Question

A tank fills completely in 2 hours if both the taps are open. If only one of the taps is open at the given time, the smaller tap takes 3 hours more than the larger one to fill the tank. How much time does each tap take to fill the tank completely?

Answer 1

Let the larger tap take x hours.

Smaller tap takes x + 3 hours.

Both fill the tank in 2 hours if they are open together.

Amount of water filled by larger tap in 1 hour = `1/x`

Amount of water filled by smaller tap in 1 hour = `1/(x + 3)`

Amount of water filled in 1 hour = `1/2`

`1/x + 1/(x + 3) = 1/2`

`(x + 3 + x)/(x(x + 3)) = 1/2`

`(2x + 3)/(x^2 + 3x) = 1/2`

2(2x + 3) = x² + 3x

4x + 6 = x² + 3x

0 = x² + 3x − 4x − 6

x² − x − 6 = 0 

x² − 3x + 2x − 6 = 0     ...`[(-3 xx 2 = -6), (-3 + 2 = -1)]`

x(x − 3) + 2(x − 3) = 0

(x − 3) (x + 2) = 0

x − 3 = 0 or x + 2 = 0

∴ x = 3 = 0 or x = −2

Thus, the larger tap takes 3 hours while the smaller tap takes 3 + 3 = 6 hours.