Are you looking for Boats and Streams Questions with solutions to practice for competitive exams? You’ve come to the right place!
Boats and Streams is one of the most important and high-scoring topics in quantitative aptitude. It is a crucial topic asked across major competitive entrance and recruitment examinations, including SSC, Banking, UPSC, Railways, and Campus Placement Tests.
In this comprehensive guide, you will find carefully selected boats and streams questions with step-by-step solutions. Whether you are preparing for competitive exams or campus placements, these practice sets will help you solve questions faster and with absolute accuracy.
Let’s dive in and elevate your quantitative problem-solving skills!
A man can row a boat at a speed of 6 km/h in still water. If the speed of the stream is 2 km/h, what is the downstream speed of the boat?
A. 4 km/h
B. 6 km/h
C. 8 km/h
D. 12 km/h
8 km/h
Explanation: The downstream speed D is the sum of the speed of the boat in still water (u) and the speed of the stream (v).
The formula is D = u + v. Substituting the given values:
u = 6\text{ km/h} v = 2\text{ km/h} D = 6 + 2 = 8\text{ km/h}.
Thus, the correct option is C.
The formula is D = u + v. Substituting the given values:
u = 6\text{ km/h} v = 2\text{ km/h} D = 6 + 2 = 8\text{ km/h}.
Thus, the correct option is C.
A man can row a boat at a speed of 6 km/h in still water. If the speed of the stream is 2 km/h, what is the downstream speed of the boat?
A. 4 km/h
B. 6 km/h
C. 8 km/h
D. 12 km/h
8 km/h
Explanation:
When a boat moves in the same direction as the stream, it is moving downstream.
The downstream speed (D) is calculated by adding the speed of the boat in still water (u) and the speed of the stream (v).
The formula is given by:
D = u + v Given:
Speed of the boat in still water, u = 6\text{ km/h}
Speed of the stream, v = 2\text{ km/h}
Substituting these values into the formula:
D = 6 + 2 = 8\text{ km/h}
Therefore, the correct option is C.
When a boat moves in the same direction as the stream, it is moving downstream.
The downstream speed (D) is calculated by adding the speed of the boat in still water (u) and the speed of the stream (v).
The formula is given by:
D = u + v Given:
Speed of the boat in still water, u = 6\text{ km/h}
Speed of the stream, v = 2\text{ km/h}
Substituting these values into the formula:
D = 6 + 2 = 8\text{ km/h}
Therefore, the correct option is C.
The downstream speed of a boat is 14 km/h and its upstream speed is 8 km/h. What is the speed of the boat in still water?
A. 22 km/h
B. 3 km/h
C. 6 km/h
D. 11 km/h
11 km/h
Explanation:
When both the downstream speed (D) and upstream speed (U) are known, the speed of the boat in still water (u) is calculated as the average of these two speeds.
The formula is given by:
u = \frac{D + U}{2} Given:
Downstream speed, D = 14\text{ km/h}
Upstream speed, U = 8\text{ km/h}
Substituting these values into the formula:
u = \frac{14 + 8}{2} = \frac{22}{2} = 11\text{ km/h}
Therefore, the correct option is D.
When both the downstream speed (D) and upstream speed (U) are known, the speed of the boat in still water (u) is calculated as the average of these two speeds.
The formula is given by:
u = \frac{D + U}{2} Given:
Downstream speed, D = 14\text{ km/h}
Upstream speed, U = 8\text{ km/h}
Substituting these values into the formula:
u = \frac{14 + 8}{2} = \frac{22}{2} = 11\text{ km/h}
Therefore, the correct option is D.
A boat travels downstream at 18 km/h and upstream at 10 km/h. Find the speed of the stream.
A. 4 km/h
B. 14 km/h
C. 8 km/h
D. 2 km/h
4 km/h
Explanation:
When both the downstream speed (D) and upstream speed (U) are known, the speed of the stream (v) is calculated as half of the difference between these two speeds.
The formula is given by:
v = \frac{D - U}{2}
Given:
Downstream speed, D = 18\text{ km/h}
Upstream speed, U = 10\text{ km/h}
Substituting these values into the formula:
v = \frac{18 - 10}{2} = \frac{8}{2} = 4\text{ km/h}
Therefore, the correct option is A.
When both the downstream speed (D) and upstream speed (U) are known, the speed of the stream (v) is calculated as half of the difference between these two speeds.
The formula is given by:
v = \frac{D - U}{2}
Given:
Downstream speed, D = 18\text{ km/h}
Upstream speed, U = 10\text{ km/h}
Substituting these values into the formula:
v = \frac{18 - 10}{2} = \frac{8}{2} = 4\text{ km/h}
Therefore, the correct option is A.
A man rows a certain distance downstream in 3 hours and the same distance upstream in 5 hours. If the speed of the stream is 2 km/h, find the distance.
A. 15 km
B. 24 km
C. 30 km
D. 45 km
30 km
Explanation:
Let the speed of the boat in still water be u\text{ km/h}.
Given that the speed of the stream is v = 2\text{ km/h}.
The downstream speed (D) is u + 2 and the upstream speed (U) is u - 2.
Since the distance traveled in both cases is the same, we can equate the product of speed and time:
\text{Distance} = \text{Downstream Speed} \times \text{Downstream Time} = \text{Upstream Speed} \times \text{Upstream Time} Substitute the values into the equation:
(u + 2) \times 3 = (u - 2) \times 5 Expanding both sides:
3u + 6 = 5u - 10 Rearranging the terms to solve for u:
5u - 3u = 6 + 10 2u = 16 \implies u = 8\text{ km/h}
Now, substitute u back into the distance formula:
\text{Distance} = (8 + 2) \times 3 = 10 \times 3 = 30\text{ km}
Therefore, the correct option is C.
Let the speed of the boat in still water be u\text{ km/h}.
Given that the speed of the stream is v = 2\text{ km/h}.
The downstream speed (D) is u + 2 and the upstream speed (U) is u - 2.
Since the distance traveled in both cases is the same, we can equate the product of speed and time:
\text{Distance} = \text{Downstream Speed} \times \text{Downstream Time} = \text{Upstream Speed} \times \text{Upstream Time} Substitute the values into the equation:
(u + 2) \times 3 = (u - 2) \times 5 Expanding both sides:
3u + 6 = 5u - 10 Rearranging the terms to solve for u:
5u - 3u = 6 + 10 2u = 16 \implies u = 8\text{ km/h}
Now, substitute u back into the distance formula:
\text{Distance} = (8 + 2) \times 3 = 10 \times 3 = 30\text{ km}
Therefore, the correct option is C.
A boat covers a distance of 36 km downstream in 4 hours and the same distance upstream in 9 hours. What is the speed of the boat in still water?
A. 5.0 km/h
B. 7.0 km/h
C. 6.0 km/h
D. 6.5 km/h
6.5 km/h
Explanation:
First, calculate the downstream speed (D) by dividing the distance traveled by the downstream time:
D = \frac{\text{Distance}}{\text{Time downstream}} = \frac{36}{4} = 9\text{ km/h}
Next, calculate the upstream speed (U) by dividing the distance traveled by the upstream time:
U = \frac{\text{Distance}}{\text{Time upstream}} = \frac{36}{9} = 4\text{ km/h}
The speed of the boat in still water (u) is found by taking the average of the downstream and upstream speeds:
u = \frac{D + U}{2} Substituting the calculated values:
u = \frac{9 + 4}{2} = \frac{13}{2} = 6.5\text{ km/h}
Therefore, the correct option is D.
First, calculate the downstream speed (D) by dividing the distance traveled by the downstream time:
D = \frac{\text{Distance}}{\text{Time downstream}} = \frac{36}{4} = 9\text{ km/h}
Next, calculate the upstream speed (U) by dividing the distance traveled by the upstream time:
U = \frac{\text{Distance}}{\text{Time upstream}} = \frac{36}{9} = 4\text{ km/h}
The speed of the boat in still water (u) is found by taking the average of the downstream and upstream speeds:
u = \frac{D + U}{2} Substituting the calculated values:
u = \frac{9 + 4}{2} = \frac{13}{2} = 6.5\text{ km/h}
Therefore, the correct option is D.
