Boats and Streams — Extra Practice

The speed of a boat always means its still-water speed unless told otherwise.

• Still water: water with no flow (stream speed zero). Stream: flowing water, speed $y$.
• Downstream: speed $d=x+y$. Upstream: speed $u=x-y$.
• Conversely: still-water speed $x=\frac{d+u}{2}$ and stream $y=\frac{d-u}{2}$.
• Time $=\frac{\text{distance}}{\text{relevant speed}}$ — divide by $d$ going with the stream and by $u$ going against it.
• Check: the stream cannot be faster than the boat, so $x>y$ and $u>0$.

Take the two speeds as $nk$ and $k$ from the ratio; their sum and difference give $x$ and $y$.

• Upstream takes $n$ times as long ⟹ speed ratio (down : up) $=n:1$. Then $x:y=(n+1):(n-1)$.
• Times $t_1$ (down) and $t_2$ (up), stream $a$ known: $x=a\frac{t_1+t_2}{t_2-t_1}$. Boat $b$ known: $a=b\frac{t_2-t_1}{t_2+t_1}$.
• Boat : stream from the times $t_2$ and $t_1$: $(t_2+t_1):(t_2-t_1)$.
• Upstream takes $k$ times the still-water time: upstream speed $=\frac{x}{k}$ ⟹ $y=x\cdot \frac{k-1}{k}$.

Going uses one speed and returning the other; add the two times.

• Total time there and back $T=\frac{D}{x+y}+\frac{D}{x-y}=\frac{2xD}{x^2-y^2}$ — solve it for $D$ or $y$.
• The average speed is always below $x$: $\frac{x^2-y^2}{x}$. A river never makes a round trip cheaper.
• Upstream takes $T$ more: $D=\frac{T(x^2-y^2)}{2y}$.
• Two situations (distances and times given): take $\frac{1}{\text{speed}}$ as the unknowns — two linear equations.

The slips that cost marks.

Habits for speed.

• Given the two speeds, the still-water speed is their mean and the stream is half their difference.
• Given times, first make both speeds as $\frac{\text{distance}}{\text{time}}$.
• $n$ times as long: $x:y=(n+1):(n-1)$.
• Check: $x$ is the bigger, $y$ the smaller; both speeds fit both conditions.

One-rule questions.

Exam-level questions.

Questions at the hard end of competitive exams.