Question

A boat takes \[6\] hours to travel \[8\] km upstream and \[32\] km downstream and it takes \[7\] hours to travel \[20\] km upstream and \[16\]km downstream. Find the speed of the boat in still water and the speed of the stream.
A . 6, 2
B . 6, 1
C . 6, 7
D . 6, 9

Answer 1

Hint: Speed \[ \to \]The rate of change of position of an object in any direction.
Formula of speed:
\[S = \dfrac{d}{t} = \dfrac{{meter}}{{second}}\]
\[ \Rightarrow \]d is the distance traveled in meters.
\[ \Rightarrow \]t is time taken in second.
Downstream/Upstream:
In water, the direction along the stream is called downstream and the direction against the stream is called upstream.
If the speed of a boat in still water is u km/hour and the speed of the stream is v km/hour then
\[Speed\,downstream = (u + v)km/hr\].
\[Speed\,upstream = (u - v)km/hr\].

Complete step by step solution:
\[Time = \dfrac{{Dis\tan ce}}{{speed}}\]
\[6\]hours to travel \[8\]km upstream and \[32\]km downstream,
\[6 = \] time
\[\Delta = 8\]km.
\[Speed = \dfrac{{u - v}}{{u + v}}\]
i.e. \[6 = \dfrac{8}{{u - v}} + \dfrac{{32}}{{u + v}}\] ________(1)
\[7\]hours to travel \[20\]km. upstream and \[16\]km. down stream
i.e. \[7 = \dfrac{{20}}{{u - v}} + \dfrac{{16}}{{u + v}}\] _____________(2)
Let \[\dfrac{1}{{u - v}} = a\] and \[\dfrac{1}{{u + v}} = b\]
\[6 = 8a + 32b\] _______(3)
\[7 = 20a + 16b\]______(4)
To solving the equations, we multiply by \[1\] in equation \[3\] and multiply by \[2\]in equation \[4\]and subtract so we get:
\[\begin{gathered}
  \underline {\begin{array}{*{20}{c}}
  {6 = }&{8a}& + &{32b} \\
  { - 14 = }&{ - 40a}& + &{( - 32b)}
\end{array}} \\
  \underline {\, - 8a\, = \, - 32a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\
\end{gathered} \]
 \[a = \dfrac{8}{{32}}\]
\[a = \dfrac{1}{4}\]
Put the value of a in equation (3) we get:
\[b = \dfrac{1}{8}\]
Now,
\[\dfrac{1}{{u - v}} = \dfrac{1}{4}\] and \[\dfrac{1}{{u + v}} = \dfrac{1}{8}\]
\[ \Rightarrow 4 = u - v\]
\[ \Rightarrow 8 = u + v\]
Add the above two equation we get:
\[12 = 2u\]
\[u = 6\]and
\[v = 2\] \[\left[ {8 = 6 + v \Rightarrow v = 2} \right]\]
So, the speed of the boat is still water \[ = 6\]km/hr.

So, the correct option is (a).

Note: If the speed downstream is a km/hr. and speed upstream is b km/hr. then
Speed in still water \[ = \dfrac{1}{2}\left( {a + b} \right)\] km/hr.