A boat can travel 48 km downstream in 6 hours, and the same distance upstream in 8 hours. Find the speed of the stream.
A. 1.0 km/h
B. 2.0 km/h
C. 1.5 km/h
D. 2.5 km/h
1.0 km/h
Explanation:
First, calculate the downstream speed (D) by dividing the distance by the downstream time:
D = \frac{48}{6} = 8\text{ km/h} Next, calculate the upstream speed (U) by dividing the distance by the upstream time:
U = \frac{48}{8} = 6\text{ km/h}
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} Substituting the calculated values:
v = \frac{8 - 6}{2} = \frac{2}{2} = 1.0\text{ km/h}
Therefore, the correct option is A.
First, calculate the downstream speed (D) by dividing the distance by the downstream time:
D = \frac{48}{6} = 8\text{ km/h} Next, calculate the upstream speed (U) by dividing the distance by the upstream time:
U = \frac{48}{8} = 6\text{ km/h}
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} Substituting the calculated values:
v = \frac{8 - 6}{2} = \frac{2}{2} = 1.0\text{ km/h}
Therefore, the correct option is A.
A boat can travel with a speed of 12 km/h in still water. If the speed of the stream is 4 km/h, what is the total time taken by the boat to go 32 km downstream and return 32 km upstream?
A. 5 hours
B. 8 hours
C. 6 hours
D. 7 hours
6 hours
Explanation:
First, determine the downstream speed (D) and upstream speed (U) using the given speeds in still water (u) and the stream (v).
D = u + v = 12 + 4 = 16\text{ km/h} U = u - v = 12 - 4 = 8\text{ km/h}
Next, calculate the time taken to travel downstream (T_1) for a distance of 32 km:
T_1 = \frac{\text{Distance}}{\text{Downstream Speed}} = \frac{32}{16} = 2\text{ hours} Then, calculate the time taken to return upstream (T_2) for the same distance of 32 km:
T_2 = \frac{\text{Distance}}{\text{Upstream Speed}} = \frac{32}{8} = 4\text{ hours} Finally, find the total time for the round trip by adding both times:
\text{Total Time} = T_1 + T_2 = 2 + 4 = 6\text{ hours}
Therefore, the correct option is C.
First, determine the downstream speed (D) and upstream speed (U) using the given speeds in still water (u) and the stream (v).
D = u + v = 12 + 4 = 16\text{ km/h} U = u - v = 12 - 4 = 8\text{ km/h}
Next, calculate the time taken to travel downstream (T_1) for a distance of 32 km:
T_1 = \frac{\text{Distance}}{\text{Downstream Speed}} = \frac{32}{16} = 2\text{ hours} Then, calculate the time taken to return upstream (T_2) for the same distance of 32 km:
T_2 = \frac{\text{Distance}}{\text{Upstream Speed}} = \frac{32}{8} = 4\text{ hours} Finally, find the total time for the round trip by adding both times:
\text{Total Time} = T_1 + T_2 = 2 + 4 = 6\text{ hours}
Therefore, the correct option is C.
A man rows a boat downstream at 15 km/h and upstream at 9 km/h. What is the average speed of the boat for the total journey?
A. 11.25 km/h
B. 12.00 km/h
C. 10.50 km/h
D. 13.50 km/h
11.25 km/h
Explanation:
When a boat travels equal distances downstream and upstream, the average speed (V_{avg}) is calculated using the harmonic mean formula for speed:
V_{avg} = \frac{2 \times D \times U}{D + U} Given:
Downstream speed, D = 15\text{ km/h}
Upstream speed, U = 9\text{ km/h}
Substitute these values into the formula:
V_{avg} = \frac{2 \times 15 \times 9}{15 + 9} V_{avg} = \frac{2 \times 135}{24} = \frac{270}{24} Dividing the numerator and denominator:
V_{avg} = 11.25\text{ km/h}
Therefore, the correct option is A.
When a boat travels equal distances downstream and upstream, the average speed (V_{avg}) is calculated using the harmonic mean formula for speed:
V_{avg} = \frac{2 \times D \times U}{D + U} Given:
Downstream speed, D = 15\text{ km/h}
Upstream speed, U = 9\text{ km/h}
Substitute these values into the formula:
V_{avg} = \frac{2 \times 15 \times 9}{15 + 9} V_{avg} = \frac{2 \times 135}{24} = \frac{270}{24} Dividing the numerator and denominator:
V_{avg} = 11.25\text{ km/h}
Therefore, the correct option is A.
A boat travels 30 km downstream and returns the same distance upstream in a total time of 8 hours. If the speed of the stream is 2 km/h, find the speed of the boat in still water.
A. 6 km/h
B. 10 km/h
C. 8 km/h
D. 12 km/h
8 km/h
Explanation:
Let the speed of the boat in still water be u\text{ km/h}. Given that the speed of the stream is v = 2\text{ km/h}.
The downstream speed (D) is u + 2 and the upstream speed (U) is u - 2.
The time taken to travel downstream for 30 km is \frac{30}{u + 2} hours, and the time taken to return upstream for 30 km is \frac{30}{u - 2} hours.
The total time given is 8 hours, so we can set up the equation:
\frac{30}{u + 2} + \frac{30}{u - 2} = 8 Taking out 30 as a common factor and finding a common denominator for the fractions:
30 \left( \frac{(u - 2) + (u + 2)}{(u + 2)(u - 2)} \right) = 8 30 \left( \frac{2u}{u^2 - 4} \right) = 8 \frac{60u}{u^2 - 4} = 8
Cross-multiplying to solve for u:
60u = 8(u^2 - 4) 60u = 8u^2 - 32
Dividing the entire equation by 4 to simplify:
2u^2 - 15u - 8 = 0 Factoring the quadratic equation:
2u^2 - 16u + u - 8 = 0 2u(u - 8) + 1(u - 8) = 0 (2u + 1)(u - 8) = 0 Since speed cannot be negative, u = 8\text{ km/h}.
Therefore, the correct option is C.
Let the speed of the boat in still water be u\text{ km/h}. Given that the speed of the stream is v = 2\text{ km/h}.
The downstream speed (D) is u + 2 and the upstream speed (U) is u - 2.
The time taken to travel downstream for 30 km is \frac{30}{u + 2} hours, and the time taken to return upstream for 30 km is \frac{30}{u - 2} hours.
The total time given is 8 hours, so we can set up the equation:
\frac{30}{u + 2} + \frac{30}{u - 2} = 8 Taking out 30 as a common factor and finding a common denominator for the fractions:
30 \left( \frac{(u - 2) + (u + 2)}{(u + 2)(u - 2)} \right) = 8 30 \left( \frac{2u}{u^2 - 4} \right) = 8 \frac{60u}{u^2 - 4} = 8
Cross-multiplying to solve for u:
60u = 8(u^2 - 4) 60u = 8u^2 - 32
Dividing the entire equation by 4 to simplify:
2u^2 - 15u - 8 = 0 Factoring the quadratic equation:
2u^2 - 16u + u - 8 = 0 2u(u - 8) + 1(u - 8) = 0 (2u + 1)(u - 8) = 0 Since speed cannot be negative, u = 8\text{ km/h}.
Therefore, the correct option is C.
A man can row at a speed of 10 km/h in still water. If he takes twice as long to row a certain distance upstream as he takes to row the same distance downstream, find the speed of the stream.
A. 2.5 km/h
B. 3.0 km/h
C. 3.33 km/h
D. 4.0 km/h
3.33 km/h
Explanation:
Let the speed of the stream be v\text{ km/h}.
Given that the speed of the boat in still water is u = 10\text{ km/h}.
The downstream speed (D) is 10 + v and the upstream speed (U) is 10 - v.
