## Boats and Stream Part 1

Important formulas

• Speed of boat =x Km/hr
• Speed of current=y km/hr
• Speed of the boat downstream =(x+y) km/hr
• Speed of the boat upstream =(x-y)km/hr
• Speed of boat in still water=$\frac{1}{2}(DownStream+UpStream)$
• Speed of current in still water=$\frac{1}{2}(DownStream-UpStream)$

Problem 1: A boat running downstream covers a distance of 24km in 4hours, While for covering the distance upstream it takes 6 hours what is the speed of the boat in still water?

Solution:

Downstream = $\frac{24}{4}=6km/h$

Upstream = $\frac{24}{6}=4km/h$

Speed of boat in still water =$\frac{1}{2}(DownStream+UpStream)$

=   $\frac{1}{2}(6+4)=\frac{10}{2}=5$

Problem 2: A boat running downstream cover a distance of 30km in 2 hours while coming back the boat takes 6 hours to cover the same distance. If the speed of the current is half that of the boat, What is the speed of the boat in km/h?

Solution:

Downstream = $\frac{30}{2}=15km/h$

Upstream = $\frac{30}{6}=6km/h$

Speed of boat in still water =$\frac{1}{2}(DownStream+UpStream)$

=   $\frac{1}{2}(15+5)=\frac{20}{2}=10$

Problem 3: Ram goes downstream with a boat to some destination and returns upstream to his original place in 6hours. If the speed of the boat in still water and the stream are 12km/hr and 5km/hr respectively, then find the distance of the destination from the starting position?

Solution:

Downstream =12+5=17km/h

Upstream = 12-5=7km/h

Distance=Speed*Time

Average speed = $\frac{2xy}{x+y}$  =  $\frac{2\times&space;17\times&space;7}{24}$

d = $\frac{2\times&space;17\times&space;7}{24}$$\times&space;\frac{6}{2}$ = $\frac{119}{4}=29.5km$

## Boats and Stream Part 2

Problem 1: A boat travels downstream for 14km and upstream for 9km. If the boat took total of 5 hours for its journey. What is the speed of the river flow if the speed of the boat in still water is 5 km/hr?

Solution:

Speed of boat=5km/hr

Speed of current = x

Downstream = (5+x)km/h

Upstream=(5-x)km/r

$\frac{4}{5+x}+\frac{9}{5-x}=5$

x=3km/hr

Problem 2: A boat goes 6 km an hour still water, it takes thrice as much time in going the same distance against the current comparison to the direction of the current?

Solution:

Speed of the boat=6km/h

up                       down

T       3                              1

S        1                              3

1-3=2

2———>6km/h

1———–>?$\frac{6\times&space;1}{2}=3$

Problem 3: A man can row 6 km/h in still water. If the speed of the current is 2km/hr, it takes 4 hours more in upstream than in downstream for the same distance?

Solution:

Speed of the boat=6km/h

Speed of the current=2km/h

Downstream=8km/h

Upstream = 4km/hr

$\frac{d}{4}-\frac{d}{8}=4$

$\frac{d}{8}=4$

$d=32$

method 2:

upstream                         downstream

S       4                                            8          ————->           1                 :                2

T       8                                            4         —————>          2                 :                1

2-1=4

1———>4

8———->?$\frac{4\times&space;8}{1}$=32

Problem 4: A steamer goes downstream from one port to another in 4 hours. It covers the same distance upstream in 5 hours. If the speed of the stream is 2km/hr, the distance between the two ports is?

Solution :

Speed of the stream = 2km/h

Speed of the boat=x km/hr

Downstream=x+2 km/h

upstream=x-2 km/h

d1=d2

(x+2)*4=(x-2)*5

4x+8=5x-10

x=18

Speed of boat=18 km/h

Distance = (x+2)*4 = (18+2)*4 = 20*4 = 80

Method 2:

Downstream                       Upstream

T ——->  4                  :                   5

S———>5                   :                   4

Speed of the boat = 5+4=9

=$\frac{9}{2}=4.5$

Speed of the current = 5-4=1

=$\frac{1}{2}=0.5$

0.5———->2

4.5———->?$\frac{2\times&space;45}{5}=18$

Downstream=18+2=20

upstream=18-216

Distance=20*4=80

Distance=16*5=80

## Boats and Stream Part 3

Problem 1: A boat running at the speed of 34 kmph downstream covers a distance of 4.8km in 8 minutes. The same boat while running upstream at the same speed covers the same distance in 9 minutes?

Solution:

Speed of the boat=34km/h

Downstream = 4.8km

$Speed=\frac{Distance}{Time}$

$\frac{4.8}{8}\times&space;60=36$

Downstream=36

Speed of the current=36-34=2km/h

Method 2:

$\frac{4.8}{9}\times&space;60=32$

Upstream=32

34-y=32

y=2

Speed of the current is 2

Problem 2: The speed of the boat in still water is 6km/h and that of the current is 3km/h. The boat starts from point A and rows to the point B and comes back to the point A. It takes 12 hours during this journey. How far is point A from point B?

