Boats & Streams Questions and Answers

Answer: D

The distance is the same both ways, so the speeds are inversely proportional to the times. Upstream takes 10 minutes and downstream 5 minutes, so the downstream speed is twice the upstream speed. With the boat's speed b and the stream's speed 2 m/s, the equation is as shown.

\[ b + 2 = 2(b - 2) \]

\[ b = 6 \]

The speed of the boat in still water is 6 m/s. Check: upstream 4 m/s and downstream 8 m/s, which is double.

Answer: B

Let the speed of the boat in still water be v km/hr. Going down the stream the speed is v + 4 and coming back it is v - 4, and the total time is 40 hours.

\[ \frac{120}{v+4} + \frac{120}{v-4} = 40 \]

\[ \frac{240v}{v^{2}-16} = 40 \Rightarrow v^{2} - 6v - 16 = 0 \]

This gives v = 8, so the speed in still water is 8 km/hr, option (B). Speeds of 15 and 10 would take much less than 40 hours. Check: 120 / 12 = 10 hours downstream and 120 / 4 = 30 hours upstream, and 10 + 30 = 40 hours, which agrees with the given time.

Answer: C

The boat takes 4 hours to go 28 km downstream. Equal times for 21 km downstream and 9 km upstream mean the speeds are in the ratio 21 : 9 = 7 : 3, so downstream speed is \(7k\) and upstream speed is \(3k\). \[ \begin{aligned} \frac{12}{7k} + \frac{12}{3k} &= 5\frac{5}{7} \\ &= \frac{40}{7} \end{aligned} \] \[ \begin{aligned} \frac{12 + 28}{7k} &= \frac{40}{7k} \\ &= \frac{40}{7} \Rightarrow k \\ &= 1 \end{aligned} \] Downstream speed is 7 km/h, so \[ \frac{28}{7} = 4\text{ hours} \] The other options do not match this speed.

Answer: A

The man rows at 8 km/h in still water and the current is 3 km/h: (8, 3 kmph).

Let the upstream speed be u and the downstream speed d. Write a = 1/u and b = 1/d to make the equations linear.

\[ \begin{aligned} 40a + 55b &= 13 \\ 30a + 44b &= 10 \end{aligned} \]

\[ \begin{aligned} 120a + 165b &= 39 \\ 120a + 176b &= 40 \\ 11b &= 1,\ b = \tfrac{1}{11} \end{aligned} \]

Then \(40a = 13 - 5 = 8\), so a = 1/5. The speeds are u = 5 km/h and d = 11 km/h.

\[ \begin{aligned} \text{still water} &= \tfrac{11 + 5}{2} = 8 \\ \text{current} &= \tfrac{11 - 5}{2} = 3 \end{aligned} \]

All options give 8 for the boat, so only the current differs; 4, 3.2 and 3.5 km/h do not satisfy both trips.

Answer: D

The ratio of the speed of the current to the speed of the boat is 5/6.

Read as the key does, the downstream trip takes 1/11 of the upstream time over the same distance, so the speeds are in the inverse ratio. Let b be the speed in still water and c the current.

\[ \begin{aligned} \frac{b - c}{b + c} &= \frac{1}{11} \\ 11b - 11c &= b + c \\ 10b &= 12c \Rightarrow \frac{c}{b} = \frac{5}{6} \end{aligned} \]

6/5 is the same ratio upside down (boat to current). 11/5 and 11/6 come from mixing the 11 into the answer instead of solving for it.

Answer: C

Both 1 and 2 are correct.

Statement 1: speed with the current = man + current, so the man's own speed is \(15 - 2.5 = 12.5\) km/h, and against the current he makes \(12.5 - 2.5 = 10\) km/h. True.

Statement 2: downstream speed is \(22 + 3 = 25\) km/h, and 12 minutes is 1/5 hour, so the distance is

\[ 25 \times \frac{12}{60} = 5 \text{ km} \]

True. A common slip in statement 1 is to subtract the current only once (15 − 2.5 = 12.5) and stop there.

Answer: C

The time taken is 4 hours.

With the current, the boat's speed is the still-water speed plus the current's speed:

\[ \begin{aligned} v &= 20 + 5 = 25\ \text{km/h} \\ t &= 100/25 = 4\ \text{h} \end{aligned} \]

Against the current the speed would be 15 km/h and the time about 6.7 hours. 2 and 3 hours would need speeds of 50 and 33 km/h, and 7 hours matches no speed in the question.

Answer: B

Against the stream the swimmer covers 80 km.

Downstream speed is 20 + 4 = 24 km/h and upstream speed is 20 - 4 = 16 km/h:

\[ \begin{aligned} t &= 120/24 = 5\ \text{h} \\ d &= 16 \times 5 = 80\ \text{km} \end{aligned} \]

Equivalently, in equal times distances are in the ratio of speeds, 120 \(\times\) 16/24 = 80. 90 and 75 km come from taking 18 or 15 km/h as the upstream speed.

Answer: A

The stream flows at 1.5 kmph.

Downstream speed is 27/4 = 6.75 km/h and upstream speed is 15/4 = 3.75 km/h. The stream's speed is half their difference:

\[ \frac{6.75 - 3.75}{2} = 1.5\ \text{km/h} \]

The rowing speed in still water is half the sum, 5.25 km/h. Taking the full difference (3 km/h) is the usual error; 1.8, 2.2 and 2.5 do not follow.

Answer: B

The speed of the stream is 20 km/hour.

Onward: 2 hours in still water at 40 km/h covers 80 km, then 2 hours downstream at \(40 + s\) covers \(2(40 + s)\) km, a total of 4 hours. The return takes twice as long, 8 hours: the still stretch again takes 2 hours, so the upstream stretch takes 6 hours at \(40 - s\).

\[ \begin{aligned} \frac{2(40 + s)}{40 - s} &= 6 \\ 80 + 2s &= 240 - 6s \\ 8s &= 160 \\ s &= 20 \end{aligned} \]

Check: downstream at 60 km/h for 2 hours covers 120 km, and upstream at 20 km/h the same 120 km takes 6 hours. With 40 km/h the boat could not move upstream at all, which rules out that option.