Problem (Algebra):
A 75 year old person has a $50 \%$ chance of living at least another 10 years.

A 75 year old person has a $20 \%$ chance of living at least another 15 years. An 80 year old person has a $25 \%$ chance of living at least another 10 years. What is the probability that an 80 year old person will live at least another 5 years?

Solution:
Consider a population of 100 people, each of whom is 75 years old and who behave according to the probabilities given in the question.

Each of the original 100 people has a $50 \%$ chance of living at least another 10 years, so there will be $50 \% \times 100=50$ of these people alive at age 85 .

Each of the original 100 people has a $20 \%$ chance of living at least another 15 years, so there will be $20 \% \times 100=20$ of these people alive at age 90 .

Since there is a $25 \%$ ( or $\frac{1}{4}$ ) chance that an 80 year old person will live at least another 10 years (that is, to age 90), then there should be 4 times as many of these people alive at age 80 than at age 90 .

Since there are 20 people alive at age 90 , then there are $4 \times 20=80$ of the original 100 people alive at age 80 .

In summary, of the initial 100 people of age 75, there are 80 alive at age 80,50 alive at age 85 , and 20 people alive at age 90 .

Because 50 of the 80 people alive at age 80 are still alive at age 85 , then the probability that an 80 year old person will live at least 5 more years (that is, to age 85 ) is $\frac{50}{80}=\frac{5}{8}$, or $62.5 \%$.

Suppose that the probability that a 75 year old person lives to 80 is $p$, the probability that an 80 year old person lives to 85 is $q$, and the probability that an 85 year old person lives to 90 is $r$.

We want to the determine the value of $q$.

For a 75 year old person to live at least another 10 years, they must live another 5 years (to age 80) and then another 5 years (to age 85). The probability of this is equal to $p q$. We are told in the question that this is equal to $50 \%$ or 0.5 .

Therefore, $p q=0.5$.



For a 75 year old person to live at least another 15 years, they must live another 5 years (to age 80), then another 5 years (to age 85), and then another 5 years (to age 90). The probability of this is equal to $p q r$. We are told in the question that this is equal to $20 \%$ or 0.2 .

Therefore, $p q r=0.2$

Similarly, since the probability that an 80 year old person will live another 10 years is $25 \%$, then $q r=0.25$.

Since $p q r=0.2$ and $p q=0.5$, then $r=\frac{p q r}{p q}=\frac{0.2}{0.5}=0.4$.

Since $q r=0.25$ and $r=0.4$, then $q=\frac{q r}{r}=\frac{0.25}{0.4}=0.625$.

Therefore, the probability that an 80 year old man will live at least another 5 years is 0.625 , or $62.5 \%$.

Final answer: 62.5%