Since the distance is the same in both cases, the time taken is inversely proportional to the speed. We are given that the upstream time is twice the downstream time:
\text{Time}_{\text{upstream}} = 2 \times \text{Time}_{\text{downstream}} Since \text{Time} = \frac{\text{Distance}}{\text{Speed}}, we can write:
\frac{\text{Distance}}{10 - v} = 2 \times \frac{\text{Distance}}{10 + v}
Canceling out the distance from both sides:
\frac{1}{10 - v} = \frac{2}{10 + v} Cross-multiplying to solve for v:
10 + v = 2(10 - v) 10 + v = 20 - 2v
Rearranging the terms:
v + 2v = 20 - 10 3v = 10 \implies v = \frac{10}{3} \approx 3.33\text{ km/h}
Therefore, the correct option is C.
Let the speed of the stream be v\text{ km/h}.
Given that the speed of the boat in still water is u = 10\text{ km/h}.
The downstream speed (D) is 10 + v and the upstream speed (U) is 10 - v.
Since the distance is the same in both cases, the time taken is inversely proportional to the speed. We are given that the upstream time is twice the downstream time:
\text{Time}_{\text{upstream}} = 2 \times \text{Time}_{\text{downstream}} Since \text{Time} = \frac{\text{Distance}}{\text{Speed}}, we can write:
\frac{\text{Distance}}{10 - v} = 2 \times \frac{\text{Distance}}{10 + v}
Canceling out the distance from both sides:
\frac{1}{10 - v} = \frac{2}{10 + v} Cross-multiplying to solve for v:
10 + v = 2(10 - v) 10 + v = 20 - 2v
Rearranging the terms:
v + 2v = 20 - 10 3v = 10 \implies v = \frac{10}{3} \approx 3.33\text{ km/h}
Therefore, the correct option is C.
A man can row three times as fast downstream as he can upstream. If the speed of the stream is 3 km/h, find the speed of the man in still water.
A. 4.5 km/h
B. 6.0 km/h
C. 9.0 km/h
D. 12.0 km/h
6.0 km/h
Explanation:
Let the upstream speed of the man be U.
According to the problem, the downstream speed (D) is three times the upstream speed:
D = 3 \times U Let the speed of the man in still water be u and the speed of the stream be v = 3\text{ km/h}.
We know the expressions for downstream and upstream speeds in terms of u and v:
D = u + v U = u - v
Substitute these expressions into our ratio equation:
u + v = 3(u - v) Expand the right side of the equation:
u + v = 3u - 3v Rearranging the terms to group u and v:
3v + v = 3u - u 4v = 2u \implies u = 2v
Given that the speed of the stream v = 3\text{ km/h}, substitute this value to find u:
u = 2 \times 3 = 6.0\text{ km/h}
Therefore, the correct option is B.
Let the upstream speed of the man be U.
According to the problem, the downstream speed (D) is three times the upstream speed:
D = 3 \times U Let the speed of the man in still water be u and the speed of the stream be v = 3\text{ km/h}.
We know the expressions for downstream and upstream speeds in terms of u and v:
D = u + v U = u - v
Substitute these expressions into our ratio equation:
u + v = 3(u - v) Expand the right side of the equation:
u + v = 3u - 3v Rearranging the terms to group u and v:
3v + v = 3u - u 4v = 2u \implies u = 2v
Given that the speed of the stream v = 3\text{ km/h}, substitute this value to find u:
u = 2 \times 3 = 6.0\text{ km/h}
Therefore, the correct option is B.
A boat takes a total of 7 hours to travel 36 km downstream and 32 km upstream. It also takes 7 hours to travel 24 km downstream and 40 km upstream. Find the speed of the stream.
A. 1.5 km/h
B. 2.0 km/h
C. 2.5 km/h
D. 3.0 km/h
2.0 km/h
Explanation:
Let the downstream speed be D and the upstream speed be U.
From the first condition, we have the equation:
\frac{36}{D} + \frac{32}{U} = 7 \quad \text{--- (Equation 1)} From the second condition, we have the equation:
\frac{24}{D} + \frac{40}{U} = 7 \quad \text{--- (Equation 2)} Since both equations equal 7, we can set them equal to each other:
\frac{36}{D} + \frac{32}{U} = \frac{24}{D} + \frac{40}{U}
Rearranging the terms to group D and U:
\frac{36}{D} - \frac{24}{D} = \frac{40}{U} - \frac{32}{U} \frac{12}{D} = \frac{8}{U}
Simplifying the ratio:
\frac{D}{U} = \frac{12}{8} = \frac{3}{2} \implies D = \frac{3}{2}U Now, substitute D = \frac{3}{2}U into Equation 1:
\frac{36}{\frac{3}{2}U} + \frac{32}{U} = 7 \frac{24}{U} + \frac{32}{U} = 7 \frac{56}{U} = 7 \implies U = \frac{56}{7} = 8\text{ km/h}
Since U = 8, find D:
D = \frac{3}{2} \times 8 = 12\text{ km/h} The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{12 - 8}{2} = \frac{4}{2} = 2.0\text{ km/h}
Therefore, the correct option is B.
Let the downstream speed be D and the upstream speed be U.
From the first condition, we have the equation:
\frac{36}{D} + \frac{32}{U} = 7 \quad \text{--- (Equation 1)} From the second condition, we have the equation:
\frac{24}{D} + \frac{40}{U} = 7 \quad \text{--- (Equation 2)} Since both equations equal 7, we can set them equal to each other:
\frac{36}{D} + \frac{32}{U} = \frac{24}{D} + \frac{40}{U}
Rearranging the terms to group D and U:
\frac{36}{D} - \frac{24}{D} = \frac{40}{U} - \frac{32}{U} \frac{12}{D} = \frac{8}{U}
Simplifying the ratio:
\frac{D}{U} = \frac{12}{8} = \frac{3}{2} \implies D = \frac{3}{2}U Now, substitute D = \frac{3}{2}U into Equation 1:
\frac{36}{\frac{3}{2}U} + \frac{32}{U} = 7 \frac{24}{U} + \frac{32}{U} = 7 \frac{56}{U} = 7 \implies U = \frac{56}{7} = 8\text{ km/h}
Since U = 8, find D:
D = \frac{3}{2} \times 8 = 12\text{ km/h} The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{12 - 8}{2} = \frac{4}{2} = 2.0\text{ km/h}
Therefore, the correct option is B.
Two places A and B are 100 km apart on a river. Two boats start simultaneously from A and B moving towards each other. Boat 1 starts from A moving downstream and Boat 2 starts from B moving upstream. If their speeds in still water are equal at 12 km/h and the speed of the stream is 4 km/h, after how much time will they meet?
A. 3.125 hours
B. 4.167 hours
C. 5.000 hours
D. 6.250 hours
4.167 hours
Explanation:
Let the speed of the boat in still water be u = 12\text{ km/h} and the speed of the stream be v = 4\text{ km/h}.
For Boat 1 moving downstream from A towards B, its effective speed (S_1) is:
S_1 = u + v = 12 + 4 = 16\text{ km/h} For Boat 2 moving upstream from B towards A, its effective speed (S_2) is:
S_2 = u - v = 12 - 4 = 8\text{ km/h}
Since both boats are moving towards each other, their relative speed (S_{rel}) is the sum of their individual speeds:
S_{rel} = S_1 + S_2 = 16 + 8 = 24\text{ km/h}
The total distance between A and B is given as 100\text{ km}.
The time taken to meet (T) is calculated by dividing the total distance by their relative speed:
T = \frac{\text{Total Distance}}{S_{rel}} = \frac{100}{24} = \frac{25}{6} \approx 4.167\text{ hours}
Therefore, the correct option is B.
Let the speed of the boat in still water be u = 12\text{ km/h} and the speed of the stream be v = 4\text{ km/h}.
For Boat 1 moving downstream from A towards B, its effective speed (S_1) is:
S_1 = u + v = 12 + 4 = 16\text{ km/h} For Boat 2 moving upstream from B towards A, its effective speed (S_2) is:
S_2 = u - v = 12 - 4 = 8\text{ km/h}
Since both boats are moving towards each other, their relative speed (S_{rel}) is the sum of their individual speeds:
S_{rel} = S_1 + S_2 = 16 + 8 = 24\text{ km/h}
The total distance between A and B is given as 100\text{ km}.
