Two boats A and B are stationed at two points Q and R on a flowing river. The direction of flow of river is from Q to R. When the boats A and B moved towards each other they met at P, Which is at a distance of 20 m from the point R. When A moves towards B and B moves away from A they met at S which is at a distance of 40 m from the point R. Boat B’s speed in still water is 5 times the speed at which river is flowing. Find the ratio of the speed of boat A to boat B
Solution
Correct Answer: Option D
∵ Speed × Time = Distance
Let the distance between point Q and P is x m
Let the speed of boat A and boat B and river is a, b and r m/s respectively.
Therefore,
We can say from the question that when the boats A and B moved towards each other they met at P, Which is at a distance of 20 m from the point R. So,
\(\Rightarrow \frac{{x}}{{a\; + \;r}}\; = \;\frac{{20}}{{b\;-\;r}}{\rm{\;}}\) - - - - - - - - - - - - - - - - - - - - - - - eqn (1)
We can also say that, when A moves towards B and B moves away from A they met at S which is at a distance of 40 m from the point R. hence similarly,
\(\Rightarrow \frac{{x\; + \;60}}{{a\; + \;r}}\; = \;\frac{{40}}{{b\; + \;r}}{\rm{\;}}\) - - - - - - - - - - - - - - - - - eqn (2)
Dividing eqn (1) by eqn (2) we get,
\(\Rightarrow \frac{{2x}}{{x + 60}}\; = \;\frac{{b+r}}{{b-r}}{\rm{\;}}\) - - - - - - - - - - - - - - - - - - eqn (3)
It is given in the question that Boat B’s speed in still water is 5 times the speed at which river is flowing
⇒ b = 5r
Substituting in eqn 3,
\(\frac{{2x}}{{x\; + \;60}} = \frac{{b\; + \;r}}{{b\; - \;r}}\)
\(\Rightarrow \frac{{2x}}{{x + 60}}\; = \;\frac{{5r+r}}{{5r-r}}{\rm{\;}}\)
\(\Rightarrow \frac{{2x}}{{x\; + \;60}} = 1.5\)
⇒ 2x = 1.5x + 90
⇒ 0.5 x = 90
⇒ x = 180 m
Distance between the points Q and R = 180 + 20 = 200 m
Putting value of x = 180 in eqn 1,
\(\Rightarrow \frac{{x}}{{a\; + \;r}}\; = \;\frac{{20}}{{b\;-\;r}}{\rm{\;}}\)
\(\Rightarrow \frac{{180}}{{20}} = \frac{{a\; + \;r}}{{b\; - \;r}}\)
\(\Rightarrow \frac{{a+r}}{{b-r}}=9\)
\(\Rightarrow \frac{{a\; + \;r}}{{5r\; - \;r}} = 9\)
\(\Rightarrow \frac{{a+r}}{{4r}} = 9\)
⇒ a = 35r
Ratio of speed of boat A : boat B = 35r : 5 r
= 7:1
Practice More Questions on Our App!
Download our app for free and access thousands of MCQ questions with detailed solutions