Question

A boat covers 32km upstream and 36km downstream in 7hrs. Also it covers 40km upstream and 48km downstream in 9 hours. Find the speed of the boat in still water and that of the stream?

Answer 1

Hint: When the boat goes upstream, its direction is opposite to the flow of water. In downstream the boat is in the direction of flow of water. With the given conditions formulate 2 equations using the speed of formula.

Complete step-by-step answer:
Let us assume the speed of the boat in still water to be ‘x’ km/hr.
The speed of the stream can be taken ‘y’ km/hr.
It is given that a boat goes 32km upstream and 36km downstream in 7 hour.
We know the formula of Speed = Distance / Time.
\[\therefore \] Time = Distance / Speed
When the boat goes upstream, it is the direction opposite to the direction of flow of water. Similarly, when a boat goes downstream it is in the same direction as that of water.
\[\therefore \] Speed of boat when going upstream = Speed of boat – Speed of stream = x – y
\[\therefore \] Speed when the boat goes downstream = Speed of boat + Speed of stream = x + y
Thus according to the first condition, we can formulate the expression as time = distance / speed.
\[\dfrac{32}{x-y}+\dfrac{36}{x+y}=7-(1)\]
i.e. time taken to go upstream and time taken to go downstream.
Similarly if the boat goes 40km upstream and 46km downstream in 9hrs, thus the expression will be formed as,
\[\dfrac{40}{x-y}+\dfrac{48}{x+y}=9-(2)\]
Thus we got two equations to solve,
Let us put, \[x-y=a\] and \[x+y=b\].
Thus the equation changes as,
Equation (1) becomes, \[\dfrac{32}{a}+\dfrac{36}{b}=7-(3)\]
Equation (1) becomes, \[\dfrac{40}{a}+\dfrac{48}{b}=9-(4)\]
Multiply equation (3) by 5 and equation (4) by 4.
\[\begin{align}
  & \dfrac{32\times 5}{a}+\dfrac{36\times 5}{b}=7\times 5\Rightarrow \dfrac{160}{a}+\dfrac{180}{b}=35 \\
 & \dfrac{40\times 4}{a}+\dfrac{48\times 4}{b}=9\times 4\Rightarrow \dfrac{160}{a}+\dfrac{192}{b}=36 \\
\end{align}\]
Thus let us subtract both equations. i.e. (4) – (3)
\[\begin{align}
  & \dfrac{160}{a}+\dfrac{192}{b}=36 \\
 & \dfrac{160}{a}+\dfrac{180}{b}=35 \\
\end{align}\]
- - -
\[0+\dfrac{192-180}{b}=1\]
Thus we got, \[\dfrac{192-180}{b}=1\]
\[\therefore \dfrac{12}{b}=1\] [cross product properly]
i.e. b = 12.
Now put the value of b in equation (3).
\[\dfrac{32}{a}+\dfrac{36}{12}=7\Rightarrow \dfrac{32}{a}=7-3\]
\[\therefore \dfrac{32}{a}=4\], apply cross product properly.
\[\therefore a=\dfrac{32}{4}=8\]
Thus, \[x-y=a\Rightarrow x-y=8-(5)\]
\[x+y=b\Rightarrow x+y=12-(6)\]
Now let us solve equation (5) and (6) to get the value of x and y.
Add (5) + (6), we get
\[\begin{align}
  & x-y=8 \\
 & x+y=12 \\
\end{align}\]
+ + +
\[2x=20\]
Thus we got, \[x=\dfrac{20}{2}=10\]
\[\therefore x=10\]
Put value of x in equation (5),
\[\begin{align}
  & x-y=8\Rightarrow 10-y=8 \\
 & \therefore y=10-8=2 \\
\end{align}\]
Thus we got x = 10 km/hr and y = 2 km/hr.
\[\therefore \] We got the speed of the boat = x = 10 km/hr.
Speed of the stream = y = 2 km/hr.

Note: Be careful while making the equation, find the condition for upstream and downstream in speed. While going upstream the boat is going against the direction of the stream. So it's taken (x - y). The speed of the boat will be greater than the speed of the stream.