The time taken to meet (T) is calculated by dividing the total distance by their relative speed:
T = \frac{\text{Total Distance}}{S_{rel}} = \frac{100}{24} = \frac{25}{6} \approx 4.167\text{ hours}
Therefore, the correct option is B.
A man rows downstream at 22 km/h and upstream at 14 km/h. Using mental math principles, what is the speed of the stream?
A. 3 km/h
B. 5 km/h
C. 4 km/h
D. 6 km/h
4 km/h
Explanation:
To find the speed of the stream (v) when downstream speed (D) and upstream speed (U) are given, apply the formula:
v = \frac{D - U}{2} Given:
Downstream speed, D = 22\text{ km/h}
Upstream speed, U = 14\text{ km/h}
Substitute these values into the formula:
v = \frac{22 - 14}{2} = \frac{8}{2} = 4\text{ km/h}
Therefore, the correct option is C.
To find the speed of the stream (v) when downstream speed (D) and upstream speed (U) are given, apply the formula:
v = \frac{D - U}{2} Given:
Downstream speed, D = 22\text{ km/h}
Upstream speed, U = 14\text{ km/h}
Substitute these values into the formula:
v = \frac{22 - 14}{2} = \frac{8}{2} = 4\text{ km/h}
Therefore, the correct option is C.
A boat’s downstream speed is 28 km/h and its upstream speed is 16 km/h. Quickly calculate the speed of the boat in still water.
A. 20 km/h
B. 22 km/h
C. 24 km/h
D. 26 km/h
22 km/h
Explanation:
To find the speed of the boat in still water (u) when downstream speed (D) and upstream speed (U) are known, use the average formula:
u = \frac{D + U}{2}
Given:
Downstream speed, D = 28\text{ km/h}
Upstream speed, U = 16\text{ km/h}
Substitute these values into the formula:
u = \frac{28 + 16}{2} = \frac{44}{2} = 22\text{ km/h}
Therefore, the correct option is B.
To find the speed of the boat in still water (u) when downstream speed (D) and upstream speed (U) are known, use the average formula:
u = \frac{D + U}{2}
Given:
Downstream speed, D = 28\text{ km/h}
Upstream speed, U = 16\text{ km/h}
Substitute these values into the formula:
u = \frac{28 + 16}{2} = \frac{44}{2} = 22\text{ km/h}
Therefore, the correct option is B.
A boat travels a distance upstream in 4 hours and returns the same distance downstream in 2.5 hours. If the speed of the stream is 1.5 km/h, what is the speed of the boat in still water?
A. 6.5 km/h
B. 7.5 km/h
C. 8.5 km/h
D. 9.5 km/h
6.5 km/h
Explanation:
Let the speed of the boat in still water be u\text{ km/h}.
Given that the speed of the stream is v = 1.5\text{ km/h}.
The upstream speed (U) is u - 1.5 and the downstream speed (D) is u + 1.5.
Since the distance traveled in both directions is equal, we can equate the product of speed and time:
\text{Distance} = U \times T_{upstream} = D \times T_{downstream} Substituting the given times and speeds:
(u - 1.5) \times 4 = (u + 1.5) \times 2.5 Expand both sides of the equation:
4u - 6 = 2.5u + 3.75
Rearranging the terms to isolate u:
4u - 2.5u = 3.75 + 6 1.5u = 9.75 Solving for u:
u = \frac{9.75}{1.5} = 6.5\text{ km/h}
Therefore, the correct option is A.
Let the speed of the boat in still water be u\text{ km/h}.
Given that the speed of the stream is v = 1.5\text{ km/h}.
The upstream speed (U) is u - 1.5 and the downstream speed (D) is u + 1.5.
Since the distance traveled in both directions is equal, we can equate the product of speed and time:
\text{Distance} = U \times T_{upstream} = D \times T_{downstream} Substituting the given times and speeds:
(u - 1.5) \times 4 = (u + 1.5) \times 2.5 Expand both sides of the equation:
4u - 6 = 2.5u + 3.75
Rearranging the terms to isolate u:
4u - 2.5u = 3.75 + 6 1.5u = 9.75 Solving for u:
u = \frac{9.75}{1.5} = 6.5\text{ km/h}
Therefore, the correct option is A.
A boat takes a total of 9 hours to travel 48 km upstream and 48 km downstream. It takes 8 hours to travel 32 km upstream and 64 km downstream. Find the speed of the boat in still water.
A. 8 km/h
B. 10 km/h
C. 12 km/h
D. 14 km/h
12 km/h
Explanation:
Let the upstream speed be U and the downstream speed be D.
From the first condition, the total time for 48 km upstream and 48 km downstream is 9 hours:
\frac{48}{U} + \frac{48}{D} = 9 \quad \text{--- (Equation 1)} From the second condition, the total time for 32 km upstream and 64 km downstream is 8 hours:
\frac{32}{U} + \frac{64}{D} = 8 \quad \text{--- (Equation 2)}
Let x = \frac{1}{U} and y = \frac{1}{D}. The equations become:
48x + 48y = 9 32x + 64y = 8 Multiply Equation 1 by 2 to align the coefficients of x:
96x + 96y = 18
Multiply Equation 2 by 3:
96x + 192y = 24 Subtract the first modified equation from the second modified equation:
(96x + 192y) - (96x + 96y) = 24 - 18 96y = 6 \implies y = \frac{6}{96} = \frac{1}{16}
Since y = \frac{1}{D}, we get D = 16\text{ km/h}.
Now, substitute y = \frac{1}{16} back into Equation 1:
48x + 48\left(\frac{1}{16}\right) = 9 48x + 3 = 9 \implies 48x = 6 \implies x = \frac{6}{48} = \frac{1}{8} Since x = \frac{1}{U}, we get U = 8\text{ km/h}.
The speed of the boat in still water (u) is calculated as the average of the downstream and upstream speeds:
u = \frac{D + U}{2} = \frac{16 + 8}{2} = \frac{24}{2} = 12\text{ km/h}
Therefore, the correct option is C.
Let the upstream speed be U and the downstream speed be D.
From the first condition, the total time for 48 km upstream and 48 km downstream is 9 hours:
\frac{48}{U} + \frac{48}{D} = 9 \quad \text{--- (Equation 1)} From the second condition, the total time for 32 km upstream and 64 km downstream is 8 hours:
\frac{32}{U} + \frac{64}{D} = 8 \quad \text{--- (Equation 2)}
Let x = \frac{1}{U} and y = \frac{1}{D}. The equations become:
48x + 48y = 9 32x + 64y = 8 Multiply Equation 1 by 2 to align the coefficients of x:
96x + 96y = 18
Multiply Equation 2 by 3:
96x + 192y = 24 Subtract the first modified equation from the second modified equation:
(96x + 192y) - (96x + 96y) = 24 - 18 96y = 6 \implies y = \frac{6}{96} = \frac{1}{16}
Since y = \frac{1}{D}, we get D = 16\text{ km/h}.
Now, substitute y = \frac{1}{16} back into Equation 1:
48x + 48\left(\frac{1}{16}\right) = 9 48x + 3 = 9 \implies 48x = 6 \implies x = \frac{6}{48} = \frac{1}{8} Since x = \frac{1}{U}, we get U = 8\text{ km/h}.
The speed of the boat in still water (u) is calculated as the average of the downstream and upstream speeds:
u = \frac{D + U}{2} = \frac{16 + 8}{2} = \frac{24}{2} = 12\text{ km/h}
Therefore, the correct option is C.
A motorboat can travel at 15 km/h in still water. It travels 36 km downstream and then returns 36 km upstream. If the speed of the current is 3 km/h, and the boat operator takes a 1-hour rest after completing the downstream journey before returning upstream, what is the total elapsed time for the entire trip?
