A tap fills a tank in half an hour. The same tap fills a tank in a third of an hour. How long will they take to fill the same tank altogether?

Let's denote the rate at which the tap fills the tank as x tanks per hour.

From the information given, we can set up the following equations:

1. x * 0.5 = 1 (the tap fills the tank in half an hour)
2. x * (1/3) = 1 (the tap fills the tank in a third of an hour)

Solving these equations, we find that x = 2 tanks per hour.

Therefore, the tap can fill the tank in 1/2 hour or 0.5 hours.

To find out how long it will take for both taps to fill the tank together, we add their rates:

1/0.5 + 1/(1/3) = 2 + 3 = 5 tanks per hour

Therefore, it will take 1/5 hours for both taps to fill the tank together, which is equal to 12 minutes.

The first tap fills 1/2 of the tank in 1 hour
The second tap fills 1/3 of the tank in 1 hour
So, both the taps will fill 1/2 + 1/3 = 5/6 of the tank in 1 hour
To fill the entire tank, they will take 1/(5/6) = 6/5 hours
= 1 hour and 12 minutes