Boats and streams

1. The basic model and its assumptions

A boat moves relative to the water, while the water itself moves relative to the bank. The speed observed from the bank therefore combines two motions. When the boat travels with the current, its motion is downstream and the current adds to its speed. When it travels against the current, its motion is upstream and the current reduces its speed.

Let b denote the boat’s speed in still water and s the stream speed. Let D and U denote downstream and upstream speeds measured relative to the bank. Then D = b + s and U = b − s. Ordinary arithmetic questions assume a uniform current, constant rowing or engine effort, straight-line travel along the stream, and no stoppages unless specified.

Upstream progress requires b > s. If b = s, a boat pointed directly upstream remains stationary relative to the bank. If b < s, it drifts downstream despite pointing upstream. A negative value of b − s describes the direction of motion; it is not a usable upstream speed for reaching a destination against the current.

These are idealised assumptions rather than complete descriptions of river navigation. Real currents vary across the channel and with discharge, depth and season. In an examination, however, do not introduce these complications unless the question explicitly provides them.

• Still-water speed means speed relative to the water, not relative to the riverbank.
• A freely drifting raft has approximately zero speed relative to the surrounding water and moves downstream at stream speed.
• Use consistent units: 30 minutes = 0.5 hour; 1 m/s = 3.6 km/h.

2. Recovering speeds from distance and time

The most reliable starting point is to calculate each ground speed separately. If a boat covers distance d downstream in time t_d and the same distance upstream in time t_u, then D = d/t_d and U = d/t_u. Adding D = b + s and U = b − s gives b = (D + U)/2. Subtracting gives s = (D − U)/2.

For example, a boat covers 24 km downstream in 2 hours and 24 km upstream in 3 hours. Its downstream speed is 12 km/h and upstream speed is 8 km/h. Hence b = 10 km/h and s = 2 km/h. These values can be verified by reconstructing both journey times.

The same recovery method works for unequal distances. If the boat covers 30 km downstream in 2 hours and 18 km upstream in 3 hours, D = 15 km/h and U = 6 km/h. Therefore, b = 10.5 km/h and s = 4.5 km/h. Equal distances are necessary for certain ratio shortcuts, not for the basic addition-and-subtraction method.

• For equal distance d: b = (d/2)(1/t_d + 1/t_u).
• For equal distance d: s = (d/2)(1/t_d − 1/t_u).
• Check that D ≥ U and that the recovered stream speed is non-negative under the stated model.

3. Ratios, round trips and average speed

For equal distances, time is inversely proportional to speed. Thus t_u/t_d = D/U = (b + s)/(b − s). If upstream time is three times downstream time, then D/U = 3. Solving b + s = 3(b − s) gives b = 2s. A ratio can determine the relationship between boat and stream speeds without determining either speed numerically.

More generally, if D:U = m:n, then b:s = (m + n):(m − n). The same expression applies when t_u:t_d = m:n for equal distances. Always check the order of the time ratio: upstream time corresponds to the smaller speed.

For a round trip between two points distance d apart, total time T = d/(b + s) + d/(b − s) = 2db/(b² − s²). Average speed is total distance divided by total time, so it equals (b² − s²)/b, or 2DU/(D + U). This is the harmonic mean of the two directional speeds because the distances are equal.

Suppose b = 10 km/h, s = 2 km/h and each leg is 24 km. The trip takes 2 + 3 = 5 hours, giving an average speed of 48/5 = 9.6 km/h. The arithmetic mean, 10 km/h, is incorrect because the boat spends longer at the lower speed. For fixed boat speed and distance, a current increases round-trip time compared with still water.

• Equal distances: use the harmonic mean for average speed.
• Equal times: the arithmetic mean of the two speeds gives average speed.
• For equal-distance legs, t_u − t_d = 2ds/(b² − s²).

4. Less direct questions: rafts, meetings and sufficiency

In raft questions, distinguish powered motion from passive drifting. A raft carried by a uniform current travels distance x in time x/s. If a boat and raft start together and travel downstream, their separation grows at (b + s) − s = b. The current cancels because both objects experience the same water motion.

A useful special result concerns an object dropped from a boat. Suppose the boat continues for time t before turning back, maintains the same still-water speed in both directions, and recovers the freely drifting object in a uniform current. It takes another t after turning to catch the object. In the water’s frame, the object is stationary and the boat retraces an equal relative distance at the same speed.

For two powered boats moving towards each other in the same uniform stream, with one travelling downstream and the other making upstream progress, closing speed is (b₁ + s) + (b₂ − s) = b₁ + b₂. This cancellation does not justify ignoring current in every problem: it depends on a shared current and the stated directions.

Data-sufficiency questions require checking the number of independent relationships. Knowing only D = 15 km/h gives b + s = 15, which cannot determine b and s separately. Knowing D = 15 km/h and U = 9 km/h gives two independent equations and determines both. Similarly, an equal-distance time ratio alone establishes b:s but not their absolute values.

• Draw directions before forming a relative-speed equation.
• Do not apply uniform-current cancellation when objects are in different currents.
• Treat river-crossing questions as a separate vector-motion extension rather than blindly using b + s and b − s.

5. A practical CSAT solving method

Begin by writing the unknowns and labelling each journey upstream or downstream. Convert times and distances into compatible units. Then use distance = speed × time to form equations. Whenever both directional speeds can be calculated directly, recover b and s immediately rather than introducing unnecessary algebra.

Look for structural shortcuts only after identifying the conditions. Equal distances permit inverse time ratios and the harmonic-mean formula. A common current may cancel in relative motion. Numerical answer options can also be substituted into the original conditions, but an option must satisfy every condition, not merely one.

Finish with a reasonableness check. Downstream travel should be faster than upstream travel at unchanged effort. For an equal-distance round trip in a non-zero current, average speed should be below still-water speed. If the question includes a rest halt, include it in elapsed journey time but not in the calculation of moving speed unless the wording requires an overall average.

• Common error: taking D − U as stream speed instead of dividing the difference by two.
• Common error: averaging unequal-distance journey times before calculating speeds.
• Common error: treating 1 hour 20 minutes as 1.20 hours rather than 4/3 hours.
• Exam discipline: prefer one correct model and a quick verification over memorising many isolated formulas.

Previous year questions

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Practice questions