Question

A boat covers 16 km upstream and 24 km downstream in 6 hours while it covers 12 km upstream and 36 km downstream in the same time. What is the speed of the boat upstream and downstream?

Answer 1

Hint: Focus on the point that while a boat covers downstream, the speed of the current is added to the boat while in case of the upstream, speed of the current is subtracted from that of the boat.

Complete step-by-step answer:
Let’s start with what is speed. Speed is a scalar quantity defined as the distance travelled by a particle or object per unit time.
Generally, we deal with two kinds of speeds. One is instantaneous, and the other is the average speed. For uniform motion, both are identical.
Average speed is defined as the total distance covered by a body divided by the time taken by the body to cover it.
$\therefore {{v}_{avg}}=\dfrac{\text{distance covered}}{\text{time taken}}$
$\Rightarrow \text{time taken}=\dfrac{\text{distance covered}}{{{v}_{avg}}}$

Now, starting with the solution to the above question. Let the original average speed of the boat be x Km/hr, and the speed of the current be y km/hr.
It is given in the question that when a boat covers 16 km upstream and 24 km downstream it takes a total of 6 hours to complete the journey. We know $\text{time taken}=\dfrac{\text{distance covered}}{{{v}_{avg}}}$ , so, we get
$\text{time taken}=\dfrac{\text{distance covered downstream }}{{{v}_{avg}}+{{v}_{current}}}+\dfrac{\text{distance covered upstream }}{{{v}_{avg}}-{{v}_{current}}}$
$\Rightarrow 6=\dfrac{\text{24 }}{x+y}+\dfrac{\text{16 }}{x-y}$
$\Rightarrow 6=\dfrac{24\left( x-y \right)\text{+16}\left( x+y \right)\text{ }}{{{x}^{2}}+{{y}^{2}}}$
$\Rightarrow 6\left( {{x}^{2}}-{{y}^{2}} \right)=40x-8y..........(i)$
It is also given in the question that when a boat covers 12 km upstream and 36 km downstream it takes a total of 6 hours to complete the journey.
$\text{time taken}=\dfrac{\text{distance covered downstream }}{{{v}_{avg}}+{{v}_{current}}}+\dfrac{\text{distance covered upstream }}{{{v}_{avg}}-{{v}_{current}}}$
$\Rightarrow 6=\dfrac{36}{x+y}+\dfrac{\text{12 }}{x-y}$
$\Rightarrow 6=\dfrac{\text{36}\left( x-y \right)\text{+12}\left( x+y \right)\text{ }}{{{x}^{2}}+{{y}^{2}}}$
$\Rightarrow 6\left( {{x}^{2}}-{{y}^{2}} \right)=48x-24y..........(ii)$
Now if we divide equation (ii) by equation (i), we get
$1=\dfrac{48x-24y}{40x-8y}$
$\Rightarrow 1=\dfrac{6x-3y}{5x-y}$
On cross-multiplication, we get
$5x-y=6x-3y$
$\Rightarrow x=2y$
Now we will substitute the value of x in equation (i). So, we get
$6\left( 4{{y}^{2}}-{{y}^{2}} \right)=80y-8y$
\[\Rightarrow 18{{y}^{2}}=72y\]
And we know the current speed cannot be zero. So, we have
\[\Rightarrow y=\dfrac{72}{18}=4\text{ }Km/hr\]
And we know that x is twice of y. Therefore, the speed of the boat is 8 km/hr.
Therefore, the speed of the boat upstream and downstream is 4 km/hr and 12 km/hr respectively..

Note: Always try to keep the quantities according to a standardised unit system, this helps us to solve the question in an error-free manner. Also, it is prescribed to write each and every statement given in the question in mathematical form as it ensures that we are not missing any information given in the question.