If a is rate downstream and b is rate upstream

Rate in still water = $\frac{1}{2}(a+b)$

Rate of current = $\frac{1}{2}(a-b)$

=> Rate in still water = $\frac{1}{2}(20+10)$ = 15 km/h

=> Rate of current = $\frac{1}{2}(20-10)$ = 5 km/h

Let speed upstream is x km/h.

Then, speed downstream = 3x km/h.

Speed in still water = $\frac{1}{2}(3x+4)$ km/h = 2x km/h.

2x = $\frac{28}{3}x=\frac{14}{3}$

Speed of upstream = $\frac{14}{3}$ km/hr,  Speed of downstream 14 km/hr.

speed of the current = $\left(\frac{1}{2}\right)\left[14-\frac{14}{3}\right]=\frac{14}{3}$ and $\frac{2}{3}$ km/h

Let the distance between A and B be x km.

Speed downstream = 6 km/h, speed upstream = 2 km/h.

$\frac{x}{6}+\frac{x}{2}$ = 4

$\Rightarrow \frac{2x}{3}$ = 4

$\Rightarrow$ x = 6

Distance AB = 6 km.

Let rate upstream = x km/h and rate downstream = y km/h.

Then, $\frac{24}{x}+\frac{36}{y}=36$   ......... (i)

and $\frac{36}{x}+\frac{24}{y}=\frac{13}{2}$  ......... (ii)

Adding (i) and (ii), we get :

$60\left(\frac{1}{x}+\frac{1}{y}\right)=\frac{25}{2}$  or  $\left(\frac{1}{x}+\frac{1}{y}\right)=\frac{5}{24}$  ........ (iii)

Subtracting (i) from (ii), we get :

$12\left[\left(\frac{1}{x}\right)-\left(\frac{1}{y}\right)\right]=\frac{1}{2}$  or $\left(\frac{1}{x}\right)-\left(\frac{1}{y}\right)=\frac{1}{24}$    ....... (iv)

Adding (iii) and (iv), we get :

= $\frac{2}{x}=\frac{6}{24}$ or x = 8

so  ($\left(\frac{1}{8}\right)+\left(\frac{1}{y}\right)=\frac{1}{24}$

=> $\frac{1}{y}=\frac{5}{24}-\frac{1}{8}=\frac{1}{12}$

$\Rightarrow$ y = 12

Speed upstream = 8 km/h,  Speed downstream = 12 km/h.

Hence, rate of current = $\frac{1}{2}(12-8)$ km/h = 2 km/h

First of all, we know that
speed of current = 1/2(speed downstream - speed upstream)

So we need to calculate speed downstream and speed upstream first.

Speed = Distance / Time [important]

Speed Upstream = $\frac{15}{3\frac{3}{4}}$

$15\times \frac{4}{15}$ = 4 km/hr

Speed  Downstream= $\frac{5}{2\frac{1}{2}}$

$5\times \frac{2}{5}$ =  2 km/hr

So speed of current = $\frac{1}{2}$  ( 4-2  ) = 1 km/hr

speed upstream = $\frac{13}{5}$ km/h

speed downstream $\frac{28}{5}$ km/h

Velocity of stream = $\frac{1}{2}\left [\left(\frac{28}{5}\right)-\left(\frac{13}{5}\right)\right]$ = 1.5 km/h

Rate upstream = $\frac{750}{675}=\frac{10}{9}$ m/sec

Rate downstream $\frac{750}{450}=\frac{5}{3}$ m/sec.

Rate in still water = ($\frac{1}{2}\left [ \left (\frac{10}{9}\right)+\left(\frac{5}{3}\right)\right]=\frac{25}{18}$ m/sec

= $\frac{25}{18}\times \frac{18}{5}$ km/h = 5 km/h

Rate upstream =$\left(\frac{7}{42}\right)\times 60$ km/h = 10 km/h. Speed of stream = 3 km/h.

Let speed in still water is x km/hr

Then, speed upstream = (x —3) km/hr.

x - 3 = 10 or  x = 13 km/h.

Speed downstream (10 + 6) km/hr 16 km/hr.

Time taken to cover 80 km downstream = $\frac{80}{16}$ hrs = 5 Hrs.

Let speed downstream = x km/h

Then Speed upstream = 2x km/h

So ratio will be,

$\frac{2x+x}{2}:\frac{2x-x}{2}$

$\frac{3}{2}:\frac{x}{2}=3:1$

If you solved this question yourself, then trust me you have a all very clear with the basics of this chapter.

If not then lets solve this together.

