Almost all questions on Boats and Streams have the boat and stream travelling ‘parallel’ to each other i.e. the boat and stream are travelling in same direction or opposite direction. In Arithmetic based questions, rarely do we see boat and streams in perpendicular direction. Only if you dig really deep into tougher questions, you might come across one such question. I personally dont think such questions are relevant for CAT or QA Aptitude based exams, but then if a question setter needs to make a difficult question, this idea of boat travelling perpendicular to stream i.e. across a river, is an easy one to frame a question on. One very common such doubt is explained here.

Q.

P and Q are two points on either shore of a river such that PQ is perpendicular to the flow of the river i.e. the two points are directly opposite each other on the two shores of a river. A boat started crossing the river from P and tried to row straight to point Q. But due to the current of the river, he landed at a point R, downstream of point Q such that distance PR is 260 meters. Next day again the boatman started from P and tried to row straight to Q, but today with speed of boat, relative to river, being 10/7 times previous day’s speed of boat (relative to river). This day he landed at point S such that distance PS is 250 meters. Points Q, S and R are on the same shore of the river and the width of the river is constant such that the two shores are parallel to each other. If the flow of river was the same on both the days, find the width of the river.

TITA type i.e. Type In The Answer type

If we consider distance as PR or PS, then the speed to be taken is speed of boat relative to an observer standing on the shore. But in this case, relative speed is a vector, which is not really a topic relevant for CAT or QA aptitude and is more a question of Physics.

However, if we consider distance as PQ, then this distance is covered by the boat at it’s speed relative to the current. Or in simpler terms … speed relative to the current is simply the speed of boat in still water. To repeat, if distance = PQ, then speed = speed of boat in still water or speed of boat relative to current, both mean the same.

And the distance QR or QS is covered by the boat, not due to it’s speed. This is covered only and only because of the speed of river.

Thus,

Distance = PQ, Speed = B, speed of boat in still water (or speed relative to current)

Distance = QR or QS, Speed = R, speed of river

Distance = PR or PS, Speed = speed of boat in the moving river, or speed of boat relative to observer on shore, which is not something we learn in Arithmetic.

And while the boat covers distance PQ, the current drives the boat a distance of QR. This sentence should make the solution very easy.

240

Refer to the above figure. As explained in the Hint …

In the time, the boat, with its speed relative to current, covers PQ … in the same time, the river drives the boats a distance of QR. Since time is same,

$$\frac{\text{PQ}}{\text{Speed of boat relative to current}}=\frac{\text{QR}}{\text{Speed of current}}$$

And similar relation for second case is

$$\frac{\text{PQ}}{\text{Speed of boat relative to current}}=\frac{\text{QS}}{\text{Speed of current}}$$

Dividing equation 1 by equation 2 and given that speed of boat relative to current the second day was 10/7 of the previous day, we get

$$\frac{10}{7}=\frac{\text{QR}}{\text{QS}}$$

$$\frac{100}{49}=\frac{260^2\;-\;\text{PQ}^2}{260^2\;-\;\text{PQ}^2}$$

Solving we get PQ = 240.

SHORTCUT:

260 being the hypotenuse should immediately have bought the triplet (5, 12, 13) in your mind and the two perpendicular sides, PQ and QR would have been 100 and 240, but we yet dont know which is which.

250 being the hypotenuse should immediately have bought the triplet (3, 4, 5) in your mind and the two perpendicular sides, PQ and QS would have been 150 and 200, but we yet dont know which is which.

Now, PQ is the same in former triangle and also in latter triangle. But there is no common value to {100, 240} and {150, 200}

Think of another triplet with hypotenuse being a multiple of 25. Pause, think and then continue.

The triplet is (7, 24, 25). And this will result in PQ and QS being 70 and 240, not necessarily in the same order.

Now, since 240 is common to {100, 240} and {70, 240}, its kind of direct that PQ = 240 and QR = 100 and QS = 70. This 100 and 70 also aligns well with the ratio 10/7 given in the question. This is enough confirmation that PQ = 240 will be the answer.