Question

The 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 stream and that of the boating till water.

Answer 1

Hint – Here we will proceed by using the formulas Downstream speed :$a = u + v$ , upstream speed: $b = u - v$. Then the speed of boat in still water $u = \dfrac{1}{2}\left( {a + b} \right)$, the speed of the boat in still water: $v = \dfrac{1}{2}\left( {a - b} \right)$.

Complete step-by-step solution -
Let the speed of the boat in still water = $x$ km per hour.
And, the speed of stream $ = y$ km per hour.
Speed of boat at downstream
$ \Rightarrow \left( {x + y} \right)$ km per hour.
Speed of boat at upstream
$ \Rightarrow \left( {x - y} \right)$ km per hour.
$\because $ $time = \dfrac{{dis\tan ce}}{{speed}}$
Time taken to cover 30 km upstream $ \Rightarrow \dfrac{{30}}{{x - y}}$
Time taken to cover 44 km downstream $ \Rightarrow \dfrac{{44}}{{x + y}}$
According to the first condition,
$ \Rightarrow \dfrac{{30}}{{x - y}} + \dfrac{{44}}{{x + y}} = 10$
Time taken to cover 40 km upstream $ \Rightarrow \dfrac{{40}}{{x - y}}$
Time taken to cover 55 km downstream $ \Rightarrow \dfrac{{55}}{{x + y}}$
According to the second condition,
$ \Rightarrow \dfrac{{40}}{{x - y}} + \dfrac{{55}}{{x + y}} = 13$
Let $\dfrac{1}{{x - y}} = u$ and $\dfrac{1}{{x + y}} = v$
$ \Rightarrow 30u + 44v = 10$ …. (1)
$ \Rightarrow 40u + 55v = 13$ …. (2)
Multiply (1) by 5 and (2) by 4 and subtract both
$ \Rightarrow \left( {150u + 220v = 50} \right) - \left( {160u + 220v = 52} \right)$
$
   \Rightarrow - 10u = - 2 \\
   \Rightarrow u = \dfrac{1}{5} \\
 $
Put $u = \dfrac{1}{5}$ in equation (1)
$
   \Rightarrow 30 \times \dfrac{1}{5} + 44v = 10 \\
   \Rightarrow 44v = 4 \\
   \Rightarrow v = \dfrac{1}{11} \\
 $
$ \Rightarrow u = \dfrac{1}{{x - y}} = \dfrac{1}{5}$
$ \Rightarrow x - y = 5$ …. (3)
$ \Rightarrow v = \dfrac{1}{{x + y}} = \dfrac{1}{{11}}$
$ \Rightarrow x + y = 11$ …. (4)
Adding equation (3) and equation (4), we get
$ \Rightarrow x - y = 5$ + ($x + y = 11$)
$ \Rightarrow x = 8$
Put $x = 8$ in (3)
$
  x - y = 5 \\
   \Rightarrow 8 - 5 = y \\
   \Rightarrow 3 = y \\
 $
Hence, the speed of the boat in still water = 8 km per hour and the speed of stream = 3 km per hour.

Note – Whenever we come up with this type of problem, one must know that when an object moves downstream in a river, the river current supports the object’s movement and hence the net speed of the object is the sum of its speed in still water and the speed of the river current. On the other hand, the current acts against the object’s movement if the object is travelling upstream, hence the net speed of the object is still water subtracted by the speed of the river current.