AFCAT Numerical Ability · Speed, Time and Distance

Boats and Streams: Downstream, Upstream and Still Water

Downstream the stream adds to the boat's own speed and upstream it takes away, so the still-water speed and the stream's speed are half the sum and half the difference.

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Why this matters

AFCAT asks this more than any other part of the chapter: eleven past-year questions, from 2015 to 2019. A fifth of the page is HARD, and the hard ones are equations in time: equal times, a round trip in a total time, or two trips giving two equations.

Concept 1 of 6: Downstream and upstream speeds

The water carries the boat. Going with the stream, the two speeds add; going against it, the stream takes some of the boat's speed away.

Definition

Downstream speed $= b + s$ ; upstream speed $= b - s$ , where $b$ is the boat's speed in still water and $s$ the stream's.
Time $=$ distance $\div$ the speed for that direction.
From one of them and $s$ : $b =$ downstream $- s$ , or $b =$ upstream $+ s$ .
Downstream minus upstream is $2s$ .

With and against the stream

v_{\text{down}} = b + s, \qquad v_{\text{up}} = b - s

Worked example

A boat does 14 km/h in still water and the stream flows at 4 km/h. How long does it take to go 90 km downstream and come back?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Speed, Time and Distance · Boats and Streams

The speed of a boat in still water is 9 km/hr and the speed of the stream is 3 km/hr. The difference between the time taken to go 36 km upstream and 36 km downstream is:

Still water is not downstream

'Speed of the boat' in these questions means its speed in still water,

b

. A boat that goes 20 km/h downstream on a 3 km/h stream has

b = 17

, not 20.

Drill 17 more on downstream and upstream speeds

Concept 2 of 6: Still water and stream from the two speeds

The downstream and upstream speeds sit the same distance

s

either side of the boat's own speed. So

b

is their average and

s

is half their difference.

Definition

$b = \dfrac{\text{down} + \text{up}}{2}$ and $s = \dfrac{\text{down} - \text{up}}{2}$ .
Find each speed first: distance $\div$ time for that direction.
Times in minutes: 10 km in 40 minutes is 15 km/h.
'The boat's speed is twice the stream's': $b = 2s$ , so downstream is $3s$ and upstream is $s$ .

Half the sum, half the difference

b = \dfrac{d + u}{2}, \qquad s = \dfrac{d - u}{2}

Worked example

A boat goes 30 km downstream in 2 h and 28 km upstream in 4 h. Find its speed in still water and the stream's speed.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 04-213Easy

Example 2 · Speed, Time and Distance · Boats and Streams

A man rows upstream a distance of 9 km or downstream a distance of 18 km taking 3 h each time. The speed of the boat in still water is

Half the difference

With 15 km/h down and 7 km/h up, the stream is 4 km/h, not 8. The difference 8 counts the stream twice, once added and once taken away.

Drill 31 more on still water and stream from the two speeds

Concept 3 of 6: Equal times and ratios of time

Equal times mean the distances are in the ratio of the speeds. 'Twice as long upstream as downstream' means the downstream speed is twice the upstream speed.

Definition

Same time: $\dfrac{d_1}{b + s} = \dfrac{d_2}{b - s}$ , so the distances are in the ratio of the speeds.
$k$ times as long upstream as downstream over the same distance: $b + s = k(b - s)$ , so $b : s = (k + 1) : (k - 1)$ .
' $p$ km down in the time of $q$ km up': down $:$ up $= p : q$ .
$b : s$ given as $m : n$ : downstream is $m + n$ parts and upstream $m - n$ parts.

Equal times

\dfrac{d_1}{b + s} = \dfrac{d_2}{b - s}

Worked example

A boat whose still-water speed is 12 km/h goes 32 km downstream in the time it takes to go 16 km upstream. Find the stream's speed.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (I) 2015 — Numerical Ability · 04-225Moderate

Example 3 · Speed, Time and Distance · Boats and Streams

The speed of a boat in still water is 10 km/h. If it can travel 26 km downstream and 14 km upstream in the same time, the speed of the stream is

Twice the time is half the speed

'Takes twice as long upstream' makes the downstream speed double the upstream speed. Writing

b - s = 2(b + s)

turns the stream backwards.

Drill 13 more on equal times and ratios of time

Concept 4 of 6: Out and back in a total time

Out and back over the same distance, the two times add up to the total. With the boat's speed unknown, the same equation becomes a quadratic.

Definition

$\dfrac{d}{b + s} + \dfrac{d}{b - s} = T$ . With $b$ and $s$ known, solve for $d$ .
With $d$ , $s$ and $T$ known: $\dfrac{2bd}{b^2 - s^2} = T$ , a quadratic in $b$ ; keep the positive root above $s$ .
A difference of times: $\dfrac{d}{b - s} - \dfrac{d}{b + s} = \dfrac{2sd}{b^2 - s^2}$ .
The average speed for the round trip is $\dfrac{2(b + s)(b - s)}{2b} = \dfrac{b^2 - s^2}{b}$ , the equal-distance average of the two speeds.

