Q: A boat covers 32km upstream and 36 km downstream in 7 hours. Also it cobers 40 km upstream and 48 km downstream in 9 hours . Find the speed of boat in still water and that of stream
MY SOLUTION:
Let the speed of boat be x km/hr and speed of stream be y km/hr
speed of boat when going upstream: $(x-y)$ km/hr
speed of boat when going downstream: $(x+y)$ km/hr
Case 1: when total time taken is 7 hrs
when going upstream: $t_1 = \frac{32}{x-y}$
when going downstream: $t_2 = \frac{36}{x+y}$
And $t_1+t_2=7$ hours
$$ \implies \frac{32}{x-y}+\frac{36}{x+y}=7 $$
Let $\frac{1}{x} = m$ and $\frac{1}{y}= n$
$$ \implies 32m-32y+36m+36n=7 \implies 68m+4n=7 $$ ... eq(i)
Case 2: when total time is 9 hours
When going upstream: $t_1= \frac{40}{x-y}$
When going downstream: $t_2=\frac{48}{x+y}$
AND $t_1+t_2=9$ hours
$$ \implies \frac{40}{x-y}+\frac{48}{x+y}=9 $$
As $\frac{1}{x} = m$ and $\frac{1}{y}= n$
$$ \implies 88m-8n=9 $$ ...eq(ii)
multiplying eq(i) with 2
$ 136m+8n=14$ ..eq(iii)
and solving eq(iii) and eq(ii) by elimination: [eq(iii)+eq(i)]
$224m=23$
This gives a decimal value and not a natural number.
the correct answer is $x=10$ km/hr and $y=2$ km/hr and the answer took $\frac{1}{x+y}=a$ & $\frac{1}{x-y}=b$
Let the speed of the boat in still water = x kmph Speed of the stream = y kmph
i ) relative speed of the boat in downstream
= ($ x$ + $y$ ) kmph
Distance travelled =$ d_1 $=$ 36$
Time = $t_1 $hr
$t_1 $=$\frac { d1} { s_1}$
$t1 $= $\frac {36}{ ( x + y ) }$
ii) relative speed of the boat in upstream = $( x - y )$ kmph
Distance =$ d_2 $=$ 32$ km Time =$ t_2 $
$t_2 $=$\frac{ 32}{ ( x - y )}$
Therefore ,
Total time = $7$ hr
$t_1$ +$ t_2 $=$ 7$hr
$\frac{36 }{( x + y )}$ +$ \frac{32}{( x - y )} = 7$ ----( 1 )
iii) second time ,
Relative speed of the boat in downstream =$ ( x + y )$ kmph
$d_3 = 48$ km
Time = $t_3$
$t_3 = \frac{48}{ ( x + y )}$
iv ) in upstream Relative speed of the boat =$ ( x - y ) $kmph
time =$ t_4$ hr
$d_4 = 40$km
$t_4 = \frac{40}{( x - y )}$
Total time =$ 9$ hr
$\frac{48}{( x + y )} + \frac{40}{( x - y )} = 9$ ---( 2 )
Let $\frac{1 }{( x + y ) }= a$ ,
$\frac{1}{( x - y )} = b$
Then rewrite ( 1 ) and ( 2 ) we get
$36 a + 32 b = 7$ -----( 3 )
$48a + 40b = 9$ ------( 4 )
Multiply ( 4 ) with 4 and equation ( 3 ) with 5 and
$192a + 160b = 36$ ---( 5 )
$180a + 160b = 35$ -----( 6 )
Subtract ( 6 ) from ( 5 )
we get
$a = \frac{1}{ 12 }$
put $a = \frac{1}{12}$ in ( 3 )
we get ,
$b = 1/ 8$
Now $\frac{1}{ ( x + y ) }= 1/ 12 $
$\frac{1}{ ( x - y ) }= 1/ 8$
Therefore ,
$x + y = 12 $----( 7 )
$x - y = 8 $----- ( 8 )
add ( 7 ) and ( 8 )
$2x = 20$
$x = 10$
put x = 10 in ( 7 ) we get
$y = 2$
Speed of the boat in
still water =$ x $= $10$ kmph speed of the stream= $y $= $2$kmph