A. 4.5 hours
B. 5.0 hours
C. 5.5 hours
D. 6.0 hours
6.0 hours
Explanation:
First, determine the downstream speed (D) and upstream speed (U) using the speed of the boat in still water (u = 15\text{ km/h}) and the speed of the stream (v = 3\text{ km/h}):
D = u + v = 15 + 3 = 18\text{ km/h} U = u - v = 15 - 3 = 12\text{ km/h}
Next, calculate the time taken to travel 36 km downstream (T_1):
T_1 = \frac{\text{Distance}}{\text{Downstream Speed}} = \frac{36}{18} = 2.0\text{ hours} Then, calculate the time taken to travel 36 km upstream (T_2):
T_2 = \frac{\text{Distance}}{\text{Upstream Speed}} = \frac{36}{12} = 3.0\text{ hours}
Account for the 1-hour rest interval between the two journeys (T_{rest} = 1.0\text{ hour}).
Calculate the total elapsed time by summing all components:
\text{Total Time} = T_1 + T_2 + T_{rest} = 2.0 + 3.0 + 1.0 = 6.0\text{ hours}
Therefore, the correct option is D.
First, determine the downstream speed (D) and upstream speed (U) using the speed of the boat in still water (u = 15\text{ km/h}) and the speed of the stream (v = 3\text{ km/h}):
D = u + v = 15 + 3 = 18\text{ km/h} U = u - v = 15 - 3 = 12\text{ km/h}
Next, calculate the time taken to travel 36 km downstream (T_1):
T_1 = \frac{\text{Distance}}{\text{Downstream Speed}} = \frac{36}{18} = 2.0\text{ hours} Then, calculate the time taken to travel 36 km upstream (T_2):
T_2 = \frac{\text{Distance}}{\text{Upstream Speed}} = \frac{36}{12} = 3.0\text{ hours}
Account for the 1-hour rest interval between the two journeys (T_{rest} = 1.0\text{ hour}).
Calculate the total elapsed time by summing all components:
\text{Total Time} = T_1 + T_2 + T_{rest} = 2.0 + 3.0 + 1.0 = 6.0\text{ hours}
Therefore, the correct option is D.
A boat covers a certain distance upstream in 9 hours. However, when the speed of the stream increases by 2 km/h, the boat takes 12 hours to cover the same distance upstream. Find the original speed of the stream, given that the speed of the boat in still water remains constant at 10 km/h.
A. 2 km/h
B. 4 km/h
C. 6 km/h
D. 8 km/h
2 km/h
Explanation:
Let the original speed of the stream be v\text{ km/h}.
Given that the speed of the boat in still water is u = 10\text{ km/h}.
The initial upstream speed (U_1) is:
U_1 = 10 - v The initial upstream time is T_1 = 9\text{ hours}. Thus, the distance (d) can be expressed as:
d = (10 - v) \times 9
When the speed of the stream increases by 2 km/h, the new stream speed becomes v + 2.
The new upstream speed (U_2) is:
U_2 = 10 - (v + 2) = 10 - v - 2 = 8 - v The new upstream time is T_2 = 12\text{ hours} for the same distance:
d = (8 - v) \times 12 Since the distance is constant in both cases, equate the two expressions for d:
(10 - v) \times 9 = (8 - v) \times 12
Divide both sides of the equation by 3 to simplify:
(10 - v) \times 3 = (8 - v) \times 4 Expand both sides:
30 - 3v = 32 - 4v Rearranging the terms to solve for v:
4v - 3v = 32 - 30 v = 2\text{ km/h}
Therefore, the correct option is A.
Let the original speed of the stream be v\text{ km/h}.
Given that the speed of the boat in still water is u = 10\text{ km/h}.
The initial upstream speed (U_1) is:
U_1 = 10 - v The initial upstream time is T_1 = 9\text{ hours}. Thus, the distance (d) can be expressed as:
d = (10 - v) \times 9
When the speed of the stream increases by 2 km/h, the new stream speed becomes v + 2.
The new upstream speed (U_2) is:
U_2 = 10 - (v + 2) = 10 - v - 2 = 8 - v The new upstream time is T_2 = 12\text{ hours} for the same distance:
d = (8 - v) \times 12 Since the distance is constant in both cases, equate the two expressions for d:
(10 - v) \times 9 = (8 - v) \times 12
Divide both sides of the equation by 3 to simplify:
(10 - v) \times 3 = (8 - v) \times 4 Expand both sides:
30 - 3v = 32 - 4v Rearranging the terms to solve for v:
4v - 3v = 32 - 30 v = 2\text{ km/h}
Therefore, the correct option is A.
A boat can travel a certain distance downstream in 3 hours and returns the same distance in 5 hours. If the speed of the stream is doubled, how long will it take the boat to travel the same distance downstream?
A. 1.5 hours
B. 2.0 hours
C. 2.5 hours
D. 3.0 hours
2.5 hours
Explanation:
Let the speed of the boat in still water be u and the original speed of the stream be v.
The downstream speed is D_1 = u + v and the upstream speed is U = u - v.
Since the distance traveled in both directions is equal, we can equate the product of speed and time:
(u + v) \times 3 = (u - v) \times 5 Expanding both sides:
3u + 3v = 5u - 5v Rearranging the terms to find the relationship between u and v:
5v + 3v = 5u - 3u \implies 8v = 2u \implies u = 4v
Now, when the speed of the stream is doubled, the new stream speed becomes v' = 2v.
The new downstream speed (D_2) becomes:
D_2 = u + v' = 4v + 2v = 6v From the original downstream condition, the total distance (d) is:
d = D_1 \times 3 = (4v + v) \times 3 = 5v \times 3 = 15v
Now, calculate the new downstream time (T_2) to cover the same distance d = 15v with the new downstream speed D_2 = 6v:
T_2 = \frac{d}{D_2} = \frac{15v}{6v} = \frac{15}{6} = 2.5\text{ hours}
Therefore, the correct option is C.
Let the speed of the boat in still water be u and the original speed of the stream be v.
The downstream speed is D_1 = u + v and the upstream speed is U = u - v.
Since the distance traveled in both directions is equal, we can equate the product of speed and time:
(u + v) \times 3 = (u - v) \times 5 Expanding both sides:
3u + 3v = 5u - 5v Rearranging the terms to find the relationship between u and v:
5v + 3v = 5u - 3u \implies 8v = 2u \implies u = 4v
Now, when the speed of the stream is doubled, the new stream speed becomes v' = 2v.
The new downstream speed (D_2) becomes:
D_2 = u + v' = 4v + 2v = 6v From the original downstream condition, the total distance (d) is:
d = D_1 \times 3 = (4v + v) \times 3 = 5v \times 3 = 15v
Now, calculate the new downstream time (T_2) to cover the same distance d = 15v with the new downstream speed D_2 = 6v:
T_2 = \frac{d}{D_2} = \frac{15v}{6v} = \frac{15}{6} = 2.5\text{ hours}
Therefore, the correct option is C.
A man rows a boat 24 km down a river and then back again. The total time taken for the round trip is 10 hours. He finds that he can row 3 km downstream in the same time as 2 km upstream. Find the speed of the stream.
A. 0.5 km/h
B. 1.0 km/h
C. 1.5 km/h
D. 2.0 km/h
1.0 km/h
Explanation:
Let the downstream speed be D and the upstream speed be U.
We are given that the time taken to row 3 km downstream is equal to the time taken to row 2 km upstream:
\frac{3}{D} = \frac{2}{U} \implies \frac{D}{U} = \frac{3}{2} \implies D = \frac{3}{2}U The total time taken for the round trip of 24 km downstream and 24 km upstream is 10 hours:
\frac{24}{D} + \frac{24}{U} = 10
Substitute D = \frac{3}{2}U into the total time equation:
\frac{24}{\frac{3}{2}U} + \frac{24}{U} = 10 \frac{16}{U} + \frac{24}{U} = 10 \frac{40}{U} = 10 \implies U = \frac{40}{10} = 4\text{ km/h}
Now, find the downstream speed D:
D = \frac{3}{2} \times 4 = 6\text{ km/h}
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{6 - 4}{2} = \frac{2}{2} = 1.0\text{ km/h}
Therefore, the correct option is B.
