lifetime of the 4p state of atom A is 21.4 ns and the lifetime of the 3s state of atom B is 14.4 ns.

At t = 77.5 ns the rate at which a sample of atom A in the 4p state is decaying is 5.20% of the rate at which a sample of atom B in the 3s state is decaying.

What's the ratio of the amounts of atom A in the 4p state and atom B in the 3s state at t = 0 s?

I have absolutely no idea how to start this, so just a shove in the right direction would be appreciated.

Say that we have an amount of A at t = 0 s, \(\displaystyle A_0\) and an amount of B at t = 0 s, \(\displaystyle B_0\). Let's say that the half-life of A is \(\displaystyle \tau _A\) and that of B is \(\displaystyle \tau _B\).

At a given time t we know that \(\displaystyle A = A_0 e^{- \tau_A ~t}\) and \(\displaystyle B = B_0 e^{- \tau _B ~ t}\). And finally, at t = 77.5 s \(\displaystyle \dfrac{dA}{dt} = 0.0520 \dfrac{dB}{dt}\)

So, what are the derivatives? How can you relate these to the amounts of A and B?

See if you can fill in the details. If you need more help, just ask.

-Dan