There are $36$ girls in a class. $19$ of the girls have dark hair and $22$ of them have blue eyes. How many girls have both dark hair and blue eyes?
My attempt was this: Let $d_w$ be girls with dark hair without blue eyes, $d_b$ be girls with dark hair with blue eyes, $nd_w$ be girls with non-dark hair without blue eyes and $nd_b$ be girls with non-dark hair with blue eyes.
Then $d_w + d_b + nd_w + nd_b =36 ; d_w + d_b=19 ; d_b+nd_b=22$
But I can't find $d_b$. What am I doing wrong in this method?
There is a range of possible answers.
It may be that all 19 dark-haired girls are among those who have blue eyes, giving us the answer 19. (We can't have more than this as we don't have enough dark-haired girls).
The minimum can be found by seeing how many girls we'd need in total if there were no dark-haired, blue-eyed girls. Then the 22 and the 19 must have no intersection and there would need to be 41 girls altogether. As in fact we have only 36, there must be at least 5 girls in both categories.
So the answer to the question is "between 5 and 19, inclusive". It is not possible to pin down the answer to a single number with the information given.