Downstream = x km/hr; Upstream = y km/hr.
Speed in still water = $\displaystyle \frac{1}{2}$ ( x + y ) km/hr.
Speed of the current = $\displaystyle \frac{1}{2}$ (Downstream - Upstream).
Speed of the current = $\displaystyle \frac{1}{2}$ (10 – 5) = 2.5 Km/hr.
Speed of the stream = $\displaystyle \frac{1}{2}$ (Downstream - Upstream).
Speed of the stream = $\displaystyle \frac{1}{2} \times \left(\frac{16}{3} \ - \ \frac{7}{3}\right) \ = \ \frac{1}{2} \times \left(\frac{9}{3}\right) \ => \ \frac{1}{2} \times 3$ = 1.5 km/hr.
Let speed upstream be ‘a’ km/hr.
Then downstream = 2a [time to speed inversely proportional].
Speed in still water = $\displaystyle \frac{1}{2}$ (2a + a) = 15 km/hr.
=> 3a = 30 => a = $\displaystyle \frac{30}{3}$ = 10 or 10 km/hr.
Upstream = 10 km/hr, then downstream = 20 km/hr.
Let distance be "a" km. Time 2 $\displaystyle \frac{1}{2}$ hours = $\displaystyle \frac{5}{2}$ hours. Downstream = 12 km/hr; upstream = 8 km/hr.
$\displaystyle \frac{a}{8} \ + \ \frac{a}{12} \ = \ \frac{3a \ + \ 2a}{24} \ = \ \frac{5}{2}$ hours (on cross multiplying we get).
=> 10a = 120 => a = 120 ÷ 10 = 12 km.
Distance = 12 km.
Let distance be = "a" km.
Speed downstream = 6 km/hr; upstream = 4 km/hr; Time = 3 $\displaystyle \frac{1}{2}$ hours = $\displaystyle \frac{7}{2}$ hours.
Then distance = $\displaystyle \frac{a}{6} \ + \ \frac{a}{4} \ = \ \frac{7}{2}$ hours.
= $\displaystyle \frac{2a \ + \ 3a}{12} \ = \ \frac{7}{2}$ => 10a = 84 => a = 8.4 or 8.4 km, distance covered = 8.4 x 2 = 16.8 km.
Let speed downstream be "a" km/hr; Upstream = 8 km/hr.
Then speed in still water = $\displaystyle \frac{1}{2}$ (a + 8) = 12 km.
a + 8 = 24 => Downstream = a = 24 – 8 = 16 km/hr.
Let speed upstream be ‘b’ km/hr; Downstream = 12 km/hr.
Speed in Still water = $\displaystyle \frac{1}{2}$ (12 + b) = 10 $\displaystyle \frac{1}{2}$ km/hr.
12 + b = 21 => b = 21 – 12 = 9 km/hr.
Speed upstream = 9 km/hr.
Time to speed will be inversely proportional.
As speed upstream is 10 km/hr then downstream will be 20 km/hr
Speed of the current = $\displaystyle \frac{1}{2}$ (20 - 10) = 5 km/hr.
Total time taken = 3 hours.
Difference in time = 1 hour.
Upstream = 2 hours, then Downstream = 1 hour. [time to speed inversely proportional]
Speed ratio Down : Up = 2 : 1
Let speed downstream = 2a km/hr; upstream = a km/hr.
Speed in Stillwater = $\displaystyle \frac{1}{2}$ (2a + a) = 10 km/hr.
3a = 20 => a = $\displaystyle \frac{20}{3}$ km/hr.
Speed of current = speed in still water – upstream.
Speed of current = 10 - $\displaystyle \frac{20}{3} = \frac{30 - 20}{3} = \frac{10}{3}$ = 3.33 km/hr.
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1. In water the direction along the stream is called ‘Downstream’. And the direction against is called ‘Upstream’.
2. If speed of a boat in still water is u km/hr and the speed of the stream is v km/hr then
Speed Down Stream = (u + v) km/hr
Upstream = (u – v) km/hr.
3. If the speed Downstream is a km/hr and the speed of the upstream is b km/hr. then,
Speed in still water = $\displaystyle \frac{1}{2}$ (a + b) km/hr.
OR
$\displaystyle \frac{\text{Speed Downstream} + \text{Speed Upstream}}{2}$
OR
Speed Downstream - Speed of the current
OR
Speed Upstream + Speed of the current
Rate of stream/current/velocity = $\displaystyle \frac{1}{2}$ (a - b) km/hr
OR
Speed Downstream – Speed in still water
OR
Speed in still water – Speed Upstream
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