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A boat running upstream takes 8 hours 48 min to cover a certain distance, while it takes 4 hours to cover the same distance downstream. What is the ratio between the speed of the boat and the speed of the water current respectively?
(a) 2:1
(b) 3:1
(c) 8:3
(d) 4:3

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: First, before proceeding for this, we need to suppose the speed of the boat in upstream be x km/hr and speed for the downstream as y km/hr. Then, the condition given in the question clearly says that the distance covered for upstream in 8 hrs 48 min is equal to the distance covered for downstream in 4 hrs. Then, now we should be aware of the formulas for the speed of the boat and speed of the water stream as $\dfrac{x+y}{2}$and $\dfrac{y-x}{2}$respectively. Then, by using the ratio calculated above, we get the ratio between the speed of the boat and the speed of the water current.

Complete step-by-step answer:
In this question, we are supposed to find the ratio between the speed of the boat and the speed of the water current respectively when a boat running upstream takes 8 hours 48 min to cover a certain distance, while it take 4 hours to cover the same distance downstream.
So, before proceeding for this, we need to suppose the speed of the boat in upstream be x km/hr and speed for the downstream as y km/hr.
Now, the condition given in the question clearly says that the distance covered for upstream in 8 hrs 48 min is equal to the distance covered for downstream in 4 hrs.
Now, by using the above condition we get the equation as multiplication of speed for downstream to its time and multiplication of upstream speed with its time as:
$x\times 8\dfrac{48}{60}=y\times 4$
Then, we will solve the above expression to get the value of the ratio x:y as:
$\begin{align}
  & 8\dfrac{4}{5}x=4y \\
 & \Rightarrow \dfrac{44}{5}x=4y \\
 & \Rightarrow \dfrac{x}{y}=\dfrac{4\times 5}{44} \\
 & \Rightarrow \dfrac{x}{y}=\dfrac{5}{11} \\
\end{align}$
So, now we should be aware of the formulas for the speed of the boat and speed of the water stream as:
Then, speed of the boat is given by $\dfrac{x+y}{2}$.
Then, speed of the water stream is given by $\dfrac{y-x}{2}$.
So, now by dividing the speed of the boat with the speed of the water current to get the ratio as:
$\dfrac{\dfrac{x+y}{2}}{\dfrac{y-x}{2}}=\dfrac{x+y}{y-x}$
Now, divide the numerator and denominator by y to get the required ratio as:
$\dfrac{\dfrac{x}{y}+1}{1-\dfrac{x}{y}}$
Then, by substituting the value of $\dfrac{x}{y}$as $\dfrac{5}{11}$ calculated above to get the ratio as:
\[\begin{align}
  & \dfrac{\dfrac{5}{11}+1}{1-\dfrac{5}{11}}=\dfrac{\dfrac{5+11}{11}}{\dfrac{11-5}{11}} \\
 & \Rightarrow \dfrac{16}{6} \\
 & \Rightarrow \dfrac{8}{3} \\
\end{align}\]
So, we get the ratio between the speed of the boat and the speed of the water current respectively as 8:3.
Hence, option (c) is correct.

Note: Now, to solve these types of the questions we need to know some of the basic formulas for the time, speed and distance. So, we must know the relation between speed(s), time (t) and distance (d) as:
$s=\dfrac{d}{t}$