So I have these estimation of survival probabilities for men:
The probability that a man lives at least $70$ years is $80\%$.
The probability that a man lives at least $80$ years is $60\%$.
The probability that a $80$ years old man lives at least $90$ years is $40\%$.
The first question was what is the probability that a man lives at least $80$ if he has just celebrated his $70^{th}$ birthday?
What I did : I let $A$ be a man who reached $70$, and let $B$ be a man who reached $80$
$$P(A)=0.80$$ $$P(B)=0.60$$
$$P(B|A)=\frac{P(A \cap B)}{P(A)}=\frac{P(B)}{P(A)}=\frac{0.60}{0.80}=0.75$$
My question is what is the probability that this man lives at least $90$ years?
Following your notation, let $C$ be the unconditional probability of living to $90$ years. Then you are given $$\Pr[C \mid B] = 0.4,$$ that is to say, given that a man has lived to $80$ years, the probability of living to $90$ is $0.4$. Then what you want to find is $$\Pr[C \mid A] = \frac{\Pr[C \cap A]}{\Pr[A]} = \frac{\Pr[C]}{\Pr[A]}.$$ But $$\Pr[C \mid B] = \frac{\Pr[C \cap B]}{\Pr[B]} = \frac{\Pr[C]}{\Pr[B]},$$ so $$\Pr[C] = \Pr[C \mid B]\Pr[B]$$ and $$\Pr[C \mid A] = \frac{\Pr[C]}{\Pr[A]} = \frac{\Pr[C \mid B]\Pr[B]}{\Pr[A]}.$$ Now you can substitute all the values you have to get the answer.