Let the downstream speed be D and the upstream speed be U.
We are given that the time taken to row 3 km downstream is equal to the time taken to row 2 km upstream:
\frac{3}{D} = \frac{2}{U} \implies \frac{D}{U} = \frac{3}{2} \implies D = \frac{3}{2}U The total time taken for the round trip of 24 km downstream and 24 km upstream is 10 hours:
\frac{24}{D} + \frac{24}{U} = 10
Substitute D = \frac{3}{2}U into the total time equation:
\frac{24}{\frac{3}{2}U} + \frac{24}{U} = 10 \frac{16}{U} + \frac{24}{U} = 10 \frac{40}{U} = 10 \implies U = \frac{40}{10} = 4\text{ km/h}
Now, find the downstream speed D:
D = \frac{3}{2} \times 4 = 6\text{ km/h}
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{6 - 4}{2} = \frac{2}{2} = 1.0\text{ km/h}
Therefore, the correct option is B.
A boat travels 36 km downstream and 24 km upstream in 6 hours. It can also travel 24 km downstream and 36 km upstream in 6.5 hours. Find the speed of the boat in still water.
A. 10 km/h
B. 12 km/h
C. 14 km/h
D. 16 km/h
10 km/h
Explanation:
Let the downstream speed be D and the upstream speed be U.
From the first condition, the total time for 36 km downstream and 24 km upstream is 6 hours:
\frac{36}{D} + \frac{24}{U} = 6 \quad \text{--- (Equation 1)} From the second condition, the total time for 24 km downstream and 36 km upstream is 6.5 hours:
\frac{24}{D} + \frac{36}{U} = 6.5 \quad \text{--- (Equation 2)}
Let x = \frac{1}{D} and y = \frac{1}{U}. The equations become:
36x + 24y = 6 24x + 36y = 6.5
Adding both equations:
60x + 60y = 12.5 \implies x + y = \frac{12.5}{60} = \frac{25}{120} = \frac{5}{24} Subtracting Equation 1 from Equation 2:
-12x + 12y = 0.5 \implies y - x = \frac{0.5}{12} = \frac{1}{24}
Now, add the sum equation (x + y = \frac{5}{24}) and the difference equation (y - x = \frac{1}{24}) to find y:
2y = \frac{5}{24} + \frac{1}{24} = \frac{6}{24} = \frac{1}{4} \implies y = \frac{1}{8} Since y = \frac{1}{U}, we get U = 8\text{ km/h}.
Substitute y = \frac{1}{8} into the sum equation:
x + \frac{1}{8} = \frac{5}{24} \implies x = \frac{5}{24} - \frac{3}{24} = \frac{2}{24} = \frac{1}{12}
Since x = \frac{1}{D}, we get D = 12\text{ km/h}.
The speed of the boat in still water (u) is calculated as the average of the downstream and upstream speeds:
u = \frac{D + U}{2} = \frac{12 + 8}{2} = \frac{20}{2} = 10\text{ km/h}
Therefore, the correct option is A.
Let the downstream speed be D and the upstream speed be U.
From the first condition, the total time for 36 km downstream and 24 km upstream is 6 hours:
\frac{36}{D} + \frac{24}{U} = 6 \quad \text{--- (Equation 1)} From the second condition, the total time for 24 km downstream and 36 km upstream is 6.5 hours:
\frac{24}{D} + \frac{36}{U} = 6.5 \quad \text{--- (Equation 2)}
Let x = \frac{1}{D} and y = \frac{1}{U}. The equations become:
36x + 24y = 6 24x + 36y = 6.5
Adding both equations:
60x + 60y = 12.5 \implies x + y = \frac{12.5}{60} = \frac{25}{120} = \frac{5}{24} Subtracting Equation 1 from Equation 2:
-12x + 12y = 0.5 \implies y - x = \frac{0.5}{12} = \frac{1}{24}
Now, add the sum equation (x + y = \frac{5}{24}) and the difference equation (y - x = \frac{1}{24}) to find y:
2y = \frac{5}{24} + \frac{1}{24} = \frac{6}{24} = \frac{1}{4} \implies y = \frac{1}{8} Since y = \frac{1}{U}, we get U = 8\text{ km/h}.
Substitute y = \frac{1}{8} into the sum equation:
x + \frac{1}{8} = \frac{5}{24} \implies x = \frac{5}{24} - \frac{3}{24} = \frac{2}{24} = \frac{1}{12}
Since x = \frac{1}{D}, we get D = 12\text{ km/h}.
The speed of the boat in still water (u) is calculated as the average of the downstream and upstream speeds:
u = \frac{D + U}{2} = \frac{12 + 8}{2} = \frac{20}{2} = 10\text{ km/h}
Therefore, the correct option is A.
The speed of a boat in still water is 15 km/h and the speed of the stream is 3 km/h. If the boat travels a certain distance upstream and the same distance downstream, what is the ratio of the time taken upstream to the time taken downstream?
A. 3 : 2
B. 2 : 1
C. 4 : 3
D. 5 : 3
3 : 2
Explanation:
First, determine the upstream speed (U) and downstream speed (D) using the given speed of the boat in still water (u = 15\text{ km/h}) and the speed of the stream (v = 3\text{ km/h}):
U = u - v = 15 - 3 = 12\text{ km/h} D = u + v = 15 + 3 = 18\text{ km/h}
Let the constant distance in both directions be d.
The time taken to travel upstream (T_{upstream}) and downstream (T_{downstream}) are given by:
T_{upstream} = \frac{d}{12}, \quad T_{downstream} = \frac{d}{18} Now, find the ratio of the time taken upstream to the time taken downstream:
\frac{T_{upstream}}{T_{downstream}} = \frac{\frac{d}{12}}{\frac{d}{18}} = \frac{18}{12} = \frac{3}{2}
Therefore, the correct option is A.
First, determine the upstream speed (U) and downstream speed (D) using the given speed of the boat in still water (u = 15\text{ km/h}) and the speed of the stream (v = 3\text{ km/h}):
U = u - v = 15 - 3 = 12\text{ km/h} D = u + v = 15 + 3 = 18\text{ km/h}
Let the constant distance in both directions be d.
The time taken to travel upstream (T_{upstream}) and downstream (T_{downstream}) are given by:
T_{upstream} = \frac{d}{12}, \quad T_{downstream} = \frac{d}{18} Now, find the ratio of the time taken upstream to the time taken downstream:
\frac{T_{upstream}}{T_{downstream}} = \frac{\frac{d}{12}}{\frac{d}{18}} = \frac{18}{12} = \frac{3}{2}
Therefore, the correct option is A.
A motorboat travels a distance of 45 km downstream and then returns the same distance upstream. If the entire journey takes 8 hours, and the speed of the stream is 3 km/h, what is the speed of the motorboat in still water?
A. 10 km/h
B. 12 km/h
C. 15 km/h
D. 18 km/h
12 km/h
Explanation:
Let the speed of the boat in still water be u\text{ km/h}.
Given that the speed of the stream is v = 3\text{ km/h}.
The downstream speed (D) is u + 3 and the upstream speed (U) is u - 3.
The time taken to travel downstream for 45 km is \frac{45}{u + 3} hours, and the time taken to return upstream for 45 km is \frac{45}{u - 3} hours.
The total time given is 8 hours, so we can set up the equation:
\frac{45}{u + 3} + \frac{45}{u - 3} = 8 Taking out 45 as a common factor and finding a common denominator for the fractions:
45 \left( \frac{(u - 3) + (u + 3)}{(u + 3)(u - 3)} \right) = 8 45 \left( \frac{2u}{u^2 - 9} \right) = 8 \frac{90u}{u^2 - 9} = 8
Cross-multiplying to solve for u:
90u = 8(u^2 - 9) 90u = 8u^2 - 72
Rearranging the quadratic equation into standard form:
8u^2 - 90u - 72 = 0 Dividing the entire equation by 2 to simplify:
4u^2 - 45u - 36 = 0 Factoring the quadratic equation by splitting the middle term:
4u^2 - 48u + 3u - 36 = 0 4u(u - 12) + 3(u - 12) = 0 (4u + 3)(u - 12) = 0
Since speed cannot be negative, u = 12\text{ km/h}.
