Question

It can take 12 minutes to fill a water tank using two pipes. If the pipe of smaller diameter is used for 9 minutes
and the pipe of larger diameter for 4 minutes, only, half the tank can be filled.

The time taken by the pipe of smaller diameter to fill the tank is

The time taken by the pipe of larger diameter to fill the tank is

If both the
pipes are working together for 10 minutes, then the tank will filled by

The ratio
of time taken by the pipes can
be​

Answer 1

Let the rate to fill the smaller pipe be 1/a units/min
let the rate to fill the larger pipe be 1/b units/min
combined rate = 1/a + 1/b = (a+b)/(ab)

so 1/( (a+b)/(ab) ) = 12
ab/(a+b) = 12
ab = 12a + 12b

9(1/a) + 4(1/b) = 1/2 <------ (1 unit was the full tank)
9/a + 4/b = 1/2
(4a + 9b)/(ab) = 1/2
ab = 8a + 9b

then
12a + 12b = 8a + 18b
4a = 6b
a = 3b/2

sub into ab = 12a + 12b
(3b/2)(b) = 12(3b/2) + 12b
multiply by 2
3b^2 = 36b +24b
b^2 = 20b
b = 20 and a = 2b/2 = 30

It takes 30 min for smaller pipe to fill the tank, and 20 minutes for the larger.

Carry on to answer your other questions.