Question

A boat travels with a speed of \[15{\rm{km/hr}}\] in still water. In a river flowing at \[5{\rm{km/hr}}\]. 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: Here, we will find the ratio of average speed to the speed in still water. First, we will find the speed of the upstream and the speed of the downstream to find the time taken by the boat in both the streams. Then we will use the average speed formula to find the average speed of the boat. We will then find the ratio of the average speed to the speed in still water. A ratio is defined as the comparison between two quantities with the same dimensions.

Formula Used:
 We will use the following formula:
1.Speed of the Upstream is given by the formula \[\left( {u - v} \right)\]
2.Speed of the Downstream is given by the formula \[\left( {u + v} \right)\] where \[u,v\] are the speed of the boat in still water, speed of the stream respectively.
3.Speed is given by the formula \[{\rm{Speed}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Time}}}}\]
4.Average speed is given by the formula Average speed \[ = \] Total distance travelled \[ \div \] total time taken.

Complete step-by-step answer:
Let \[u\] be the speed of the boat in still water and \[v\] be the speed of the stream.
We are given that the speed of the boat in still water is \[15{\rm{km/hr}}\] and the speed of the boat in stream is \[5{\rm{km/hr}}\]
So, we get \[u = 15{\rm{km/hr}}\] and \[v = 5{\rm{km/hr}}\]
Now, we will find the speed of the upstream
Speed of the Upstream is given by the formula \[\left( {u - v} \right){\rm{km/hr}}\]
 Speed of the upstream \[ = \left( {15 - 5} \right){\rm{km/hr}}\]
Subtracting the terms, we get
\[ \Rightarrow \] Speed of the upstream\[ = 10{\rm{km/hr}}\]
Now, we will find the speed of the downstream
Speed of the Downstream is given by the formula \[\left( {u + v} \right){\rm{km/hr}}\]
Speed of the downstream\[ = \left( {15 + 5} \right){\rm{km/hr}}\]
Subtracting the terms, we get
\[ \Rightarrow \] Speed of the downstream \[ = 20{\rm{km/hr}}\]
Now, we will be using the speed formula to find the time taken by the boat
Let \[d\] be the distance travelled by the boat.
Using the formula \[{\rm{Speed}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Time}}}}\] or \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\], we get
\[ \Rightarrow \] Time taken by the boat in upstream \[ = \dfrac{d}{{10}}\]
\[ \Rightarrow \] Time taken by the boat in downstream \[ = \dfrac{d}{{20}}\]
Thus the total distance travelled \[ = d + d = 2d\]
Thus the total time taken \[ = \dfrac{d}{{10}} + \dfrac{d}{{20}}\]
Now, we will find the average speed of the boat
Average speed is given by the formula Average speed \[ = \] Total distance travelled \[ \div \] total time taken.
\[ \Rightarrow \] Average Speed \[ = \dfrac{{2d}}{{\dfrac{d}{{10}} + \dfrac{d}{{20}}}}\]
By taking LCM, we get
\[ \Rightarrow \] Average Speed \[ = \dfrac{{2d}}{{\dfrac{{2d}}{{20}} + \dfrac{d}{{20}}}}\]
\[ \Rightarrow \] Average Speed \[ = \dfrac{{2d}}{{\dfrac{{3d}}{{20}}}}\]
\[ \Rightarrow \] Average Speed \[ = \dfrac{{2\left( {20} \right)}}{3}\]
Multiplying the terms in the denominator, we get
\[ \Rightarrow \] Average Speed \[ = \dfrac{{40}}{3}{\rm{km/hr}}\]
Now, we will find the ratio of average speed to the speed in still water. Therefore, we get
\[{\rm{Required \,ratio}} = \dfrac{{\dfrac{{40}}{3}}}{{15}}\]
\[ \Rightarrow {\rm{Required\, ratio}} = \dfrac{{40}}{{3\left( {15} \right)}}\]
Multiplying the terms, we get
\[ \Rightarrow {\rm{Required\, ratio}} = \dfrac{{40}}{{45}}\]
By dividing, we get
\[ \Rightarrow {\rm{Required\, ratio}} = \dfrac{8}{9}\]
Therefore, the ratio of average speed to the speed in still water is \[8:9\]. Thus Option(C) is the correct answer.

Note: We should know that speed of the stream is the rate at which the water flows. If the boat is the same as in the direction of the stream, then it is said to be upstream and if the boat is in the opposite direction of the stream, then it is said to be downstream. We should also know that the distance travelled remains the same in the upstream irrespective of the direction of the stream. While finding the ratio, we should note that both the quantities are in the same dimensions.