Suppose he move 4 km downstream in x hours
Then,
Speed downstream = 4x
km/hr
Speed upstream = 3x
km/hr
∴48(4/x)+48(3/x)=14 or x=12

So, Speed downstream = 8 km/hr
Speed upstream = 6 km/hr
∴ Rate of the stream
= 12
(8 - 6) km/hr
= 1 km/hr

Boats and streams use the concept of relative speed, where we have one moving object on a surface which is in motion.

In this case, Speeds are added when moving in the same direction, and subtracted when moving in the opposite direction.



In the question above, we are given that for every 4 kms traveled downstream, he moves 3 kms upstream i.e the ratio of speeds \(S_d : S_u = 4 : 3\).


Let \(S_d = 4x\) and \(S_u = 3x\)

\(Total \space Time = \frac{d_u}{S_u} + \frac{d_d}{S_d}\)

\(14 = \frac{48}{4x} + \frac{48}{3x} = \frac{12}{x} + \frac{16}{x} = \frac{28}{x}\)

Therefore x = 2.


Substituting for x, \(S_d = 8\) and \(S_u = 6\)


\(S_s = \frac{S_d \space - \space S_u}{2}= \frac{8 - 6}{2} = 1\)



Option A



It is good to remember to derive the equations when doing questions on Boats and Streams

There are 4 equations based on this.


If \(S_b\) is the Speed of Boat in still water and \(S_s\) is the speed of the stream, then


Speed Downstream, \(S_d = S_b \space + \space S_s\) .... (1)

Speed Upstream, \(S_d = S_b \space - \space S_s\) ..... (2)


Adding Equations (1) and (2), we get \(S_b = \frac{S_d \space + \space S_u}{2}\) ... (3)


Subtracting the Equations, we get \(S_s = \frac{S_d \space - \space S_u}{2}\) ... (4)


Arun Kumar
_________________

You could go through the trouble of setting up equations and solving them. But if you think about it, you can use the fact that the distances with and against the stream are the same to make the problem much easier.

Since the distances are the same, the rates will be inversely proportional to the times. (Note: this wouldn't work if the distances were different each way.) Since the ratio for rates is 4:3, the times will be 3:4. Therefore, the time will be:

> Time, with stream: \(\frac{3}{7} × 14 = 6\) hours

> Time, against stream: \(\frac{4}{7} × 14 = 8\) hours

By the formula \(Rate = \frac{Distance}{Time}\), you get

> Rate, with stream: \(\frac{48}{6} = 8\) km/h

> Rate, against stream: \(\frac{48}{8} = 6\) km/h

Don't be fooled into thinking the answer is 2. The stream adds for the with-stream rate and subtracts for the against-stream rate. So, the no-stream rate must be 7 km/h and the speed of the stream must be (A) 1 km/h.
_________________

Let R = Speed of boat in still water

Let W = Speed of Stream

Given, that when time is Constant, he can travel 4 km in the same time he can travel 3 km

Rule: given constant time, the distance traveled is directly proportional to the speed traveled at


Ratio of——-speed with stream : speed against stream = 4 / 3

Now, the person is traveling the Same Distance of 48 km with the stream and 48 km back against the Stream in a round trip. Thus, Distance is Constant across these 2 parts of the Round Trip.

Rule: when Distance is Constant, the SPEED traveled at is INVERSELY Proportional to the TIME taken to travel over that Distance.

Ratios are also Inversely Proportional.


Speed With Stream : Speed Against Stream = 4 : 3

Time With Stream : Time Against Stream = 1/4 : 1/3 = 3 : 4

This means 3/7 of the Total travel Time of 14 hours is spent traveling WITH the stream—- (3/7) * 14 = 6 hours

And 4/7 of the Total Time of 14 hours is spent traveling AGAINST the stream —— (4/7) * 14 = 8 hours


Rule: Speed = (Distance traveled) / (travel Time taken)


Speed With Stream = R + W = 48 km / 6 hr = 8 km/hr

Speed Against Stream = R - W = 48 km / 8 hr = 6 km/hr


Rule: the following Speeds are in an Arithmetic Progression with the common difference = W = Speed of Stream


Speed WITH Stream = R + W = 8

Speed in Still Water = R = ?

Speed AGAINST Stream = R - W = 6

In any A.P., any 2 consecutive terms will have a Common Difference = d —— in this A.P. involving the Speeds, the Speed in Still Water = R = will equal:

-(W) from the Speed WITH the Stream —- and —- +(W) above the Speed AGAINST the Stream:

8 - W = R = W + 6

8 - W = W + 6

2 = 2*W

W = 1 km/hr = Speed of Stream

-Answer A-

Posted from my mobile device