Boats and Streams: Formulas, Tricks and Solved Questions

Boats and Streams questions use the speed of a boat in still water and the speed of the stream to calculate upstream speed, downstream speed, distance or time. This page covers the main formulas, relative-speed rules, equal-distance cases, unit conversions and short methods for solving boat aptitude questions accurately.

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What Are Boats and Streams?

In Boats and Streams problems, the effective speed of a boat changes according to the direction of the current. Downstream speed is the sum of the boat's still-water speed and stream speed, while upstream speed is their difference.

Let the boat's speed in still water be b and the stream's speed be s. Then downstream speed is b + s, and upstream speed is b - s, provided b is greater than s. For any journey, time = distance ÷ speed. If downstream speed is D and upstream speed is U, then b = (D + U) ÷ 2 and s = (D - U) ÷ 2.

Boats and Streams Formula & Tricks

Important Formulas

Downstream speed
D = b + s

When the boat moves in the direction of the stream, the stream adds to the boat's still-water speed.

Upstream speed
U = b - s

When the boat moves against the stream, the stream reduces the boat's still-water speed.

Boat speed in still water
b = (D + U) / 2

Add the downstream and upstream speeds and divide by 2.

Stream speed
s = (D - U) / 2

Subtract upstream speed from downstream speed and divide by 2.

Time, distance and speed
Time = Distance / Speed

Use the effective upstream or downstream speed according to the direction of travel.

Average speed for equal distances
Average speed = 2DU / (D + U)

For equal distances travelled at speeds D and U, use the harmonic mean, not the ordinary average.

Quick Tricks

Use sum and difference directly

If downstream and upstream speeds are given, their half-sum gives the boat speed in still water and their half-difference gives the stream speed.

Example: For D = 18 km/h and U = 10 km/h, b = (18 + 10)/2 = 14 km/h and s = (18 - 10)/2 = 4 km/h.
Equal-distance average speed

For a boat travelling the same distance upstream and downstream, average speed is 2DU/(D + U). The arithmetic mean (D + U)/2 is not correct because the travel times are different.

Example: For D = 12 km/h and U = 8 km/h, average speed = 2 × 12 × 8/(12 + 8) = 9.6 km/h.
Compare times for the same distance

For a fixed distance, time is inversely proportional to speed. Therefore, the ratio of upstream time to downstream time is downstream speed to upstream speed.

Example: If D = 15 km/h and U = 10 km/h, then upstream time : downstream time = 15 : 10 = 3 : 2.

Boats and Streams Concepts

Downstream and Upstream Motion

Downstream motion is along the current, so the effective speed is b + s; upstream motion is against the current, so the effective speed is b - s.

Here b denotes the boat's speed in still water and s denotes the stream's speed. If a boat covers 48 km downstream at 16 km/h, its time is 48/16 = 3 hours. If it covers the same distance upstream at 12 km/h, its time is 48/12 = 4 hours.

Example: For b = 14 km/h and s = 4 km/h, downstream speed = 18 km/h and upstream speed = 10 km/h.

Finding Boat and Stream Speeds

When both effective speeds are known, the boat's still-water speed is their average and the stream speed is half their difference.

If D is downstream speed and U is upstream speed, then b = (D + U)/2 and s = (D - U)/2. The downstream speed must be greater than the upstream speed when the stream is flowing in the same direction as usual.

Example: For downstream speed 22 km/h and upstream speed 14 km/h, b = 18 km/h and s = 4 km/h.

Distance and Time in a Given Direction

Calculate journey time by dividing the distance by the effective speed in that direction.

Use D = b + s for downstream travel and U = b - s for upstream travel. Keep the units consistent: distance in kilometres with speed in kilometres per hour gives time in hours; distance in metres with speed in metres per second gives time in seconds.

Example: A boat travels 30 km upstream at 10 km/h. Its time is 30/10 = 3 hours.

Equal-Distance Upstream and Downstream Journey

For equal distances, the average speed is 2DU/(D + U), where D and U are downstream and upstream speeds.

