According to 17th century data average life span was 26 years, however chance to live for no more than 8 years was 1/2. From that point estimate average life span of people, who have been living for not less than 8 years? Specify the range of possible values, considering only natural (non-decimal) numbers of years.
I have tried to estimate the mentioned range using extreme cases: such as all people who have survived for no more than 8 years actually lived 0 or 8 years, however I'm not sure whether the values I've got from that approach form range limits itself, and if so, how to explain the way of getting every value of range mentioned. The estimation method I've used is following: Generally speaking, the average life longevity is expected value, so to find maximum value for people, who lived not less than 8 years we should minimize contribution of people who lived not more than 8 years in the expected value, so we consider them all dying at 0 ages: $0 \cdot \frac{1}{2} + x \cdot \frac{1}{2}=26 \rightarrow x=52$, where x is average life span of people that lived more than 8 years, the same goes for min(x)=44, where we consider that all people, who have lived for not more than 8 years, actually died after living for 8 years, maximizing their contribution in expected value.
Looks good to me.
50% of people do not live more than eight years. All of these people could die at age 0 (upper bound) or they could all die at age exactly 8 (lower bound). Because we are only asked to find a range of possible values, this is all we require.
In the upper bound case, your math above it correct. The expected lifespan of the remaining 50% of the population is 52 ($2 \cdot 26$) years old.
In the lower bound case we have $0.5 \cdot 8 + 0.5 \cdot x = 26 \rightarrow x = 44$.