distance travelled = d₄ = 40
Speed = Distance / Time
⇒ t₄ = 48 / (x - y)
∴ t₃ + t₄= 9 hour
⇒ { 48 / ( x + y )} + { 40 / ( x - y ) }…… (ii)
Let { 1 / ( x +y )} = a and { 1 / ( x - y )} = b
Now, evaluate (i) and (ii)
36a + 32b = 7 …… (iii)
48a + 40b = 9 …… (iv)
Multiply (iv) with 4, and (iii) with 5
192a + 160b = 36 …… (v)
180a + 160b = 35 …… (vi)
On subtracting (vi) from (v)
192a - 180a = 36 - 35
⇒ 12a = 1
⇒ a = 1 / 12,
Putting the value of a in (v)
192 * 1 / 12 + 160b = 36
160b = 36 - 16
⇒ b = 20 / 160
⇒ b = 1 / 8
Now,
1 / (x + y ) = 1 / 12 [ ∴ a = 1 / 12 ]
⇒ ( x + y ) = 12 ……… (vii)
And,
1 / (x - y ) = 1 / 8 [ ∴ b = 1 / 8 ]
⇒ ( x - y ) = 8 ……… (viii)
After adding (vii) and (viii),
⇒ x + y + x - y = 12 + 8
⇒ 2x = 20
⇒ x = 10
On substituting the value of x in (vii)
⇒ ( x + y ) = 12
⇒ 10 + y = 12
⇒ y = 2
Hence,
Speed of the boat in still water = 10 km/hr
And, Speed of the stream = 2 km/hr