Question

A boat goes 64 km upstream and 72 km downstream in 14 hr. It goes 80 km upstream and 96 km downstream in 18 hrs. Find the speed of the boat in still water and the speed 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 formula of speed. Solve the equations and get the speed of boat and stream.

Complete step-by-step answer:

Let us assume the speed of the boat in still water to be ‘x’.
The speed of the stream can be taken as ‘y’.
It is given that a boat goes 64 km upstream and 72 km downstream in 14 hrs.
We know the formula that, Speed = Distance / Time.
\[\therefore \] Time = Distance / speed.
When the boat goes upstream, it is 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
Similarly 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.
\[\dfrac{64}{x-y}+\dfrac{72}{x+y}=14-(1)\]
i.e. Time taken to go upstream and time taken to go downstream.
Similarly, if boat goes 80 km upstream and 96 km downstream in 18 hrs, then the expression will be formed as
\[\dfrac{80}{x-y}+\dfrac{96}{x+y}=18-(2)\]
Let us put, x – y = a and x + y = b.
Therefore equation (1) becomes, \[\dfrac{64}{a}+\dfrac{72}{b}=14-(3)\].
Equation (2) becomes, \[\dfrac{80}{a}+\dfrac{96}{b}=18-(4)\].
Multiply equation (3) by (4) and equation (4) by (3).
Equation (3) \[\Rightarrow \dfrac{256}{a}+\dfrac{288}{b}=56\].
Equation (4) \[\Rightarrow \dfrac{240}{a}+\dfrac{288}{b}=54\].
Now subtract the equations.
\[\begin{align}
  & \dfrac{256-240}{a}=56-54 \\
 & \therefore 16=2a \\
 & \therefore a=8 \\
\end{align}\]
Now, put a =8 in equation (3) and find b.
\[\begin{align}
  & \dfrac{64}{8}+\dfrac{72}{b}=14 \\
 & \therefore \dfrac{72}{b}=14-8 \\
 & \therefore b=\dfrac{72}{6}=12 \\
\end{align}\]
Thus we got, a = 8 and b = 12.
x – y = 8 and x + y = 12.
Solve them to get values of x and y.
Adding both expressions,


x – y = 8
x + y =12
2x = 20
\[\therefore \] x = 10
Put, x = 10 in x – y = 8.
\[\begin{align}
  & \therefore 10-y=8 \\
 & \therefore y=2 \\
\end{align}\]
Thus, x = 10 km/hr, y = 2 km/hr.
\[\therefore \] Speed of the boat = x = 10 km/hr.
Speed of the stream = y = 2 km/hr.

Note: Be careful when you make the equation. Find conditions for upstream and downstream in speed. While going upstream that 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 water.