Boats & Streams Questions and Answers

Q: The speeds of a boat in still water and of the stream are in the ratio 4 : 1. The boat covers 30 km downstream in 3 hours. How long will it take to cover the same distance upstream ?

Correct answer: 5 hoursLet the speeds be \(4k\) and \(k\), so downstream is \(5k\) and upstream \(3k\).\[5k = \frac{30}{3} = 10 \quad\Longrightarrow\quad k = 2\]The upstream speed is \(3k = 6\) km/hr, so\[\text{Time} = \frac{30}{6} = 5 \text{ hours}\]

Q: A man rows at 5 km/hr in still water and at 3 km/hr against the current. What is his speed along the current ?

Correct answer: 7 km/hrThe upstream speed is \(b - s\), so\[5 - s = 3 \quad\Longrightarrow\quad s = 2 \text{ km/hr}\]\[\text{Downstream speed} = b + s = 5 + 2 = 7 \text{ km/hr}\]The still-water speed always lies exactly midway between the two.

Q: A man can swim at 3 km/hr in still water. If the stream flows at 2 km/hr, how long will he take to swim to a place 10 km upstream and back ?

Correct answer: 12 hours\[\text{Upstream speed} = 3 - 2 = 1 \text{ km/hr}, \qquad \text{Downstream speed} = 3 + 2 = 5 \text{ km/hr}\]\[\text{Total time} = \frac{10}{1} + \frac{10}{5} = 10 + 2 = 12 \text{ hours}\]Almost the whole journey is spent on the upstream leg, because the swimmer barely outpaces the current.

Q: A boat covers a certain distance downstream in 1 hour and returns in 1 hour 30 minutes. If the stream flows at 3 km/hr, what is the speed of the boat in still water ?

Correct answer: 15 km/hrThe distance is the same each way, so speed multiplied by time must match:\[(b + 3) \times 1 = (b - 3) \times 1.5\]\[b + 3 = 1.5b - 4.5 \quad\Longrightarrow\quad 7.5 = 0.5b \quad\Longrightarrow\quad b = 15 \text{ km/hr}\]The distance is then \(18 \times 1 = 18\) km, and the return at 12 km/hr indeed takes 1.5 hours.

Q: A boat covers 30 km upstream and 44 km downstream in 10 hours, and 40 km upstream and 55 km downstream in 13 hours. What is the speed of the stream ?

Correct answer: 3 km/hrWrite \(p = \tfrac{1}{u}\) and \(q = \tfrac{1}{d}\), which makes both conditions linear.\[30p + 44q = 10, \qquad 40p + 55q = 13\]Multiplying the first by 4 and the second by 3 and subtracting gives \(11q = 1\), so \(d = 11\) km/hr. Substituting back, \(30p = 6\), so \(u = 5\) km/hr.\[s = \frac{11 - 5}{2} = 3 \text{ km/hr}\]

Q: A boat whose speed in still water is 10 km/hr travels 30 km downstream and returns, the stream running at 5 km/hr. What is its average speed for the whole journey ?

Correct answer: 7.5 km/hr\[\text{Downstream time} = \frac{30}{15} = 2 \text{ h}, \qquad \text{Upstream time} = \frac{30}{5} = 6 \text{ h}\]\[\text{Average speed} = \frac{\text{total distance}}{\text{total time}} = \frac{60}{8} = 7.5 \text{ km/hr}\]Note that this is well below the still-water speed of 10 km/hr: the slow return leg occupies most of the time, so it dominates the average.

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121.

A boat has a speed of 14 kmph in standing water. The stream has a speed of 1.2 kmph. What time does the boat take to go to a distance of 4864 km and come back? (hours)

A.700
B.350
C.1000
D.4200

Answer: A

A boat travels at the speed of 14 km/hr in still water.

=> Speed of stream = 1.2 kmph

=> Speed of boat downstream becomes 14 + 1.2 = 15.2 kmph

=> Speed of boat upstream becomes 14 - 1.2 = 12.8 kmph

Total time taken in the journey = Journey downstream + Journey upstream

Therefore, 4864(1/12.8 + 1/15.2) = 700 hours

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122.

The speeds of a boat in still water and of the stream are in the ratio 4 : 1. The boat covers 30 km downstream in 3 hours. How long will it take to cover the same distance upstream ?

A.4 hours
B.5 hours
C.6 hours
D.7.5 hours

Answer: C

Correct answer: 5 hours

Let the speeds be \(4k\) and \(k\), so downstream is \(5k\) and upstream \(3k\).

\[5k = \frac{30}{3} = 10 \quad\Longrightarrow\quad k = 2\]

The upstream speed is \(3k = 6\) km/hr, so

\[\text{Time} = \frac{30}{6} = 5 \text{ hours}\]

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123.

