Boats and Stream Part 7

Problem 1: A boat can cover 40 km upstream and 60 km downstream together in 13 hours. Also, it can cover 50 km upstream and 72 km downstream together in 16 hours. What is the speed of the boat in still water?

Solution:

Speed of the boat = x km/h

Speed of the current = y km/h

Downstream=x+y km/h

Upstream =x-y km/h

\frac{40}{x-y}+\frac{60}{x+y}=13             \frac{50}{x-y}+\frac{72}{x+y}=16

Solving this will take much time

Shortcut

Up                      DO                  T

(40                        60                  13)*5

(50                        72                   16)*4

200                      300                 65

200                       288                64

Downstream = 300-288=12km/h

65-64=1

12———>1hr

60———->?\frac{1\times 60}{12}=5hr

72———->?\frac{1\times 72}{12}=6hr

13-5=8

\frac{40}{8}=5

Upstream=5km/h

Speed of the boat still in water=\frac{12+5}{2}=\frac{17}{2}=8.5km/h

Problem 2: A boat can cover 20 km upstream and 32 km downstream together in 3 hours. Also, it can cover 40 km upstream and 48 km downstream together in 5 and half hours. What is the speed of the current?

Solution:

Up                 DO                  T

(20                  32                   3)*2

40                  48                   5\frac{1}{2}

40                 64                     6

40                 48                    5\frac{1}{2}

64-48=16               6 –  5\frac{1}{2}=0.5

16———->0.5

1————–>? 32 is downstream

64———->?\frac{64}{32}=2hr

6-2=4

\frac{40}{4}=10 is upstream

Speed of the current is = \frac{32-10}{2}=\frac{22}{2}=11

Boats and Stream Part 6

Problem 1: A boat takes 26 hours for traveling downstream from point A to point B and coming back to point C midway between A and B. If the velocity of the stream is 4km/h and speed of the boat still water is 10 km/hr, What is the distance between A and B?

Solution:

Speed of the boat=10km/h

Speed of the current=4km/h

Downstream=10+4=14km/h

Upstream=10-4=6km/h

Midway means \frac{1}{2}

The distance between A and B is 2d and the distance between C and B is d

\frac{2d}{14}+\frac{d}{6}=26

\frac{13d}{42}=26

d=84

Total distance=2d=2\times 84=168km

If they asked what is the total distance covered by boat=3d=3*84=252km

Problem 2: A boat takes 5 hours for traveling downstream from point A to point B and coming back to point C at  \frac{3}{4}th of the total distance between A and B from point B. If the velocity of the stream is 3kmph and the speed of the boat in still water is 9kmph what is the distance between A and B?

Solution:

Speed of the boat =9km/h

Speed of the current =3km/h

Downstream=9+3=12km/h

Upstream=9-3=6km/h

\frac{3}{4}th means total distance is 4d and the point C is at 3d distance from B.

\frac{4d}{12}+\frac{3d}{6}=5

\frac{d}{3}+\frac{d}{2}=5

d=6

The distance between A and B is 4d=4*6=24km

Problem 3: A boat travels downstream from point A to B and comes back to point C half distance between A and B is 18 hours. If the boat is still water is 7km/hr and distance AB=80KM, then find the downstream speed?

Solution:

Speed of the boat=7km/h

Speed of the current =x km/h

Downstream=7+x

Upstream=7-x

\frac{80}{7+x}+\frac{40}{7-x}=18

Substitute 3 in the place of X

\frac{80}{10}+\frac{40}{4}=18

18=18

So, x=3.

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 14\times 2}{16}=3.5hours

Average speed of a person traveling on a cycle=\frac{2\times 6\times 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

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 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 x}{3x}=\frac{5}{3}

5x———>?\frac{5\times 5x}{3x}=\frac{25}{3}

Time=\frac{30\times 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 \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 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 60=36

Downstream=36

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

Method 2:

\frac{4.8}{9}\times 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 speed=\frac{2ab}{a+b}

=\frac{2\times 9\times 3}{12\times 2}\times 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