A round trip on a river

\dfrac{d}{b + s} + \dfrac{d}{b - s} = T

Worked example

A boat rows at 10 km/h in still water on a stream of 2 km/h. It goes out and back in 5 h. How far does it go out?

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2015 · AFCAT (II) 2015 — Numerical Ability · 04-218Easy

Example 4 · Speed, Time and Distance · Boats and Streams

A boat travels upstream from

B

A

and downstream from

A

B

in 3 h. If the speed of the boat in still water is 9 km/h and the speed of the current is 3 km/h, the distance between

A

and

B

The round trip average is below b

b = 6

and

s = 2

the round trip averages

5\tfrac13

km/h, not 6. The slow upstream leg lasts longer than the quick one.

Drill 7 more on out and back in a total time

Concept 5 of 6: Two trips, two equations

Two trips with different distances give two equations. Treat

\tfrac1u

and

\tfrac1d

, the reciprocals of the upstream and downstream speeds, as the unknowns; then the equations are linear.

Definition

Let $x = \tfrac1u$ and $y = \tfrac1d$ . The trips give $p_1 x + q_1 y = T_1$ and $p_2 x + q_2 y = T_2$ .
Scale the equations so one unknown has the same coefficient, then subtract.
Then $u = \tfrac1x$ , $d = \tfrac1y$ , and $b$ , $s$ are half their sum and half their difference.

Two trips

p_1 x + q_1 y = T_1, \quad p_2 x + q_2 y = T_2, \quad x = \tfrac1u,\; y = \tfrac1d

Worked example

A boat goes 12 km upstream and 36 km downstream in 6 h, and 20 km upstream and 24 km downstream in 7 h. Find the boat's and the stream's speeds.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

AFCAT · 2016 · AFCAT (II) 2016 — Numerical Ability · 04-215Hard

Example 5 · Speed, Time and Distance · Boats and Streams

A man can row 30 km upstream and 44 km downstream in 10 h. He can also row 40 km upstream and 55 km downstream in 13 h. Find the rate of the current.

Solve for the reciprocals

Taking

u

and

d

as the unknowns leaves

\tfrac{12}{u} + \tfrac{36}{d} = 6

, which is not linear. With

x = \tfrac1u

and

y = \tfrac1d

it is.

Drill 4 more on two trips, two equations

Concept 6 of 6: Boats meeting, chasing and crossing the river

Two boats heading towards each other on a river: one rides the stream and one fights it, so their closing speed is

(b_1 + s) + (b_2 - s)

, and the stream cancels out.

Definition

Two boats from opposite ends: closing speed $= b_1 + b_2$ . For identical boats it is downstream plus upstream, $2b$ .
One boat chasing another downstream: the stream carries both, so the gap closes at $b_1 - b_2$ .
Rowing straight across: time $=$ width $\div$ rowing speed, and the stream carries the boat downstream by $s \times$ that time.

Two boats closing

v_{\text{close}} = (b_1 + s) + (b_2 - s) = b_1 + b_2

Worked example

A boat goes 16 km/h downstream and 10 km/h upstream. Two such boats start from the two ends of a 78 km stretch towards each other. When do they meet?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 6 · Speed, Time and Distance · Boats and Streams

A boat downstream takes

4

hours to travel

60

km. The same boat upstream takes

6

hours to travel

60

km. Two such boats start from opposite ends of a

120

km river stretch and travel towards each other. How long do they take to meet?

The stream cancels for boats that meet

One boat gains the stream and the other loses it, so the meeting time depends only on the boats' own speeds. Adding the stream to both counts it twice.

Drill 5 more on boats meeting, chasing and crossing the river

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (6)

Downstream and upstream speeds
With and against the stream
$v_{\text{down}} = b + s, \qquad v_{\text{up}} = b - s$
Still water and stream from the two speeds
Half the sum, half the difference
$b = \dfrac{d + u}{2}, \qquad s = \dfrac{d - u}{2}$
Equal times and ratios of time
Equal times
$\dfrac{d_1}{b + s} = \dfrac{d_2}{b - s}$
Out and back in a total time
A round trip on a river
$\dfrac{d}{b + s} + \dfrac{d}{b - s} = T$
Two trips, two equations
Two trips
$p_1 x + q_1 y = T_1, \quad p_2 x + q_2 y = T_2, \quad x = \tfrac1u,\; y = \tfrac1d$
Boats meeting, chasing and crossing the river
Two boats closing
$v_{\text{close}} = (b_1 + s) + (b_2 - s) = b_1 + b_2$

Watch out for (6)

Still water is not downstream
→ Downstream and upstream speeds
Half the difference
→ Still water and stream from the two speeds
Twice the time is half the speed
→ Equal times and ratios of time
The round trip average is below b
→ Out and back in a total time
Solve for the reciprocals
→ Two trips, two equations
The stream cancels for boats that meet
→ Boats meeting, chasing and crossing the river

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