Suppose that 80% of families own a DVD player and that 70% of families that own a computer. What is the range of possible percentages of families that own both?
I was able to see that the highest possible percentage of families that own both was 70%, since only 70% of families own a computer. However, I guessed the lower bound of the range by adding both percentages to get 80%+70% = 150% and subtracting 100% from it since it is impossible for over 100% of families to own both to get 50% of families that own both, which happened to be correct. Can someone please explain if this is the correct way to obtain the lower end of the range, and if so, why? Also, is there a general formula for the lower bound if the percentages are x% and y%?
It can very simply be determined by a line diagram.
Draw a line representing $100\%$ between $A$ and $B$
For maximum % owning both, draw the lines for both families from $A$ to $B$
$0\,0\,0\,0\,0\,0\,0\,0\,0\,0 $ 100% line
$0\,0\,0\,0\,0\,0\,0\,0\quad$ A 80%
$0\,0\,0\,0\,0\,0\,0\quad$ B 70%
Overlap 70%
$0\,0\,0\,0\,0\,0\,0\,0\,0\,0 $ 100% line
$0\,0\,0\,0\,0\,0\,0\,0\quad\quad$ A 80%
$\quad\quad0\,0\,0\,0\,0\,0\,0 \;\;\,$ B 70%
Overlap 50%