Therefore, the correct option is B.
Let the speed of the boat in still water be u\text{ km/h}.
Given that the speed of the stream is v = 3\text{ km/h}.
The downstream speed (D) is u + 3 and the upstream speed (U) is u - 3.
The time taken to travel downstream for 45 km is \frac{45}{u + 3} hours, and the time taken to return upstream for 45 km is \frac{45}{u - 3} hours.
The total time given is 8 hours, so we can set up the equation:
\frac{45}{u + 3} + \frac{45}{u - 3} = 8 Taking out 45 as a common factor and finding a common denominator for the fractions:
45 \left( \frac{(u - 3) + (u + 3)}{(u + 3)(u - 3)} \right) = 8 45 \left( \frac{2u}{u^2 - 9} \right) = 8 \frac{90u}{u^2 - 9} = 8
Cross-multiplying to solve for u:
90u = 8(u^2 - 9) 90u = 8u^2 - 72
Rearranging the quadratic equation into standard form:
8u^2 - 90u - 72 = 0 Dividing the entire equation by 2 to simplify:
4u^2 - 45u - 36 = 0 Factoring the quadratic equation by splitting the middle term:
4u^2 - 48u + 3u - 36 = 0 4u(u - 12) + 3(u - 12) = 0 (4u + 3)(u - 12) = 0
Since speed cannot be negative, u = 12\text{ km/h}.
Therefore, the correct option is B.
A boat travels 54 km downstream and 40 km upstream in 7 hours. It can also travel 36 km downstream and 60 km upstream in 8 hours. Find the speed of the stream.
A. 1.5 km/h
B. 2.0 km/h
C. 2.5 km/h
D. 4.0 km/h
4.0 km/h
Explanation:
Let the downstream speed be D and the upstream speed be U.
From the first condition, the total time for 54 km downstream and 40 km upstream is 7 hours:
\frac{54}{D} + \frac{40}{U} = 7 \quad \text{--- (Equation 1)} From the second condition, the total time for 36 km downstream and 60 km upstream is 8 hours:
\frac{36}{D} + \frac{60}{U} = 8 \quad \text{--- (Equation 2)}
Let x = \frac{1}{D} and y = \frac{1}{U}. The equations become:
54x + 40y = 7 36x + 60y = 8 Divide Equation 2 by 6 to simplify:
6x + 10y = \frac{4}{3}
Multiply this simplified equation by 9 to align the coefficient of x with Equation 1:
54x + 90y = 12 Subtract Equation 1 from this new equation:
(54x + 90y) - (54x + 40y) = 12 - 7 50y = 5 \implies y = \frac{5}{50} = \frac{1}{10}
Since y = \frac{1}{U}, we get U = 10\text{ km/h}.
Substitute y = \frac{1}{10} into Equation 1:
54x + 40\left(\frac{1}{10}\right) = 7 54x + 4 = 7 \implies 54x = 3 \implies x = \frac{3}{54} = \frac{1}{18}
Since x = \frac{1}{D}, we get D = 18\text{ km/h}.
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{18 - 10}{2} = \frac{8}{2} = 4.0\text{ km/h}
Therefore, the correct option is D.
Let the downstream speed be D and the upstream speed be U.
From the first condition, the total time for 54 km downstream and 40 km upstream is 7 hours:
\frac{54}{D} + \frac{40}{U} = 7 \quad \text{--- (Equation 1)} From the second condition, the total time for 36 km downstream and 60 km upstream is 8 hours:
\frac{36}{D} + \frac{60}{U} = 8 \quad \text{--- (Equation 2)}
Let x = \frac{1}{D} and y = \frac{1}{U}. The equations become:
54x + 40y = 7 36x + 60y = 8 Divide Equation 2 by 6 to simplify:
6x + 10y = \frac{4}{3}
Multiply this simplified equation by 9 to align the coefficient of x with Equation 1:
54x + 90y = 12 Subtract Equation 1 from this new equation:
(54x + 90y) - (54x + 40y) = 12 - 7 50y = 5 \implies y = \frac{5}{50} = \frac{1}{10}
Since y = \frac{1}{U}, we get U = 10\text{ km/h}.
Substitute y = \frac{1}{10} into Equation 1:
54x + 40\left(\frac{1}{10}\right) = 7 54x + 4 = 7 \implies 54x = 3 \implies x = \frac{3}{54} = \frac{1}{18}
Since x = \frac{1}{D}, we get D = 18\text{ km/h}.
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{18 - 10}{2} = \frac{8}{2} = 4.0\text{ km/h}
Therefore, the correct option is D.
A boat travels 40 km downstream in 4 hours and 30 km upstream in 6 hours. If the speed of both the boat in still water and the stream are increased by 2 km/h each, how long will it take the boat to travel 49 km downstream?
A. 3.0 hours
B. 3.5 hours
C. 4.0 hours
D. 4.5 hours
3.5 hours
Explanation:
First, calculate the original downstream speed (D_1) and upstream speed (U_1) from the given data:
D_1 = \frac{\text{Distance downstream}}{\text{Time downstream}} = \frac{40}{4} = 10\text{ km/h} U_1 = \frac{\text{Distance upstream}}{\text{Time upstream}} = \frac{30}{6} = 5\text{ km/h}
Next, determine the original speed of the boat in still water (u) and the original speed of the stream (v) using the standard formulas:
u = \frac{D_1 + U_1}{2} = \frac{10 + 5}{2} = 7.5\text{ km/h} v = \frac{D_1 - U_1}{2} = \frac{10 - 5}{2} = 2.5\text{ km/h}
When the speed of both the boat and the stream are increased by 2 km/h each, the new speed of the boat in still water (u') and new stream speed (v') become:
u' = 7.5 + 2 = 9.5\text{ km/h} v' = 2.5 + 2 = 4.5\text{ km/h}
Calculate the new downstream speed (D_2):
D_2 = u' + v' = 9.5 + 4.5 = 14.0\text{ km/h} Finally, calculate the time taken to travel 49 km downstream with the new downstream speed:
\text{Time} = \frac{\text{New Distance}}{D_2} = \frac{49}{14} = 3.5\text{ hours}
Therefore, the correct option is B.
First, calculate the original downstream speed (D_1) and upstream speed (U_1) from the given data:
D_1 = \frac{\text{Distance downstream}}{\text{Time downstream}} = \frac{40}{4} = 10\text{ km/h} U_1 = \frac{\text{Distance upstream}}{\text{Time upstream}} = \frac{30}{6} = 5\text{ km/h}
Next, determine the original speed of the boat in still water (u) and the original speed of the stream (v) using the standard formulas:
u = \frac{D_1 + U_1}{2} = \frac{10 + 5}{2} = 7.5\text{ km/h} v = \frac{D_1 - U_1}{2} = \frac{10 - 5}{2} = 2.5\text{ km/h}
When the speed of both the boat and the stream are increased by 2 km/h each, the new speed of the boat in still water (u') and new stream speed (v') become:
u' = 7.5 + 2 = 9.5\text{ km/h} v' = 2.5 + 2 = 4.5\text{ km/h}
Calculate the new downstream speed (D_2):
D_2 = u' + v' = 9.5 + 4.5 = 14.0\text{ km/h} Finally, calculate the time taken to travel 49 km downstream with the new downstream speed:
\text{Time} = \frac{\text{New Distance}}{D_2} = \frac{49}{14} = 3.5\text{ hours}
Therefore, the correct option is B.
A boat can travel at 12 km/h in still water. If the boat takes 3 hours more to travel 48 km upstream than to travel the same distance downstream, what is the speed of the stream?
A. 2 km/h
B. 3 km/h
C. 4 km/h
D. 6 km/h
4 km/h
Explanation:
Let the speed of the stream be v\text{ km/h}.
