Two water taps together can fill a tank in hours. The tap with longer diameter takes 2 hours less than the tap with smaller one to fill the tank separately. Find the time in which each tap can fill the tank separately.

Or

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 time taken by smaller tap to fill the tank completely = x

hours

The volume of the tank filled by a smaller tap in 1 hour =

The time taken by larger tap to fill the tank completely = x - 2

hours

The volume of the tank filled by a larger tap in 1 hour

Time taken by the tank to fill hours

hours

The volume of the tank filled by a smaller tap in hours

The volume of the tank filled by a larger tap in hours

⇒ 15(x - 2) + 15x = 8x(x-2)

⇒15(x – 2 + x) = 8x(x – 2)

⇒ 15(2x - 2) = 8x² - 16

⇒8x² – 46x + 30 = 0

⇒2(4x² – 23x + 15) = 0

⇒4x² – 23x + 15 = 0

⇒4x² – 20x - 3x + 15 = 0

⇒ 4x(x -5) - 3(x - 5) = 0

⇒ (4x-3) (x -5) = 0

At x = 5

Time taken by smaller tap to fill the tank completely = 5 hours

Time taken by larger tap to fill tank completely = 5 – 2 = 3 hours

At

Time taken by smaller tap to fill the tank completely hours

Time taken by larger tap to fill the tank completely

 hours

Time is not possible to be negative.

So, is not possible.

OR

Let, speed of the stream = x km/h

And the speed of the boat = y km/h

So, the speed of the upstream = (y – x) km/h

And the speed of the downstream = (y + x) km/h

…(1)

…(2)


Let,



The equations will be:

30u + 44v - 10 = 0

40u + 55v - 13 = 0

By cross multiplication method we know,



Here a₁ = 30 , b₁ = 44 , c₁ = -10
a₂= 40, b₂ = 55, c₂ = -13

So,










Now,



⇒ 10 = 2(y - x)

⇒ 10 = 2y - 2x

⇒ 5 = y - x  .... (3)

⇒ 5 + x = y 

And



⇒ y + x = 11  ..... (4)

Put the value of y in eq (4),

⇒ 5 + x + x = 11

⇒ 5 + 2x = 11

⇒2x  = 6

⇒ x = 3

Put the value of x in (3) to get,

⇒  5 = y - 3

⇒ y = 8

So, speed of the stream = 3 km/h

And the speed of the boat = 8 km/h