Boats and Streams

Competitive Mathematics · Time, Work and Distance

6 Part3 worked examples33 practice questions

You can swim at 4 km/h in a still pond. Now you are in a river flowing at 1 km/h. Swim with the flow and the river pushes you: your real speed is $4 + 1 = 5$ km/h. Swim against it and the river holds you back: $4 - 1 = 3$ km/h. That is the whole secret of the chapter: the still-water speed $x$ and the stream speed $y$ make two speeds, and the rest is plain speed-time-distance.

साथ : x + y, विरुद्ध : x - y, x = \frac{d + u}{2}, y = \frac{d - u}{2}

The four basics

The two speeds and travel times

With = add, against = subtract

Downstream and upstream speeds, the still-water and stream speeds from them, and the time.

Terms and direct calculation

Still water, stream, downstream, upstream

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 $= \frac{distance}{relevant speed}$ — 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 $u > 0$ .

Example 1 · Basic

A man rows downstream at 14 km/h and upstream at 8 km/h. Find his still-water speed and the stream speed.

Solution, step by step

1 $x = \frac{14 + 8}{2} = 11$ km/h.
2 $y = \frac{14 - 8}{2} = 3$ km/h (check: $11 + 3 = 14$ , $11 - 3 = 8$ ✓)

Answer 11 and 3 km/h

Example 2 · Basic

A boat with a still-water speed of 15 km/h goes 104 km against a 2 km/h stream. How long does it take?

Solution, step by step

1Upstream speed $= 15 - 2 = 13$ km/h.
2Time $= \frac{104}{13} = 8$ hours

Answer 8 hours

Example 3 · Standard

A boat goes 72 km downstream in 6 hours and the same distance upstream in 9 hours. Find its still-water speed and the stream speed.

Solution, step by step

1Downstream $= \frac{72}{6} = 12$ ; upstream $= \frac{72}{9} = 8$ km/h.
2 $x = 10$ , $y = 2$ km/h

Answer 10 and 2 km/h

BasicTry it yourself first

A boat's speed in still water is 6 km/h and the stream flows at 2 km/h. What is its speed downstream?

BasicTry it yourself first

A boat's speed in still water is 9 km/h and the stream flows at 4 km/h. What is its speed downstream?

StandardTry it yourself first

A boat's speed in still water is 14 km/h and the stream flows at 3 km/h. What is its speed downstream?

BasicTry it yourself first

A boat's speed in still water is 8 km/h and the stream flows at 3 km/h. What is its speed upstream?

BasicTry it yourself first

A boat's speed in still water is 12 km/h and the stream flows at 5 km/h. What is its speed upstream?

StandardTry it yourself first

A boat's speed in still water is 18 km/h and the stream flows at 7 km/h. What is its speed upstream?

BasicTry it yourself first

A man can row downstream at 14 km/h and upstream at 8 km/h. What is his speed in still water?

BasicTry it yourself first

A man can row downstream at 15 km/h and upstream at 9 km/h. What is his speed in still water?

StandardTry it yourself first

A man can row downstream at 20 km/h and upstream at 12 km/h. What is his speed in still water?

StandardTry it yourself first

A man can row downstream at 13 km/h and upstream at 7 km/h. What is his speed in still water?

AdvancedTry it yourself first

A man can row downstream at 22 km/h and upstream at 10 km/h. What is his speed in still water?

BasicTry it yourself first

A man can row downstream at 16 km/h and upstream at 10 km/h. What is the speed of the stream?

BasicTry it yourself first

A man can row downstream at 18 km/h and upstream at 12 km/h. What is the speed of the stream?

StandardTry it yourself first

A man can row downstream at 19 km/h and upstream at 9 km/h. What is the speed of the stream?

StandardTry it yourself first

A man can row downstream at 21 km/h and upstream at 13 km/h. What is the speed of the stream?

AdvancedTry it yourself first

A man can row downstream at 25 km/h and upstream at 15 km/h. What is the speed of the stream?

BasicTry it yourself first

A boat with still-water speed 12 km/h goes 90 km downstream in a 3 km/h stream. How many hours does it take?

BasicTry it yourself first

A boat with still-water speed 18 km/h goes 120 km downstream in a 2 km/h stream. How many hours does it take?

StandardTry it yourself first

A boat with still-water speed 15 km/h goes 81 km upstream in a 6 km/h stream. How many hours does it take?

StandardTry it yourself first

A boat with still-water speed 14 km/h goes 96 km upstream in a 2 km/h stream. How many hours does it take?

AdvancedTry it yourself first

A boat with still-water speed 20 km/h goes 150 km downstream in a 5 km/h stream. How many hours does it take?

AdvancedTry it yourself first

A boat with still-water speed 17 km/h goes 105 km upstream in a 2 km/h stream. How many hours does it take?

BasicTry it yourself first

A boat covers 96 km downstream in 6 hours and the same distance upstream in 12 hours. Find the speed of the boat in still water.

StandardTry it yourself first

A boat covers 90 km downstream in 5 hours and the same distance upstream in 9 hours. Find the speed of the stream.

StandardTry it yourself first

A boat covers 84 km downstream in 6 hours and the same distance upstream in 12 hours. Find the speed of the boat in still water.

AdvancedTry it yourself first

A boat covers 120 km downstream in 8 hours and the same distance upstream in 15 hours. Find the speed of the stream.

AdvancedTry it yourself first

A boat covers 54 km downstream in 3 hours and the same distance upstream in 6 hours. Find the speed of the boat in still water.

AdvancedTry it yourself first

A boat covers 100 km downstream in 5 hours and the same distance upstream in 10 hours. Find the speed of the stream.

StandardTry it yourself first

A boat goes 18 km downstream in 36 minutes and 18 km upstream in 45 minutes. Find the speed in still water.

StandardTry it yourself first

A boat goes 10 km downstream in 30 minutes and 10 km upstream in 40 minutes. Find the speed of the stream.

AdvancedTry it yourself first

A boat goes 3 km downstream in 10 minutes and 8 km upstream in 60 minutes. Find the speed of the stream.

AdvancedTry it yourself first

A boat goes 9 km downstream in 18 minutes and 6 km upstream in 30 minutes. Find the speed in still water.

AdvancedTry it yourself first

A boat goes 5 km downstream in 12 minutes and 5 km upstream in 20 minutes. Find the speed of the stream.

Ratios of times and speeds

Same distance ⟹ speed inverse to time

When a “how many times as long” or a time ratio is given, you get the ratio of downstream to upstream speeds.

$n$ times as long, and ratios

x = a \frac{n + 1}{n - 1}

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

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There-and-back and two equations

Total time, difference and

x^{2} - y^{2}

Total time to go and return, its average speed, and two unknowns from two situations.

Going and returning

\frac{D}{x + y} + \frac{D}{x - y} = T

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

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Traps and fast tricks

Slips · tricks

Where marks go in boat-and-stream questions.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Sum-difference and ratios

Habits for speed.

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The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Boats and Streams

The two speeds and travel times

Terms and direct calculation

Ratios of times and speeds

$n$ times as long, and ratios

There-and-back and two equations

Going and returning

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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