What's the Half-Life for a Sample with Both Gamma and Beta Decay?

[ksac]

Homework Statement

A sample of radioactive nuclei of a certain element
can decay only by gamma -emission and beta -emission. If
the half-life for gamma -emission is 24 minutes and that
for beta -emission is 36 minutes, what's the half-life for the sample?


2. The attempt at a solution

I don't really know how to go about it.

First i thought that may be the half life of the sample should be the half life of beta decay as gamma radiation does not transform the sample into a new element, but only into it's nuclear isomer; but the answer isn't 36 minutes. So clearly it means a new nuclear species by all means.

Then I calculated the decay constant, lambda, for both, took the average and converted the constant to half life.
that still doesn't match the answer given.

How do i do this?

 

[ksac]

Oh i just realized I shouldn't be averaging the decay constants.
Decay constant = Number of decays per unit time.
So the sum of the two decay constants would give me the total decays in unit time (Gamma or bets).
0.693 times the inverse of that would be my half life.

Forgive me for being too quick to share my confusion. :)