51. A motorboat can cover a distance of 14.4 km downstream and 6.4 km upstream in 4 hours. It can also cover 3 km downstream and 1.2 km upstream in 48 minutes. What is the speed of the boat in still water(in km/hr)?
एक नाव 4 घंटे में 14.4 किमी धारा के अनुकूल और 6.4 किमी धारा के प्रतिकूल दूरी तय कर सकती है तथा वह 48 मिनट में 3 किमी धारा के अनुकूल और 1.2 किमी धारा प्रतिकुल तय कर सकती हैं। स्थिर पानी में नाव की चाल (किमी/घंटा) में कितनी है?
Option “C” is correct.
Given:
A motorboat covers a distance of 14.4 km downstream and 6.4 km upstream in 4 hours.
It also covers 3 km downstream and 1.2 km upstream in 48 minutes.
Concept:
The direction along the stream is called downstream.
The direction against the stream is called upstream.
Formula used:
Downstream speed of the boat is
⇒ Speed of boat + speed of stream
Speed of a boat in upstream
⇒ Speed of boat – speed of stream
Speed = distance/time
Calculation:
Let x = speed of motor boat in still water and a = speed of stream
Then the speed of motor boat in downstream be (x + a) and in upstream be (x – a)
Distance covered in downstream = 14.4 km
Distance covered in upstream = 6.4 km
Total time taken in whole journey= 4 hr
For total time,
⇒ [14.4/(x + a)] + [6.4/(x – a)] = 4
⇒ [18/(x + a)] + [8/(x – a)] = 5 —-(i)
After that it covers 3 km in downstream and 1.2 km in upstream in 48 minutes
Now for total time,
⇒ [3/(x + a)] + [1.2/(x – a)] = 48/60
⇒ [30/(x + a)] + [12/(x – a)] = 480/60
⇒ [15/(x + a)] + [6/(x – a)] = 4 —-(ii)
Now multiply equation (i) by 3 and equation (ii) by 4, and subtract equation (i) from (ii) we get
Equation (iii) shows the downstream speed of motor boat.
Now put the value in equation (i) from equation (iii) we get
⇒ 18/6 + [8/(x – a)] = 5
⇒ 8/(x – a) = 5 – 3
⇒ x – a = 4 —-(vi)
On subtracting equation (vi) from (iii) we get
⇒ 2a = 2
⇒ a = 1 km/h
Now the speed of motor boat in still water
⇒ Total downstream speed of motor boat – speed of stream
⇒ 6 – 1 = 5 km/hr
∴ The speed of motor boat in still water is 5 km/hr.
52. A girl swims triple the distance in the direction of the current as compared to the distance she covers while swimming against the current. If her speed in still water is 3.5 km/hr and time taken by her in the direction of current and against the current is 36 minutes and 48 minutes respectively, then what will be the sum of speeds of swimming with and against the current?
एक लड़की धारा प्रवाह के खिलाफ तैरने के दौरान जितनी दूरी तय करती है, उसकी तुलना में वह धारा प्रवाह की दिशा में दूरी को तीन गुना कर देती है। यदि स्थिर पानी में उसकी गति 3.5 किमी/घंटा है और धारा प्रवाह की दिशा में और धारा प्रवाह खिलाफ 36 मिनट और 48 मिनट का समय लिया जाता है, तो वर्तमान के साथ और खिलाफ तैराकी की गति का योग क्या होगा?
Option “B” is correct.
SHORTCUT TRICK
Sum of downstream and upstream speeds = D + U
= B + S + B – S
= 2 × B
= 2 × 3.5
= 7 km/hr
IMPORTANT POINT
Sometimes extra data in the question is there to waste the time.
ALTERNATE METHOD
Given:
Speed of girl = 3.5 km/hr
Time taken by girl in downstream = 36 mins.
Time taken by girl in upstream = 48 mins.
Formula Used:
Distance = Speed × Time
D = B + S
U = B – S
B = (D + U)/2
S = (D – U)/2
where, D → Downstream speed, U → Upstream speed, B → Speed of boat in still water, S → Speed of stream.
Calculations:
Let the speed of current be x km/hr.
Let the downstream and upstream speed be D km/hr and U km/hr respectively.
Let the downstream and upstream distance be d1 km and d2 km respectively.