A man rows at 5 km/hr in still water and at 3 km/hr against the current. What is his speed along the current ?

A.6 km/hr
B.7 km/hr
C.8 km/hr
D.9 km/hr

Answer: C

Correct answer: 7 km/hr

The upstream speed is \(b - s\), so

\[5 - s = 3 \quad\Longrightarrow\quad s = 2 \text{ km/hr}\]

\[\text{Downstream speed} = b + s = 5 + 2 = 7 \text{ km/hr}\]

The still-water speed always lies exactly midway between the two.

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124.

A man can swim at 3 km/hr in still water. If the stream flows at 2 km/hr, how long will he take to swim to a place 10 km upstream and back ?

A.9.25 hours
B.10 hours
C.11 hours
D.12 hours

Answer: C

Correct answer: 12 hours

\[\text{Upstream speed} = 3 - 2 = 1 \text{ km/hr}, \qquad \text{Downstream speed} = 3 + 2 = 5 \text{ km/hr}\]

\[\text{Total time} = \frac{10}{1} + \frac{10}{5} = 10 + 2 = 12 \text{ hours}\]

Almost the whole journey is spent on the upstream leg, because the swimmer barely outpaces the current.

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125.

A boat covers a certain distance downstream in 1 hour and returns in 1 hour 30 minutes. If the stream flows at 3 km/hr, what is the speed of the boat in still water ?

A.12 km/hr
B.15 km/hr
C.18 km/hr
D.20 km/hr

Answer: C

Correct answer: 15 km/hr

The distance is the same each way, so speed multiplied by time must match:

\[(b + 3) \times 1 = (b - 3) \times 1.5\]

\[b + 3 = 1.5b - 4.5 \quad\Longrightarrow\quad 7.5 = 0.5b \quad\Longrightarrow\quad b = 15 \text{ km/hr}\]

The distance is then \(18 \times 1 = 18\) km, and the return at 12 km/hr indeed takes 1.5 hours.

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126.

A motorboat, whose speed in 15 km/hr in still water goes 30 km downstream and comes back in a total of 4 hours 30 minutes. The speed of the stream (in km/hr) is:

Answer: B

Write \( b \) for the boat's speed in still water and \( r \) for the stream's. Then

\[ \text{downstream} = b + r, \qquad \text{upstream} = b - r \]

\[ 4 \text{ h } 30 \text{ min} = 4.5 \text{ h} \]

\[ \frac{30}{15 + r} + \frac{30}{15 - r} = 4.5 \]

Combining the fractions over the common denominator \( (15+r)(15-r) = 225 - r^{2} \):

\[ \frac{30(15 - r) + 30(15 + r)}{225 - r^{2}} = \frac{900}{225 - r^{2}} = 4.5 \]

\[ 225 - r^{2} = 200 \quad\Longrightarrow\quad r^{2} = 25 \quad\Longrightarrow\quad r = 5 \text{ km/h} \]

The \( r \) terms in the numerator cancel, which is what turns a messy equation into a simple one.

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127.

A boat covers 30 km upstream and 44 km downstream in 10 hours, and 40 km upstream and 55 km downstream in 13 hours. What is the speed of the stream ?

A.2 km/hr
B.3 km/hr
C.4 km/hr
D.5 km/hr

Answer: C

Correct answer: 3 km/hr

Write \(p = \tfrac{1}{u}\) and \(q = \tfrac{1}{d}\), which makes both conditions linear.

\[30p + 44q = 10, \qquad 40p + 55q = 13\]

Multiplying the first by 4 and the second by 3 and subtracting gives \(11q = 1\), so \(d = 11\) km/hr. Substituting back, \(30p = 6\), so \(u = 5\) km/hr.

\[s = \frac{11 - 5}{2} = 3 \text{ km/hr}\]

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128.

A boat whose speed in still water is 10 km/hr travels 30 km downstream and returns, the stream running at 5 km/hr. What is its average speed for the whole journey ?

A.6 km/hr
B.7.5 km/hr
C.8 km/hr
D.10 km/hr

Answer: B

Correct answer: 7.5 km/hr

\[\text{Downstream time} = \frac{30}{15} = 2 \text{ h}, \qquad \text{Upstream time} = \frac{30}{5} = 6 \text{ h}\]

\[\text{Average speed} = \frac{\text{total distance}}{\text{total time}} = \frac{60}{8} = 7.5 \text{ km/hr}\]

Note that this is well below the still-water speed of 10 km/hr: the slow return leg occupies most of the time, so it dominates the average.

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