Given that the speed of the boat in still water is u = 12\text{ km/h}.
The downstream speed (D) is 12 + v and the upstream speed (U) is 12 - v.
The distance in both directions is 48\text{ km}.
The time taken to travel upstream is \frac{48}{12 - v} hours, and the time taken to travel downstream is \frac{48}{12 + v} hours.
According to the problem, the upstream time exceeds the downstream time by 3 hours:
\frac{48}{12 - v} - \frac{48}{12 + v} = 3 Factor out 48 and find a common denominator:
48 \left( \frac{(12 + v) - (12 - v)}{(12 - v)(12 + v)} \right) = 3 48 \left( \frac{2v}{144 - v^2} \right) = 3 \frac{96v}{144 - v^2} = 3
Divide the entire equation by 3 to simplify:
\frac{32v}{144 - v^2} = 1 Cross-multiply to solve for v:
32v = 144 - v^2 Rearranging into standard quadratic equation form:
v^2 + 32v - 144 = 0
Factoring the quadratic equation:
v^2 + 36v - 4v - 144 = 0 v(v + 36) - 4(v + 36) = 0 (v - 4)(v + 36) = 0 Since speed cannot be negative, v = 4\text{ km/h}.
Therefore, the correct option is C.
Let the speed of the stream be v\text{ km/h}.
Given that the speed of the boat in still water is u = 12\text{ km/h}.
The downstream speed (D) is 12 + v and the upstream speed (U) is 12 - v.
The distance in both directions is 48\text{ km}.
The time taken to travel upstream is \frac{48}{12 - v} hours, and the time taken to travel downstream is \frac{48}{12 + v} hours.
According to the problem, the upstream time exceeds the downstream time by 3 hours:
\frac{48}{12 - v} - \frac{48}{12 + v} = 3 Factor out 48 and find a common denominator:
48 \left( \frac{(12 + v) - (12 - v)}{(12 - v)(12 + v)} \right) = 3 48 \left( \frac{2v}{144 - v^2} \right) = 3 \frac{96v}{144 - v^2} = 3
Divide the entire equation by 3 to simplify:
\frac{32v}{144 - v^2} = 1 Cross-multiply to solve for v:
32v = 144 - v^2 Rearranging into standard quadratic equation form:
v^2 + 32v - 144 = 0
Factoring the quadratic equation:
v^2 + 36v - 4v - 144 = 0 v(v + 36) - 4(v + 36) = 0 (v - 4)(v + 36) = 0 Since speed cannot be negative, v = 4\text{ km/h}.
Therefore, the correct option is C.
A boat travels 48 km downstream and 40 km upstream in 7 hours. It can also travel 32 km downstream and 60 km upstream in 8 hours. Find the speed of the stream.
A. 2.0 km/h
B. 2.5 km/h
C. 3.0 km/h
D. 3.5 km/h
3.0 km/h
Explanation:
Let the downstream speed be D and the upstream speed be U.
From the first condition, the total time for 48 km downstream and 40 km upstream is 7 hours:
\frac{48}{D} + \frac{40}{U} = 7 \quad \text{--- (Equation 1)} From the second condition, the total time for 32 km downstream and 60 km upstream is 8 hours:
\frac{32}{D} + \frac{60}{U} = 8 \quad \text{--- (Equation 2)}
Let x = \frac{1}{D} and y = \frac{1}{U}. The equations become:
48x + 40y = 7 32x + 60y = 8 Simplify Equation 2 by dividing the entire equation by 4:
8x + 15y = 2
Multiply this simplified equation by 6 to align the coefficient of x with Equation 1:
48x + 90y = 12 Subtract Equation 1 from this new equation:
(48x + 90y) - (48x + 40y) = 12 - 7 50y = 5 \implies y = \frac{5}{50} = \frac{1}{10}
Since y = \frac{1}{U}, we get U = 10\text{ km/h}.
Substitute y = \frac{1}{10} into Equation 1:
48x + 40\left(\frac{1}{10}\right) = 7 48x + 4 = 7 \implies 48x = 3 \implies x = \frac{3}{48} = \frac{1}{16}
Since x = \frac{1}{D}, we get D = 16\text{ km/h}.
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{16 - 10}{2} = \frac{6}{2} = 3.0\text{ km/h}
Therefore, the correct option is C.
Let the downstream speed be D and the upstream speed be U.
From the first condition, the total time for 48 km downstream and 40 km upstream is 7 hours:
\frac{48}{D} + \frac{40}{U} = 7 \quad \text{--- (Equation 1)} From the second condition, the total time for 32 km downstream and 60 km upstream is 8 hours:
\frac{32}{D} + \frac{60}{U} = 8 \quad \text{--- (Equation 2)}
Let x = \frac{1}{D} and y = \frac{1}{U}. The equations become:
48x + 40y = 7 32x + 60y = 8 Simplify Equation 2 by dividing the entire equation by 4:
8x + 15y = 2
Multiply this simplified equation by 6 to align the coefficient of x with Equation 1:
48x + 90y = 12 Subtract Equation 1 from this new equation:
(48x + 90y) - (48x + 40y) = 12 - 7 50y = 5 \implies y = \frac{5}{50} = \frac{1}{10}
Since y = \frac{1}{U}, we get U = 10\text{ km/h}.
Substitute y = \frac{1}{10} into Equation 1:
48x + 40\left(\frac{1}{10}\right) = 7 48x + 4 = 7 \implies 48x = 3 \implies x = \frac{3}{48} = \frac{1}{16}
Since x = \frac{1}{D}, we get D = 16\text{ km/h}.
The speed of the stream (v) is calculated as half of the difference between the downstream and upstream speeds:
v = \frac{D - U}{2} = \frac{16 - 10}{2} = \frac{6}{2} = 3.0\text{ km/h}
Therefore, the correct option is C.
A boat travels from point A to point B downstream and returns from B to A upstream. The speed of the boat in still water is 12 km/h and the speed of the stream is 4 km/h. If the total time taken for the round trip is 6 hours, find the distance between point A and point B.
A. 24 km
B. 28 km
C. 32 km
D. 36 km
32 km
Explanation:
First, determine the downstream speed (D) and upstream speed (U) using the speed of the boat in still water (u = 12\text{ km/h}) and the speed of the stream (v = 4\text{ km/h}):
D = u + v = 12 + 4 = 16\text{ km/h} U = u - v = 12 - 4 = 8\text{ km/h}
Let the distance between point A and point B be d\text{ km}.
The time taken to travel downstream from A to B is \frac{d}{16} hours, and the time taken to return upstream from B to A is \frac{d}{8} hours.
Given that the total time for the round trip is 6 hours, we can set up the equation:
\frac{d}{16} + \frac{d}{8} = 6 Finding a common denominator (16) to combine the fractions:
\frac{d}{16} + \frac{2d}{16} = 6 \frac{3d}{16} = 6
Cross-multiplying to solve for d:
3d = 6 \times 16 3d = 96 \implies d = \frac{96}{3} = 32\text{ km}
Therefore, the correct option is C.
First, determine the downstream speed (D) and upstream speed (U) using the speed of the boat in still water (u = 12\text{ km/h}) and the speed of the stream (v = 4\text{ km/h}):
D = u + v = 12 + 4 = 16\text{ km/h} U = u - v = 12 - 4 = 8\text{ km/h}
Let the distance between point A and point B be d\text{ km}.
The time taken to travel downstream from A to B is \frac{d}{16} hours, and the time taken to return upstream from B to A is \frac{d}{8} hours.
Given that the total time for the round trip is 6 hours, we can set up the equation:
\frac{d}{16} + \frac{d}{8} = 6 Finding a common denominator (16) to combine the fractions:
\frac{d}{16} + \frac{2d}{16} = 6 \frac{3d}{16} = 6
Cross-multiplying to solve for d:
3d = 6 \times 16 3d = 96 \implies d = \frac{96}{3} = 32\text{ km}
Therefore, the correct option is C.