Distance = Speed × Time
For downstream, distance (d1) = D × 36
⇒ d1 = (3.5 + x) × 36
⇒ d1 = 36(3.5 + x)
For upstream, distance (d2) = U × 48
⇒ d2 = (3.5 – x) × 48
⇒ d2 = 48(3.5 – x)
∵ d1 = 3d2
⇒ 36(3.5 + x) = 3 × 48(3.5 – x)
⇒ (3.5 + x)/(3.5 – x) = 4
Using componendo – dividendo
⇒ 3.5/x = (4 + 1)/(4 – 1)
⇒ x = 2.1 km/hr
D = B + x
⇒ D = (3.5 + 2.1) km/hr
⇒ D = 5.6 km/hr
U = B – x
⇒ U = (3.5 – 2.1) km/hr
⇒ U = 1.4 km/hr
Sum of downstream and upstream speeds = D + U = (5.6 + 1.4) km/hr
⇒ D + U = 7 km/hr
∴ The sum of speeds of swimming with and against the current is 7 km/hr.
53. A man can row at a speed of 15/2 km/hr in still water. He goes to a certain point upstream and back to the starting point in a river which flows at 3/2 km/hr, then the average speed of the man for the total journey is:
एक आदमी स्थिर जल में 15/2 किमी/घंटे की गति से नाव चलाता है वह नदी में एक निश्चित बिंदु तक धारा के विपरीत जाता है और वापस अपने प्रारंभिक बिंदु पर आता है जिसमें धारा की गति 3/2 किमी./घंटा है, तो कुल यात्रा में आदमी की औसत गति ज्ञात कीजिये।
Option “B” is correct.
Let the total upstream and downstream distance be 2x km,
54. A boat can go 20 km downstream and 30 km upstream in 2 hours 20 minutes. Also, it can go 10 km downstream and 8 km upstream in 49 minutes. What is the speed of boat downstream in km/h?
एक नाव 20 किमी प्रवाह के अनुकूल दिशा में और 30 किमी प्रवाह के प्रतिकूल दिशा में 2 घंटे 20 मिनट में जा सकती है। इसके अतिरिक्त, यह 10 किमी प्रवाह के अनुकूल दिशा में और 8 किमी प्रवाह के प्रतिकूल दिशा में 49 मिनट में तय कर सकती है। नाव की प्रवाह के अनुकूल दिशा में गति किमी/घंटा में क्या है?
Option “D” is correct.
Time in hours = 2 + 20/60 = 7/3 hours
Time in hours = 49/60 hours
Let upstream speed and downstream speed be A kmph and B kmph respectively.
⇒ 30/A + 20/B = 7/3 —- (1)
⇒ 8/A + 10/B = 49/60
Solving,
⇒ 16/A + 20/B = 49/30 —- (2)
Subtracting,
⇒ 14/A = 7/10
⇒ A = 20
⇒ 20/B = 7/3 – 30/20
⇒ B = 24
Speed downstream = 24 kmph
55. Boat M can row 60 km downstream and 20 km upstream in the same time and Boat N can row 50 km downstream and 10 km upstream in the same time in the stream running at 10 km/h. Find the difference between the speeds of boats in still water.
नाव M 60 किमी अनुप्रवाह और 20 किमी उर्ध्वप्रवाह समान समय में चलती है और नाव N 50 किमी अनुप्रवाह और 10 किमी उर्ध्वप्रवाह समान समय में 10 किमी/घंटा वाली धारा में चलती है। स्थिर पानी में नाव की चालों का अंतर ज्ञात कीजिये
Option “A” is correct.
Let the speed of the boat M in still water be M km/h and speed of the boat N in still water be N km/h.
According to the question
⇒ 60/(M + 10) = 20/(M – 10)
⇒ 60M – 600 = 20M + 200
⇒ M = 20km/h
⇒ 50/(N + 10) = 10/(N – 10)
⇒ 10N + 100 = 50N – 500
⇒ N = 15 km/h
Required difference = 20 – 15 = 5 km/h
56. The total time taken by a boat to go 120 km upstream and came back to the starting point is 8 hours. If the speed of the stream is 25% of the speed of the boat in still water, then find the difference between the upstream speed and the downstream speed of the boat.
एक नाव द्वारा 120 किमी धारा के प्रतिकूल जाने में और प्रारंभिक बिंदु पर वापस आने में लिया गया कुल समय 8 घंटे है। यदि धारा की गति स्थिर पानी में नाव की गति की 25% है, तो धारा के प्रतिकूल और धारा के अनुकूल नाव की गति के बीच अंतर ज्ञात कीजिये।
Option “A” is correct.
Given:
Time to go 120 km upstream and come back to the starting point = 8 hours
The speed of the stream is 25% of the speed of the boat in still water
Formula Used:
Downstream Speed = Speed of boat + Speed of stream
Upstream Speed = Speed of boat – Speed of stream
Calculation:
Let the speed of the boat in still water be x km/h.
