### Boats and Stream

1. Explanation :

Rate in still water=1/2(10+7)km/hr=8.5 km/hr.
Rate of current=1/2(10-7)km/hr=1.5 km/hr.

1. Explanation :

rate downstream = (15/3 ¾)km/hr =(15*4/15)km/hr = 4km/hr.
Rate upstream =(5/2 ½)km/hr =(5*2/5)km/hr =2km/hr.
Speed of current = 1/2(4-2)km/hr = 1km/hr

1. Explanation :

Let man’s rate upstream be x kmph.then ,his rate downstream=3xkmph.
So,2x=18 or x=9.
Rate upstream=9 km/hr,rate downstream=27 km/hr
Hence,rate of stream=1/2(27-9)km/hr=9 km/hr.

1. Explanation :

Speed downstream =(7.5+1.5)km/hr=9 km/hr;
Speed upstream=(7.5-1.5)kmph=6kmph.
Let the required distance be x km.then, x/9+x/6=50/60.
2x+3x=(5/6*18)
5x=15 x=3.
Hence,the required distance is 3km

1. Explanation :

let the speed of the motarboat in still water be x kmph.then, 6/x+2 +6/x-2=33/60
11x2-240x-44=0
11x2-242x+2x-44=0
(x-22)(11x+2)=0
x=22.

1. Explanation :

let rate upstream=x km/hr and rate downstream=y km/hr.
Then,40/x +55/y =13…(i) and 30/x +44/y =10
Multiplying (ii) by 4 and (i) by 3 and subtracting ,we get:11/y=1 or y=11. Substituting y=11 in (i),we get:x=5.
Rate in still water =1/2(11+5)kmph=8kmph.
Rate of current=1/2(11-5)kmph=3kmph