Q. Time taken by boat to travel (K + 175) km upstream is 50% more than time taken by boat to travel (K + 250) km downstream. Speed of the boat in still water is four times the speed of the stream.
Find which of the following statement(s) is/are true.
I. Time taken by boat to travel (2K – 80) km downstream is 184/3 hours. Speed of stream is 3 km/h
II. Time taken by boat to travel (K – 140) km upstream is twice of time taken by boat to travel (K/2 + 50) km downstream
III. Unit digit of (K/4 + 4)99 is 9
A. II only
B. III only
C. I and II only
D. II and III only
E. None of these
Ans: None of these
Sol:
Let speed of stream = a
So, speed of stream in still water = 4 x a = 4a
Now,
[(K + 175) / (4a – a)] / [(K + 250) / (4a + a) = 3/2
10K + 1750 = 9K + 2250
Value of K = 500
I. Time taken by boat to travel (2K – 80) km
downstream is 184/3 hours. Speed of stream is
3 km/h
Downstream speed of boat = 3 + 3 x 4 = 15 km/h
Distance to travel = 2 x 1000 – 80 = 920 km
Required time = 920/15 = 184/3 hours
This statement is true
II. Time taken by boat to travel (K – 140) km
upstream is twice of time taken by boat to travel
(K/2 + 50) km downstream
Respective ratio of upstream distance and
Downstream distance travelled = (500 – 140):
(500/2 + 50) = 6:5
Ratio of upstream speed and Downstream
speed respectively = (4 – 1): (4 + 1) = 3:5
Ratio of upstream time and Downstream time =
6/3: 5/5 = 2:1
This statement is true
III. Unit digit of (K/4 + 4)99 is 9
(K/4 + 4) = 129
91 = 9
92 = 1
93 = 9
So, odd power of 9 gives 9 as a unit digit
So, required unit digit = 9
This statement is true
All statement(s) is/are true
Hence the answer is option E