Solution:

Speed of the boat=6km/hr

Speed of the current=3km/h

Upstream =6+3=9km/h

Downstream 6-3=3km/h

$\frac{d}{9}+\frac{d}{3}=12$

$\frac{4d}{9}=12$

$d=27$

Method 2:

$Average&space;speed=\frac{2ab}{a+b}$

$=\frac{2\times&space;9\times&space;3}{12\times&space;2}\times&space;12=27$

Problem 3: There are two places A and B which are separated by a distance of 100km. Two boats start from both the places at the same time towards each other. If one boat is going downstream then the other one is going upstream, If the speed of A and B is 12km/h and 13km/h. respectively. Find at how much time will they meet each other?

Solution:

Speed of boat A=12kmph

Speed of boat B=13kmph

Speed of the current=x km/h

Downstream=12+x

upstream=12-x

$Time=\frac{Distance}{Speed}$

$T=\frac{100}{12+x+13-x}=\frac{100}{25}=4$

## Boats and Stream Part 4

Problem 1: A ship sails 30km of the river towards upstream in 6 hours. How long will it take to cover the same distance downstream? If the speed of the current speed is $\frac{1}{4}$ th of the speed of the boat in still water?

Solution:

Upstream=$\frac{30}{6}=5km/h$

C=$\frac{1}{4}\times&space;B$

$\frac{C}{B}=\frac{1}{4}$

B=4X   ,   C=X

Downstream=4x+x=5x

Upstream=4x-x=3x

3x——>5

x———>?$\frac{5\times&space;x}{3x}=\frac{5}{3}$

5x———>?$\frac{5\times&space;5x}{3x}=\frac{25}{3}$

Time=$\frac{30\times&space;3}{25}=\frac{18}{5}=3.6hr$

Problem 2: The speed of the boat in still water is 17.5 km/h and that of current is 2.5km/h. The boat goes from X to Y in downstream and returns to point Z. The whole journey takes 429 minutes. The distance between Z and Y is  $\frac{2}{5}th$ of the distance between X and Y. Find the total distance covered by the boat?

Solution:

Speed of boat=17.5km/h

Speed of current=2.5km/h

Downstream=17.5+2.5=20

Upstream=17.5-2.5=15

$\frac{5x}{20}+\frac{2x}{15}=\frac{429}{60}$

$\frac{15x+8x}{60}=\frac{429}{60}$

$23x=429$

$x=\frac{429}{23}$

5x+2x=7x

$7\times&space;\frac{429}{23}=125$ approximately

Problem 3: Speed of the boat in standing water is 12km/h and speed of the stream is 3km/h. A man rows to a place at a distance of 6300 km and comes back to the starting point. The total time taken by him is?

Solution:

Speed of the boat=12km/h

Speed of the current=3km/h

Downstream=12+3=15km/h

Upstream=12-3=9km/h

$\frac{6300}{15}+\frac{6300}{9}$

420+700=1120=$\frac{1120}{60}=18hours$

## Boats and Stream Part 5

Problem 1: There is a road beside a river. Two friends started from place A, moved to a temple situated at another place B and then returned to A again. One of them moves on a cycle at a speed of 6km/h, While the other sails on a boat at a speed of 8km/h. If the river flows at a speed of 6km/h, Which of the two friends will return to place A first?

Solution:

Speed of the boat=8km/h

speed of the current=6km/h

Downstream=8+6=14km/h

Upstream=8-6=2km/h

Average speed=$\frac{2ab}{a+b}=\frac{2\times&space;14\times&space;2}{16}=3.5hours$

Average speed of a person traveling on a cycle=$\frac{2\times&space;6\times&space;6}{12}=6hours$

Therefore the person traveling on the boat will reach first.

Problem 2: Four times the downstream speed is 8 more than the 15 times the upstream speed. If the difference between downstream and upstream speed is 24km/h, then what is the ratio of speed in still water to the speed of the current?

Solution:

4*downstrean=15*upstream+8

4d-15u=8———->equ 1

d-u=24

d=24+u

Substitute d value in equation 1

4(24+u)-15u=8

96+4u-15u=8

11u=88

u=8

d = 24+u = 24+8 =32

Speed of current in still water =$\frac{32-8}{2}=12$

Speed of boat in still water=$\frac{32+8}{2}=\frac{40}{2}=20$

20   :   12 = 5  :  3

Problem 3: A boat running upstream takes 8 hours 48min to cover a certain distance, while it takes 4 hours to cover the same distance running the downstream. What is the ratio between the speed of the boat and speed of the water current respectively?

Solution:

Upstream                                       Downstream

T—>8h 48min              :                      4

$8\frac{4}{5}$                           :                     4

$\frac{44}{5}$                            :                     4

44                            :                      20

11                            :                      5

S—>  5                               :                     11

X+Y=11

X-Y=5

2X=16             8-Y=5

X=8                  Y=3

Speed of boat x = 8kmph

Speed of current y = 3kmph