(Q12).Formulate the following problems as a pair of equations and then find their solutions.

(i) A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can go 40 km upstream and 55 km downstream. Determine the speed of the stream and that of the boat in still water.

Let the speed of the boat in still water = x kmph

and the speed of the stream = y kmph

then speed in downstream = x + y

Speed in upstream = x – y

and time =

By problem,

x-y
 +
x+y
 = 10
x-y
 +
x+y
 = 13
Take
1
x-y
 = and
1
x+y
 = , then

the given equations reduce to

a + b = 10 ……… (1)

a + b = 13 ……… (2)

equation (1) × ⇒ a + b =
equation (2) × ⇒ a + b =
b =
⇒ b =
Substituting b =
1
 in equation (1) we get
30a + 44(
1
 ) = 10
⇒ 30a = 10 - ⇒ a =
30
 =
but a =
1
x-y
 =
1
 ⇒ x - y = ............(3)
b =
1
x+y
 =
1
 ⇒ x + y = ............(4)

Adding (3) and (4);

x - y =

x + y =

x =

⇒ x =

Substituting x = in x – y = 5 we get

– y = 5

⇒ y = – 5 =

∴ The solution (x, y) = (, )

Speed of the boat in still water = kmph

Speed of the stream = kmph.