Boats & Streams Questions and Answers

Answer: C

With \( b \) the still-water speed and \( r \) the stream's,

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

Still water gives \( b = 6 \) km/hr. Taking three times as long against the current means going at one third the speed:

\[ b - r = \frac{6}{3} = 2 \text{ km/hr} \]

\[ r = 6 - 2 = 4 \text{ km/hr} \]

The comparison here is with the still-water time, not the downstream time - so \( b - r \) is compared with \( b \), giving \( r = \tfrac{2b}{3} \).

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.

Answer: A

With \( b \) the still-water speed and \( r \) the stream's,

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

\[ b = \frac{(b+r) + (b-r)}{2}, \qquad r = \frac{(b+r) - (b-r)}{2} \]

\[ r = \frac{15.5 - 8.5}{2} = \frac{7}{2} = 3.5 \text{ km/hr} \]

The boat does 12 km/hr in still water. The 7 km/hr option is the raw difference, which is twice the current.

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.

Answer: C

Speed with current is 20,

speed of the man + It is speed of the current

Speed in still water = 20 - 3 = 17

Now speed against the current will be

speed of the man - speed of the current

= 17 - 3 = 14 kmph

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.

Answer: B

Rate upstream = (750/675) = 10/9 m/sec

Rate downstream (750/450) m/sec = 5/3 m/sec

Rate in still water = (1/2)*[(10/9) + (5/3)] m/sec.

= 25/18 m/sec

= (25/18) × (18/5) kmph

= 5 kmph

Answer: C

Correct answer: 10 km/hr

With \(b\) the still-water speed, the two times differ by \(1\tfrac{1}{2}\) hours:

\[\frac{36}{b-2} - \frac{36}{b+2} = 1.5\]

\[36 \cdot \frac{(b+2)-(b-2)}{b^2 - 4} = 1.5 \quad\Longrightarrow\quad \frac{144}{b^2 - 4} = 1.5\]

\[b^2 - 4 = 96 \quad\Longrightarrow\quad b^2 = 100 \quad\Longrightarrow\quad b = 10 \text{ km/hr}\]

Checking: 36 km takes 3 hours downstream at 12 km/hr and 4.5 hours upstream at 8 km/hr.

Answer: C

Let the speed of boat be x km/hr and speed of stream be ykm/hr. 

44/5 × (x -y) = 4x (x +y) 

=> 44(x-y) = 20 (x+y) 

 =>11 (x-y) = 5 (x +y) 

 => 6x =16 y 

 => x/y = 16/6 = 8/3 

 => x : y = 8 : 3

Answer: A

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

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

\[ 90 \text{ min} = 1.5 \text{ h} \]

\[ \frac{36}{10 - r} - \frac{36}{10 + r} = 1.5 \]

\[ \frac{36(10 + r) - 36(10 - r)}{100 - r^{2}} = \frac{72r}{100 - r^{2}} = 1.5 \]

\[ 72r = 150 - 1.5r^{2} \quad\Longrightarrow\quad r^{2} + 48r - 100 = 0 \]

\[ r = \frac{-48 + \sqrt{2304 + 400}}{2} = \frac{-48 + 52}{2} = 2 \text{ mph} \]

The negative root is discarded as a speed. Checking: 36 miles at 8 mph takes 4.5 hours and at 12 mph takes 3 - exactly 90 minutes apart.