| Rate downstream =$ \left(\dfrac{16}{2} \right) $kmph = 8 kmph. |
| Rate upstream =$ \left(\dfrac{16}{4} \right) $kmph = 4 kmph. |
| $\therefore$ Speed in still water =$ \dfrac{1}{2} $(8 + 4) kmph = 6 kmph. |
Let the speed of the stream $ x $ mph. Then,
Speed downstream = 10 + $ x $ mph,
Speed upstream = 10 - $ x $ mph.
| $\therefore \dfrac{36}{(10 - x)} $-$ \dfrac{36}{(10 + x)} $=$ \dfrac{90}{60} $ |
$\Rightarrow$ 72$ x $ x 60 = $90 \left(100 - x^2\right)$
$\Rightarrow$ $ x $2 + 48$ x $ - 100 = 0
$\Rightarrow$ $ x + 50\left( x - 2\right)$ = 0
$\Rightarrow$ $ x $ = 2 mph.
Pipe A can fill $\dfrac{1}{8}$ of the cistern in 1 hour.
Pipe B can empty $\dfrac{1}{12}$ of the cistern in 1 hour
Both Pipe A and B together can effectively fill $\dfrac{1}{8}-\dfrac{1}{12}=\dfrac{1}{24}$ of the cistern in 1 hour
i.e, the cistern will be full in 24 hrs.
Let the speed of the boat in still water be $ $ kmph. Then,
Speed downstream = $ x $ + 3 kmph.
Speed upstream = $ x $ - 3 kmph.
| $\therefore$ $ x $ + 3 x 1 = $ x $ - 3 x $ \dfrac{3}{2} $ |
$\Rightarrow$ 2$ x $ + 6 = 3$ x $ - 9
$\Rightarrow x $ = 15 kmph.
Let mans rate upstream be $ x $ kmph.
Then, his rate downstream = 2$ x $ kmph.
| $\therefore$ Speed in still water : Speed of stream =$ \left(\dfrac{2x + x}{2} \right) $:$ \left(\dfrac{2x - x}{2} \right) $ |
| =$ \dfrac{3x}{2} $:$ \dfrac{x}{2} $ |
= 3 : 1.
Pipe A can fill $\dfrac{1}{10}$ of the tank in 1 hr
Pipe B can fill $\dfrac{1}{40}$ of the tank in 1 hr
Pipe A and B together can fill $\dfrac{1}{10}+\dfrac{1}{40}=\dfrac{1}{8}$ of the tank in 1 hr
i.e., Pipe A and B together can fill the tank in 8 hours
speed of boat in still water = 12 km/hr
speed of the stream = 4 km/hr
Speed downstream = (12+4) = 16 km/hr
Time taken to travel 68 km downstream $=\dfrac{68}{16}=\dfrac{17}{4}$ = 4.25 hoursSpeed of the boat in still water = 22 km/hr
speed of the stream = 5 km/hr
Speed downstream = (22+5) = 27 km/hr
Distance travelled downstream = 54 km
Time taken $=\dfrac{\text{distance}}{\text{speed}} = \dfrac{54}{27}\text{ = 2 hours}$Speed downstream $=\dfrac{22}{4}\text{ = 5.5 kmph}$
Speed upstream $=\dfrac{22}{5}\text{ = 4.4 kmph}$
Speed of the boat in still water $=\dfrac{5.5+4.4}{2}\text{ = 4.95 kmph}$
Speed upstream = 7.5 kmph.
Speed downstream = 10.5 kmph.
| $\therefore$ Total time taken =$ \left(\dfrac{105}{7.5} +\dfrac{105}{10.5} \right) $hours = 24 hours. |
| Speed in still water =$ \dfrac{1}{2} $(11 + 5) kmph = 8 kmph. |
Speed downstream = (13 + 4) km/hr = 17 km/hr.
| Time taken to travel 68 km downstream =$ \left(\dfrac{68}{17} \right) $hrs = 4 hrs. |
Speed downstream = 15 km/hr
Rate of the current= 11/2 km/hr
Speed in still water = 15 - 11/2 = 131/2 km/hr
Rate against the current = 131/2 km/hr - 11/2 = 12 km/hr
Speed of the boat in still water = 12 km/hr
Speed downstream $=\dfrac{45}{3}$ = 15 km/hr
Speed of the stream = 15-12 = 3 km/hr
Speed upstream = 12-3 = 9 km/hr
Time taken to cover 45 km upstream $=\dfrac{45}{9}$ = 5 hours