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10th
Maths
Pair of Linear Equations in Two Variables
Algebraic Methods of Solving a Pair of Linear Equations
A boat covers 32 km upstrea...
Maths
A boat covers 32 km upstream and 36 km downstream in 7 hours. Also, it covers 40 km upstream and 48km downstream in 9 hours. Find the speed of the boat in still water and that of the stream.
Medium
Answer
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Let the speed of the boat in still water be x km/hr and the speed of the stream but y km/hr. Then,
Speed upstream =(x−y)km/hr
Speed downstream =(x+y) km/hr
Now, Time taken to cover 32km upstream =
x−y
32
hrs
Time taken to cover 36 km downstream =
x+y
36
hrs
But, total time of journey is 7 hours.
∴
x−y
32
+
x+y
36
=7 ..(i)
Time taken to cover 40km upstream =
x−y
40
Time taken to cover 48 km downstream =
x+y
48
In this case, total time of journey is given to be 9 hours.
∴
x−y
40
+
x+y
48
=9 (ii)
Putting
x−y
1
=u and
x+y
1
=v in equations (i) and (ii), we get
32u+36v=7⇒32u−36v−7=0 ..(iii)
40u+48v=9⇒40u−48v−9=0 ..(iv)
Solving these equations by cross-multiplication, we get
36×−9−48×−7
u
=
32×−9−40×−7
−v
=
32×48−40×36
1
⇒
−324+336
u
=
−288+280
−v
=
1536−1440
1
⇒
12
u
=
8
v
=
96
1
⇒u=
96
12
and v=
96
8
⇒u=
8
1
and v=
12
1
Now, u=
8
1
⇒
x−y
1
=
8
1
⇒x−y=8 ..(v)
and, v=
12
1
⇒
x+y
1
=
12
1
⇒x+y=12 ..(vi)
Solving equations (v) and (vi), we get x=10 and y=2
Hence, Speed of the boat in still water =10 km/hr
and Speed of the stream =2km/hr.
Answer:
10 km per hour and 2 km per hour is the correct answer