A boat takes 6 hours to cover 100 km
downstream and 30 km upstream. If the
boat goes 75 km downstream and returns
back to the starting point in 8 hours, find
the speed of the boat in still water and speed of sterm
Answers
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▶⏩ Let the speed of the boat in still water be x km/hr.
and, the speed of the boat in stream be y km/hr. Then,
↪➡ Speed upstream= ( x - y ) km/hr.
↪➡ and, speed downstream = ( x + y ) km/hr.
▶⏩ Time taken to cover 30km upstream
\boxed{ = \frac{30}{(x - y)} hrs}
=
(x−y)
30
hrs
▶⏩ Time taken to cover 24km downstream
\boxed{ = \frac{100}{(x + y)} hrs}
=
(x+y)
100
hrs
▶⏩ Total time taken = 6hrs.
\bf{ = > \frac{30}{x - y} + \frac{100}{x + y} = 6............(1)}=>
x−y
30
+
x+y
100
=6............(1)
▶⏩ Again, time taken to cover 75km upstream
\boxed{ = \frac{75}{(x - y)} hrs}
=
(x−y)
75
hrs
( Returning 75km back from 75km downstream).
▶⏩ Time taken to cover 75km downstream
\boxed{ = \frac{75}{(x + y)} hrs}
=
(x+y)
75
hrs
▶⏩ Total time taken 8hrs.
\bf{ = > \frac{75}{(x - y)} + \frac{75}{(x + y)} = 8................(2)}=>
(x−y)
75
+
(x+y)
75
=8................(2)
\bf \underline{putting \: \frac{1}{(x - y)} = u \: and \: \frac{1}{(x + y)} = v.}
putting
(x−y)
1
=uand
(x+y)
1
=v.
in equation (1) and (2), we get:-)
↪➡ 30u + 100v = 6.
↪➡ 15u + 50v = 3.....................(3)
and,
↪➡ 75u + 75v = 8.....................(4)
▶⏩ Multiply by 5 in equation (3).
we get,
↪➡ 75u + 250v = 15....................(5)
▶▶ Substract in equation (4) and (5).
75u + 250v = 15
75u + 75v = 8
(-)......(-)...........(-)
______________
=> 175v = 7.
\boxed{ = > \: v = \frac{7}{175} = \frac{1}{25} }
=>v=
175
7
=
25
1
▶⏩ Put the value of ‘v’ in equation (3).
\bf{ = > 15u + 50( \frac{1}{25} ) = 3}=>15u+50(
25
1
)=3
\bf{ = > 15u + 2 = 3.}=>15u+2=3.
\bf{ \: = > 15u = 3 - 2.}=>15u=3−2.
\bf{ = > \: u = \frac{1}{15} .}=>u=
15
1
.
\bf{ = > \frac{1}{(x - y)} = \frac{1}{15} .}=>
(x−y)
1
=
15
1
.
\boxed{ = > \: x - y = 15...............(6)}
=>x−y=15...............(6)
and,
\bf{ \: = > v = \frac{1}{25} .}=>v=
25
1
.
\bf{ = > \frac{1}{(x + y)} = \frac{1}{25} .}=>
(x+y)
1
=
25
1
.
\boxed{ = > \: x + y = 25..........(7)}
=>x+y=25..........(7)
▶⏩ Substract equation (6) and (7).
x - y = 15
x + y = 25
(-)...(-)....(-)
_________
=> -2y = -10
\boxed{ \: = > y = \frac{ - 10}{ - 2} = 5.}
=>y=
−2
−10
=5.
▶⏩ Now, put the value of ‘y’ in equation (6).
↪➡ x - 5 = 15.
↪➡ x = 15 + 5.
\bf \boxed{ \: = > x = 20.}
=>x=20.
✅✅Hence, the speed of motorboat in still water is 20 km/hr.
and , the speed of motorboat in stream is 5 km/hr.✔✔
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