Speed with current is 20,

speed of the man + It is speed of the current

Speed in still water = 20 - 3 = 17

Now speed against the current will be

speed of the man - speed of the current

= 17 - 3 = 14 km/h

It is very important to check, if the boat speed given is in still water or with water or against water. Because if we neglect it we will not reach on right answer. I just mentioned here because mostly mistakes in this chapter are of this kind only.

Lets see the question now.
Speed downstream = (16 + 5) = 21 km/h

Time = $\frac{Distance}{SPeed}=\frac{84}{21}$ = 4 hours

Let the distance is x km

Rate downstream = 5 + 1 = 6 km/h

Rate upstream = 5 - 1 = 4 km/h

$\frac{x}{6}+\frac{x}{4}=1$ [$\because \frac{Distance}{Speed}$ = time]

=> 2x + 3x = 12

=> x = $\frac{12}{5}$ = 2.4 km

Speed in still water = (1/2) * (6 + 2) km/hr = 4 km/hr

Speed of stream = $\frac{1}{2}$   (13-8) kmph = 2.5 kmph

Let the speed of the current be ‘x’ kmph

$\therefore$ Speed down stream =$\frac{distance}{time}$  $\Rightarrow$  12 + x = 45 + 3

$\therefore$  x = 3 kmph

effective speed up stream = 12 – 3 = 9

9 =$\frac{45}{x}$    $\Rightarrow$  x = 5 hrs

Speed of the boat upstream = 13 x 60/39 = 20 km/hr

Speed of the stream = 3 km/hr

Let the speed of the boat in still water = x km/hr

We have, x - 3 = 20

x = 20 + 3 = 23 km/hr

We know we can calculate it by 1/2(a+b)

=> 1/2(11+5) = 1/2(16) = 8 km/hr

Rate downstream =$\left ( \frac{16}{2} \right )km/h=8km/h$

Rate upstream =$\left ( \frac{16}{4} \right )km/h=4km/h$

Speed in still water = $\frac{1}{2}\left ( 8+4 \right )km/h=6 \, km/h$

Let the speed of the stream be x km/hr. Then,

Speed downstream = (15 + x) km/hr,

Speed upstream = (15 - x) km/hr.

$\therefore \frac{30}{15+x}+\frac{30}{15-x}=4\frac{1}{2}$

$\Rightarrow \frac{900}{225-x^{2}}=\frac{9}{2}$

9x² = 225

x² = 25

x = 5 km/hr.

Speed downstream (10+6) km/hr 16 km/hr.
Time taken to cover 80 km downstream = (80/16) hrs = 5 hrs.

Let the speed of the stream be x km/hr, Then,

Speed downstream = (10 + x) km/hr, Speed upstream = (10 – x) km/hr.

$\therefore \, \, \, \frac{20}{10+x}=\frac{12}{10-x}$

$\Rightarrow 200-20x=120+12x$

$x=\frac{80}{32}=2.5$

Speed of the boat downstream = 15 km/hr.

Speed of the boat in still water = 8 km/hr

Let the speed of the stream = y km/hr.

We have, 15 = 8 + y

y = 15 – 8 = 7 km/hr.

Speed of upstream =$\frac{3/4}{\frac{45/4}{60}}=\left ( \frac{3}{4}\times \frac{2}{16}\times 60 \right )km/h=4km/h$

Speed of downstream = $\frac{3/4}{\frac{15/2}{60}}=\left ( \frac{3}{4}\times \frac{2}{15}\times 60 \right )km/h=6km/h$

Speed in still water=$\frac{1}{2}\left ( 4+6 \right )=5km/h$

Speed of man in still water = 10 km/hr

Speed of current = 3 km/hr

Speed of man upstream = 10 – 3 = 7km/hr

Let the man's rate upstream be x kmph and that downstream be y kmph.

Then, distance covered upstream in 8 hrs 48 min = Distance covered downstream in 4 hrs.

$x\times 8\frac{4}{5}=\left ( y\times 4 \right )$

$\Rightarrow \frac{44}{5}x=4y$

$\Rightarrow y=\frac{11}{5}x$

Required ratio =$\left ( \frac{y+x}{2} \right ):\left ( \frac{y-x}{2} \right )$

$\left ( \frac{16x}{5}\times \frac{1}{2} \right ):\left ( \frac{6x}{5}\times \frac{1}{2} \right )$

$\frac{8}{5}:\frac{3}{5}=8:3$

Let the speed of the boat and that of the current be ‘x’ and ‘y’ kmph

x + y = 20

x - y = 10

x = 15 kmph and y = 5 kmph

speed of the current = 5 kmph

Speed of the man upstream = 7 km/hr.

Speed of the man downstream = 3 km/hr.

Speed of the man in still water = $\frac{1}{2}$ (Downstream Speed + Upstream Speed)

= $\frac{1}{2}$ (7 + 3) = 5 km/hr.