## Boats and Streams Questions - 1

A man can row downstream at ‘x’ km/hr, and upstream at ‘y’ km/hr, what is the speed in still water ?
• 2xy km/hr
• x km/hr
• $\displaystyle \frac{1}{2}$ ( x – y ) km/hr
• $\displaystyle \frac{1}{2}$ ( x + y ) km/hr
• 2y km/hr
Explanation

Downstream = x km/hr; Upstream = y km/hr.

Speed in still water = $\displaystyle \frac{1}{2}$ ( x + y ) km/hr.

Workspace
A man can row downstream at 10 km/hr. Find speed of the current ?
• 5 km/hr
• 2.5 km/hr
• 15 km/hr
• 1 km/hr
• 3.5 km/hr
Explanation

Speed of the current = $\displaystyle \frac{1}{2}$ (Downstream - Upstream).

Speed of the current = $\displaystyle \frac{1}{2}$ (10 – 5) = 2.5 Km/hr.

Workspace
A man can swim along the current a 16 km distance in 3 hours and against the current 7 km distance in 3 hours. Find speed of the stream ?
• 1.5 km/hr
• 2 km/hr
• 3 $\displaystyle \frac{1}{2}$ km/hr
• 2 $\displaystyle \frac{1}{2}$ km/hr
• 4 km/hr
Explanation

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.

Workspace
A man can row at 15 km/hr in still water. It takes him twice as much time as to row up as to down. Find the speed downstream ?
• 30 km/hr
• 20 km/hr
• 25 km/hr
• 18 km/hr
• 24 km/hr
Explanation

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.

Workspace
From a place ‘P’ to another place ‘Q’, a man travelled upstream at 8 km/hr, and returned to the starting place ‘P’ along the stream at 12 km/hr. Altogether it took him 2$\displaystyle \frac{1}{2}$ hours. What is the distance between P to Q ?
• 24 km
• 18 km
• 15 km
• 12 km
• 20 km
Explanation

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.

Workspace
A person travelled upstream at 4 km/hr and returned downstream at 6 km/hr, taking altogether 3$\displaystyle \frac{1}{2}$ hours. What is the total distance travelled ?
• 16.8 km
• 8.4 km
• 12 km
• 15 km
• 18 km
Explanation

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.

Workspace
Speed of a boat in still water is 12 km/hr and the speed upstream is 8 km/hr. What is the speed of the boat downstream ?
• 10 km/hr
• 14 km/hr
• 18 km/hr
• 15 km/hr
• 16 km/hr
Explanation

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.

Workspace
The speed of a boat along the stream is 12 km/hr and the speed in still water is 10$\displaystyle \frac{1}{2}$ km/hr. What is the speed upstream ?
• 8 $\displaystyle \frac{1}{2}$ km/hr
• 7 $\displaystyle \frac{1}{2}$ km/hr
• 9 km/hr
• 9 $\displaystyle \frac{1}{2}$ km/hr
• 10 km/hr
Explanation

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.

Workspace
The speed of a boat upstream is 10 km/hr. It takes twice as much time as to row up as to down. Find the rate of the current ?
• 12 km/hr
• 8 km/hr
• 6 km/hr
• 4 $\displaystyle \frac{1}{2}$ km/hr
• 5 km/hr
Explanation

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.

Workspace
The speed of a boat in still water is 10 km/hr. The boat started from a point ‘P’ to Q and returned from Q to P, taking a total time of 3 hours to row up and down. Travelling upstream took 1 hour more time than downstream. Find the rate of the current ?
• 3 km/hr
• 3.33 km/hr
• 4 km/hr
• 4 $\displaystyle \frac{1}{2}$ km/hr
• 5 km/hr
Explanation

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.

Workspace

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## Boats and Streams Important Formulas

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

Time:

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