So, the speed of stream = x × 25/100 = (x/4) km/h
The upstream speed of boat = x – (x/4) = (3x/4) km/h
The downstream speed of boat = x + (x/4) = (5x/4) km/h
According to the question:
[120/ (3x/4)] + [120/ (5x/4)] = 8
⇒ (480/3x) + (480/5x) = 8
⇒ 480[(5 + 3) /15x] = 8
⇒ 15x = 480
⇒ x = 32
⇒ The upstream speed of the boat = 32 × 3/4 = 24 km/h
⇒ The downstream speed of the boat = 32 × 5/4 = 40 km/h
∴ The required difference = 40 – 24 = 16 km/h
57.A boat can go 3.6 km upstream and 5.4 km downstream in 54 minutes, while it can go 5.4 km upstream and 3.6 km downstream in 58.5 minutes. The time (in minutes) taken by the boat in going 10 km downstream is:
एक नाव 3.6 किमी धारा के प्रतिकूल और 5.4 किमी धारा के अनुकूल 54 मिनट में जा सकती है, जबकि यह 5.4 किमी के प्रतिकूल और 3.6 किमी के अनुकूल 58.5 मिनट में जा सकती है। 10 किमी धारा के अनुकूल जाने में नाव द्वारा लिया गया समय (मिनट में) है:
Option “C” is correct.
Short Trick:
U = upstream and D = downstream
3.6(U) + 5.4(D) = 54 and 5.4(U) + 3.6(D) = 58.5
Divide by 18
2U + 3D = 30 and 3U + 2D = 32.5
Now, Multiply by 3 in (1) and by 2 in (2), and then subtract (2) from (1), then
5D = 25 min
∴10D = 50 min
Detailed Method:
Let the speed of boat be x km/hr and speed of current be y km/hr
Upstream speed = (x – y) km/hr
Downstream speed = (x + y) km/hr
According to the question
3.6/(x – y) + 5.4/(x + y) = 54/60
5.4/(x – y) + 3.6/(x + y) = 58.5/60
Let 1/(x – y) = a and 1/(x + y) = b, then
3.6a + 5.4b = 54/60 —(1)
5.4a + 3.6b = 58.5/60 —(2)
Add equation (1) and equation (2)
9 (a + b) = 112.5/60
⇒a + b = 12.5/60
⇒a + b = 5/24 —(3)
Subtract equation (2) from equation (1)
1.8 (b – a) = -4.5/60
⇒b – a = -2.5/60
⇒b – a = -1/24 —(4)
Add equation (3) and equation (4)
2b = 4/24
⇒b = 1/12
⇒(x + y) = 12 km/hr
Downstream speed = 12 km/hr
∴Time taken to cover 10 km distance with downstream = 10/12 hr or (10/12) × 60 = 50 min
58. In a stream running at 3 km/h, a motorboat goes 12 km upstream and back to the starting point in 60 min. Find the speed of the motorboat in still water. (in km/h)
3 किमी/घंटा की गति से बहने वाली एक नदी में मोटरबोट धारा के प्रतिकूल 12 किमी जाती है और 60 मिनट में वापस प्रारंभिक स्थान पर आती है। तो शांत जल में मोटरबोट की गति (किमी/घंटा में) ज्ञात कीजिए।
Option “B” is correct.
Given:
In a stream running at 3 km/h, a motorboat goes 12 km upstream and back to the starting point in 60 min
Formula used:
x = [-b ± √(b2 – 4ac)]/2a
Calculation:
Let the speed of boat in still water be x, then
⇒ 12/(x – 3) + 12/(x + 3) = 1
⇒ 12 [(x + 3 + x – 3)/(x2 – 9)] = 1
⇒ 12 × 2x = x2 – 9
⇒ x2 – 24x – 9 = 0
Compare on ax2 + bx + c = 0
a = 1, b = -24 and c = -9
As we know,
⇒ x = [-b ± √(b2 – 4ac)]/2a
⇒ x = [24 ± √(242 – 4 × 1 × (-9)]/2
⇒ x = [24 ± √(576 + 36)]/2
⇒ x = [24 ± √612]/2
⇒ x = (24 ± 6√17)/2
⇒ x = 12 ± 3√17
⇒ x = 3 (4 ± √17)
Speed of boat in still water is 3 (4 + √17)
59. Ram hires a motorboat to go to a place on the other side of a lake and come back to the starting point. He goes to his destination at a speed of 125 km/hr in still water. If the speed of stream is 25 km/hr and the distance between the 2 ends is 55 km. Find the average speed of Ram in the whole journey.
