Question

Aboat goes 24 km upstream and 28 km downstream

in 6 hours. It goes 30 km upstream and 21 km

downstream in 6 hours 30 minutes. Find the speed

of the boat in still water.

Answer 1

if speed of the boat be x and stream speed be y 

from the given question, 

24/(x-y) + 28/(x+y) = 6 
30/(x-y) + 21/(x+y) = 6.5 

Taking 1/(x-y) as X and 1/(x+y) as Y 
24X + 28Y = 6 
30X + 21Y = 6.5 

solving for X and Y==> 
X = 1/6 and Y = 1/14 

so x-y = 6 and x+y = 14 
hence 
x = 10 kmph and y = 4kmph

Answer 2

Concept:

Speed is measured because the ratio of distance to the time within which the gap was covered. Speed may be a scalar quantity because it has only direction and no magnitude.

Given:

We are provided that $24km$ upstream and $28km$ downstream in $6$ hours of boat and in $6$ hours $30$ minutes of $30km$ upstream and $21km$ downstream.

Find:

We have to search out the speed of the boat in still water.

Solution:

Firstly, we assumed that the speed of the boat in still water as $xkm/hr$.

And, The speed of the stream as$y km/hr$

As we will know that the speed of the boat in upstream $= (x - y) km/hr$

Speed of the boat in downstream $= (x + y) km/hr$

So, time taken to cover$28 km$ downstream $= \frac{28}{(x+y)} hr$ $[ time = \frac{distance}{ speed}]$

Time taken to hide $24 km$ upstream $= \frac{24}{ (x - y)} hr$ $[ time = \frac{distance}{ speed}]$

It's providing the full time of journey is $6$ hours.

So, this could expressed as $\frac{24}{ (x - y)} +\frac{ 28}{ (x + y)} = 6$...... (i)

Similarly, Time taken to hide $30 km$ upstream $= \frac{30}{ (x - y)}$ $[ time = \frac{distance}{ speed}]$

Time taken to hide $21km$ downstream $=\frac{21}{(x+y)}$ $[ time = \frac{distance}{ speed}]$

And for this case the overall time of the journey is given as $6.5$ i.e $\frac{13}{2}$ hours.

Hence, we can write $\frac{30}{(x - y)} + \frac{21}{(x + y)} =\frac{ 13}{2}$ ..... (ii)

Hence, by solving (i) and (ii) we get the desired solution

Taking, $\frac{1}{ (x - y)} = u$ and $\frac{1} {(x + y)} = v$in equations (i) and (ii) we've (after rearranging)

$24u+28v-6=0$ ...... (iii)

$30 u + 21 v-\frac{13}{2} = 0$ ....... (iv)

Solving these equations by cross multiplication we get,

$\frac{u}{28x-6.5-21x-6}=\frac{v}{24x-6.5-30x-6}=\frac{1}{24\times 21-30 \times 28}$

$u=\frac{1}{6}$ and $v=\frac{1}{14}$

Now,

$\begin{aligned}u= \frac{1}{(x - y)}& = \frac{1}{6}\\x - y &= 6\end$.... (v)

$\begin{aligned}v = \frac{1}{(x + y) }&= \frac{1}{ 14}\\x + y &= 14\end$....... (vi)

On Solving (v) and (vi)

Adding (v) and (vi), we get

$\begin{aligned}2x&=20\\ x&=10\end$

Using $x=10$ in (v), we find $y$

$\begin{aligned}10+y&=14\\ y&=4\end$

Hence, Speed of the stream $= 4km/hr$ and Speed of boat$= 10km/hr$.

#SPJ3