A radioactive nucleus can decay by two different process. The half life for the first process is t1 and that for the second is t2. Show that the effective half life t of the nucleus is given by
1/t = 1/t1 + 1/t2

It's similar to the problem of having a house to build and men to help build it. The more man available, the faster the house will be built and vice versa.

If there was only the half life, the nucleus would have decayed at a certain rate. Since there is another way of decaying as well, the nucleus decays quicker.

If you had t = t1 + t2, this won't be good, because if it took t1 amount of time alone to decay by a certain amount and t2 amount of time alone to decay through the other way by the same amount, together they shouldn't take more time than their sum.

So, you take the 'inverse' of each time, giving:

\frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2}