If each part has distance x, total distance is 2x and total time is x/D + x/U. Hence average speed = 2x/(x/D + x/U) = 2DU/(D + U). This value is less than the arithmetic mean when D and U are unequal.

Example: A boat covers equal distances at 18 km/h downstream and 12 km/h upstream. Average speed = 2 × 18 × 12/(18 + 12) = 14.4 km/h.

Speed Conversion in Boat Problems

Convert speeds before applying a formula when the distance and speed units do not match.

Use 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h. For example, 54 km/h = 54 × 5/18 = 15 m/s. The formulas D = b + s and U = b - s remain unchanged after conversion.

Example: A boat's speed of 36 km/h is 10 m/s, because 36 × 5/18 = 10.

Boats and Streams Video Lessons

Watch short topic-wise lessons for quick revision.

11 Lessons
Lesson 1 of 11 Quick Revision

Boats Downstream and Upstream Speed Formulas

Learn the formulas for calculating a boat’s downstream and upstream speeds using its still-water speed and the speed of the stream.

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Practice Boats and Streams Questions

Practise published questions related to this topic.

1Three numbers are in the ratio 9 : 7 : 5, and their LCM is 2205. Their HCF is:→ 2Successive discounts of 9% and 6% are equivalent to a single discount of:→ 3A dealer buys an article marked at 1800 after successive discounts of 25% and 62/3%. Find the cost price of the article to the dealer.→ 4An article was bought for ₹4,000. Its price was marked up by 40%. Thereafter, it was sold at a discount of 20% on the marked price. What was the per cent profit on the transaction?→ 5The cost price of a box of marker is ₹3,300. The shopkeeper gives 75% of discount on the cost price. What will be the selling price (in ₹)?→ 6A shopkeeper buys a toy for ₹500 and spends ₹50 on packaging. He then sells it to a customer at 10% profit. Find the selling price of the toy.→ 7By selling an article for ₹117 , a man loses 40% . At what price (in ₹ ) should he sell the same article to gain 40% ?→ 8The difference between the two numbers is 1365. On dividing the larger by the smaller, the quotient is 6, and the remainder is 15. Find the smaller number.→ 9An article is sold for a profit of 25 percent. What is the ratio of the selling price of the article to the cost price of the article?→ 10Simplify: ((6⁴ × 4⁵ × 6⁵))/((6² × 4² × 6³))→

Boats and Streams Quick Quiz

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Quick Revision Notes

Boats and Streams Quick Revision

Use these direct rules while solving upstream and downstream questions.

  • Downstream speed = boat speed in still water + stream speed.
  • Upstream speed = boat speed in still water - stream speed.
  • Boat speed in still water = (downstream speed + upstream speed)/2.
  • Stream speed = (downstream speed - upstream speed)/2.
  • Time = distance/speed, using the effective speed in the direction of travel.
  • For equal distances at speeds D and U, average speed = 2DU/(D + U).
  • For the same distance, upstream time : downstream time = D : U.
  • Convert units before calculation: 1 km/h = 5/18 m/s and 1 m/s = 18/5 km/h.

Boats and Streams FAQs

What is the boats and streams formula for downstream speed?

Downstream speed = boat speed in still water + stream speed, or D = b + s.

What is the boats and streams formula for upstream speed?

Upstream speed = boat speed in still water - stream speed, or U = b - s.

A boat's downstream speed is 20 km/h and upstream speed is 12 km/h. What are its still-water and stream speeds?

Still-water speed = (20 + 12)/2 = 16 km/h. Stream speed = (20 - 12)/2 = 4 km/h.

How much time does a boat take to cover 45 km upstream at 15 km/h?

Time = distance/speed = 45/15 = 3 hours.

What is the average speed for equal upstream and downstream distances?

If downstream speed is D and upstream speed is U, average speed = 2DU/(D + U). For 18 km/h and 12 km/h, it is 14.4 km/h.

A boat moves at 10 m/s in still water and the stream flows at 2 m/s. What are its upstream and downstream speeds?

Downstream speed = 10 + 2 = 12 m/s. Upstream speed = 10 - 2 = 8 m/s.

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