Aptitude - Boats and Streams - Discussion

We know,
Distance = Time x Speed.

Distance is same for both up and down streams => d = 36.
Speed of boat in still water is 10 => u = 10.

Let time for up stream be "t" and time for down stream becomes "t-(90/60)".

We need to find speed of the stream i.e; v = ? (say x).

From distance = Time x Speed => Time = Distance/Speed.

So, time for up stream t = 36/(10 - x) and,
Time for down stream (t - (90/60)) = 36/(10 + x).

For above to get 'x' value we need to subtract time for upstream and time for downstream.

=> t - (t - (90/60)) = [36/(10 - x)] - [36/(10 + x)].
=> 90/60 = 36{[1/(10 - x)] - [1/(10 + x)]}.
=> 90/60 = 36{ [(10 + x) - (10-x)]/[(10*10) - (x*x)]}.
=> 1/2 = 12 {[2x]/[100 - x2]}.
=> 100 - x2 = 12 * 2 * 2x.
=> 100 - x2 = 48x.
=> x2 + 48x - 100 = 0.

And here we got the complete equation with one variable "x" that is what we need to find so we can simply substitute options and check.

From the Option [A] will fit the above equation.

Let stream is considered as "S"

Speed Downstream = (10+S).
Speed Upstream =(10-S).

Upstream Speed - Downstream Speed = 90/60 (minutes converted to hour).

36/(10-S) - 36(10+S) = 90/60.

36(10+S) - 36(10-S) / (10+S) (10-S) = 3/2 ( Divided those) {(10+S) and (10-S) as LCM}.

360+36S-360+36S / 100-10S+10S-S^2= 3/2 (360 and 10s Subtracted).

72S / (100-S^2) = 3/2.

144S = 300 - 3S^2 (cross Multiplication).

3S^2 + 144S - 300 = 0.
S^2 + 48S - 100 = 0 (ALL are divided by 3).

S^2 + 50S - 2S -100 =0.
S( S+50) - 2(S+50) = 0.
(S+50) (S-2) = 0.

S = - 50 (Is not granted).
S = 2.

So the speed of the stream is 2.

Hi there,

I dont understand from where the 60 comes. is it a concept or derived from theose data? pls explain.

You should multiply (10 - x) and (10 + x) as to take LCM, so that you can substract the numeraters i.e. 36-36.

You knew that (a+b) x (a-b) = (a2 - b2) , (NOTE: read it as a square minus b square.)

And in order to multiply the denominators you got to multiply them with the numerators also; so, you get 36(10+x) - 36 (10-x), this is divided by (100 - x2 -i.e.square)

But what i do not understand is from where does that 60 come?

Anyone?

Speed=distance/time.

They had given that, the time difference between upstream and downstream as 90 min or 90/60 hours.

We know the relation that: speed=distance/time.

From that time=distance/speed.

Therefore: upstream time-downstream time= (upstream distance/upstream speed) - (downstream distance/downstream speed).

Which is equal to:.

90/60 hours= (36/10-X) - (36/10+X). (X=speed of stream).

From this by solving the equation we can get the X value as 2 miles/hour).

Therefore speed of stream=2 miles/hour (mph).

We have a formula,

The speed of boat = (Upstream speed + downstream speed)/2 -----> (1)
Let the actual time be x, then upstream speed = Distance/time =>36/(x-3/2),
<--3/2 came from coverting 90 min to hrs.-->
Similarly, downstream speed will be = 36/x.
Therefore, according to the Formula above (1), ( 36/(x-3/2) +36/x ) /2 = 10.
Onsolving we get Quadratic eqn as: 10x2- 51x +27 = 0.
I don't understand how this :x2 + 48x - 100 = 0 came?

Please help me to solve it.

@Rohit.

Distance/Speed = Time.

Taking x as speed of the stream.

Upstream distance/(Still_water_speed - x) or Upstream time minus Downstream_distance/(Still_water_speed - x) or Downstream time.

Equals to 90 mins/60 dividing by 60 as we want to bring all the time values to the same unit.

This thing is already mentioned in the problem.

"A boat takes 90 minutes less to travel 36 miles downstream than to travel the same distance upstream".

Since upstream takes 90 minutes more than downstream,
U(time) = 90 mins+ D( time)
Since time = Distance/speed.
D/ S = 90+D/S.
36/(10-x) = 90+36/(10+x).
36/(10-x) - 36/(10+x) = 90/60( converting into hrs)
36(1/10-x-1/10+x) = 3/2,
12((10+x-10+x)/10^2-x^2) = 1/24,
2x(24) = 10^2-x^2,
x^2+48x-10^2 = 0,
x^2+50x-2x-100 = 0,
x(x+50)-2(x+50) = 0,
(x-2)(x+50) = 0,
x=2,-50.
Since we neglect the negative number, then the answer will be 2mph.

x = 10.
36/(10-x) - 36/(10+x) = 90/60,
36/(10-x) - 36/(10-x) = 3/2.
then 12*4 = 36.
divide all no. * 12.

12/(10-x) - 12/(10-x) = 1/2.
(12/(x-y)- 12/(x+y)) / ((10-x)(10+x)) = 1/2.

Multiply by * 2 all equation.
24(10-x) - 24(10+x) = (10-x)(10+x),
240 + 24x - 240+ 24x = (10-x)(10+x),
24x + 24x = (10-x)(10+x),
48x = 100+10x-10x-x^2,
48x = 100- x^2,
48 = 100-x,
x= (100/48),
x = 2.

Take 36 common,
36(1/(10-x)-1/(10+x)) = 90/60.

36((10+x)-(10-x))/(100-x2) = 3/2 since in Denominator is in the form (a+b)(a-b) = a2-b2.
And 90/60 = 2/3.

36((10+x-10+x)/100-x2 = 3/2.
36(2x/100-x2) = 3/2.

72x/100-x2 = 3/2.
hence 72x*2 = 3(100-x2).

3x2+144x-300 = 0.
x2+48x-100 = 0.
(x+ 50)(x - 2) = 0.

x = 2 mph.
Hope you understand guys.

Let the speed of the stream be x mph.

Then sp. Downstream=(10+x) mph.

Sp. Upstream =(10-x) mph.
(36/10-x) - (36/10-x) =90/60.

Can be written as (36/10-x)- (36/10-x) = 9/6,

[360+36x-360+36x]6 = 9(10^2-x^2),

72x * 6= 900-9x^2,
9x^2+432x-900=0,
x^2 +48x -100=0.
(x+50)(x-2),
X=2mph.