राम एक झील के दूसरी ओर एक स्थान पर जाने के लिए एक मोटरबोट को किराए पर लेता हैं और शुरुआती बिंदु पर वापस आता है। वह स्थिर पानी में 125 किमी/घंटा की गति से अपने गंतव्य तक जाता है। यदि धारा की गति 25 किमी/घंटा है और 2 छोरों के बीच की दूरी 55 किमी है। पूरी यात्रा में राम की औसत गति ज्ञात कीजिये।
Option “B” is correct.
Short Trick Method/Topper’s Approach:
Let the downstream and upstream speeds be D km/hr and U km/hr respectively.
Let the speed of motorboat in still water and speed of stream be B km/hr and S km/hr respectively.
Speed = Distance/Time
D = B + S
⇒ D = (125 + 25) km/hr
⇒ D = 150 km/hr
U = B – S
⇒ U = (125 – 25) km/hr
⇒ U = 100 km/hr
For equal distance, Average speed = 2v1v2/(v1 + v2)
⇒ Average speed = 2 × 100 × 150/(100 + 150)
⇒ Average speed = 120 km/hr
∴ The average speed of Ram in the whole journey is 120 km/hr.
ALTERNATE METHOD
Given:
Distance between the 2 ends = 55 km
Speed of Ram in still water = 125 km/hr
Speed of current = 25 km/hr
Concept Used:
Time = Distance/Speed
Average speed = Total distance/Total time
Formula Used:
D = B + S
U = B – S
where, D → Downstream speed, U → Upstream speed, B → Speed of boat in still water, S → Speed of stream.
Calculations:
Let the downstream and upstream speeds be D km/hr and U km/hr respectively.
Let the speed of motorboat in still water and speed of stream be B km/hr and S km/hr respectively.
Let the total time taken in downstream and upstream be t1 and t2 hours respectively.
Speed = Distance/Time
D = B + S
⇒ D = (125 + 25) km/hr
⇒ D = 150 km/hr
U = B – S
⇒ U = (125 – 25) km/hr
⇒ U = 100 km/hr
Time = Distance/Speed
t1 = 55/150 = 11/30 hrs
⇒ t1 = (11/30) × 60 = 22 mins.
t2 = 55/100 = 11/20 hrs
⇒ t2 = (11/20) × 60 = 33 mins.
Total time in the journey = (22 + 33) mins. = 55/60 hrs.
Average speed = Total distance/Total time
⇒ Average speed = (2 × 55)/(55/60)
⇒ Average speed = 120 km/hr
∴ The average speed of Ram in the whole journey is 120 km/hr.
60. Two cities X and Y are 700 km apart. X to Y is a downstream journey on a stream which flows at a speed of 5 km/hr. Two boats A and B run to and fro between two cities. Speed of Boat A, which starts from city X is 45 km/hr, while speed of boat B, which starts from city Y at the same time is 25 km/hr. After how much time will two boats meet for the second time?
दो शहर X और Y 700 किमी दूर हैं। X से Y तक की यात्रा धारा के अनुकूल है जो 5 किमी/घंटा की गति से बहती है। दो नाव A और B दोनों शहरों से चलती हैं। शहर X से चलते हुए नाव A की गति 45 किमी/घंटा है, जबकि शहर Y से समान समय में नाव B 25 किमी/घंटा की गति से चलती है। कितने समय के बाद दोनों नाव दूसरी बार मिलेंगी?
Option “B” is correct.
Distance between city X and Y = 700 km,
Speed of boat A = 45 km/hr,
Speed of boat B = 25 km/hr,
Speed of the stream = 5 km/hr.
Since X to Y is the downstream journey;
∴ When boat A travels downstream, effective speed of boat A = 45 + 5 = 50 km/hr
Similarly, when boat B travels upstream, effective speed of boat B = 25 – 5 = 20 km/hr
∴ The relative speed (Since boats are travelling in opposite direction) = 50 + 20 = 70 km/hr
∴ Time in which boats will meet for the first time = 700/70 = 10 hours
Now boat A will reach the city Y in (700/50) = 14 hours
When boat A reaches city Y, boat B will be at a point (20 × 14) = 280 km from city Y
Now boat A has reached city Y, it will return to city X then and thus its direction will change.
∴ After 14 hours, both A and B will be travelling upstream;
∴ When boat A travels upstream, effective speed of boat A = 45 – 5 = 40 km/hr
Similarly, when boat B travels upstream, effective speed of boat B = 25 – 5 = 20 km/hr
∴ The relative speed (Since boats are travelling in same direction) = 40 – 20 = 20 km/hr
Since B is ahead of A by 280 km;
∴ A will catch B after (280/20) = 14 hours
∴ Boat A and B will meet after (14 + 14) = 28 hours
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