Question

A boat travels with a speed of \[15{\rm{km}}/{\rm{hr}}\] in still water. In a river flowing at \[5{\rm{km}}/{\rm{hr}}\], the boat travels some distance downstream and then returns. The ratio of average speed to the speed in still water is
A) \[8:3\]
B) \[3:8\]
C) \[8:9\]
D) \[9:8\]

Answer 1

Hint:
We will first assume the distance covered to be any variable and then we will find the speed of the boat while moving downstream. From there, we will calculate the time taken in the downstream. We will find the speed of the boat while moving upstream. From there, we will calculate the time taken in the upstream. Then we will calculate the value of the average speed of the boat using this and hence, we will find the ratio of the average speed to the speed in the still water.

Formula used:
We will use the formula \[{\rm{time}} = \dfrac{{{\rm{distance}}}}{{{\rm{speed}}}}\] to solve the question.

Complete Step by Step Solution:
Let \[d\] be the distance by the boat.
It is given that:
The speed of the boat in still water is \[15{\rm{km}}/{\rm{hr}}\] and the speed of the river is \[5{\rm{km}}/{\rm{hr}}\].
Speed of the boat while moving downstream is equal to the sum of speed of the boat and the speed of the river.
Speed of the boat while moving downstream \[ = 15 + 5 = 20{\rm{km}}/{\rm{hr}}\]
Now, we will substitute the value of the distance travel and speed in the formula \[{\rm{time}} = \dfrac{{{\rm{distance}}}}{{{\rm{speed}}}}\] to calculate the time taken in the downstream. Therefore, we get
Time taken in downstream \[ = \dfrac{d}{{20}}\] hours
Speed of the boat while moving upstream is equal to the difference of speed of the boat and the speed of the river.
Speed of the boat while moving upstream \[ = 15 - 5 = 10km/hr\]
Now, we will substitute the value of the distance travel and speed in the formula \[{\rm{time}} = \dfrac{{{\rm{distance}}}}{{{\rm{speed}}}}\] to calculate the time taken in the upstream. Therefore, we get
Time taken in upstream \[ = \dfrac{d}{{10}}\] hours
Now, we will calculate the average speed which is equal to the ratio of the total distance covered to the total time taken.
Hence, average speed \[ = \dfrac{{d + d}}{{\dfrac{d}{{20}} + \dfrac{d}{{10}}}}\]
Now, adding the terms of the numerator and denominator, we get
\[ \Rightarrow \] Average speed \[ = \dfrac{{2d}}{{\dfrac{{d + 2d}}{{20}}}} = \dfrac{{2d \times 20}}{{3d}}\]
On further simplification, we get
 \[ \Rightarrow \] Average speed \[ = \dfrac{{40}}{3}\]
We know the speed of the boat in still water is equal to \[15{\rm{km}}/{\rm{hr}}\].
Now, we will calculate the ratio of the average speed to the speed of the boat in still water.
Therefore,
\[ \Rightarrow \] Average speed \[ \div \] Speed in still water \[ = \dfrac{{\dfrac{{40}}{3}}}{{15}}\]
On further simplification, we get
\[ \Rightarrow \] Average speed \[ \div \] Speed in still water \[ = \dfrac{8}{{3 \times 3}}\]
On multiplying the terms, we get
\[ \Rightarrow \] Average speed \[ \div \] Speed in still water \[ = \dfrac{8}{9}\]

Hence, the correct option is option C.

Note:
Here, we need to keep in mind that the distance covered by any object is always equal to the product of the speed of that object and the time taken by that object to complete that distance. Also, the average speed of any object is equal to the ratio of the total distance covered to total time taken by that object to cover total distance.