Set theory text question 1) Atleast 70% participants had a
2) Atleast 75% participants had b
3) Atleast 80% participants had c
4) Atleast 85% participants had d
What is the least percentage of participants who had all a, b, c and d
Example of a real life situation where least percentage applies.
 A: Let P, Q, R, S be the properties not A, not B, not C, not D.
Then $30\%$ have P, $25\%$ have Q, $20\%$ have R, and $15\%$ have S. We want to maximize the number of people who have at least one of P, Q, R, S. This max is $90$, achieved by making P, Q, R, and S pairwise disjoint.  So A, B, C, D have at least $10\%$ in common. 
As to real-life example, I will pass, wishing to avoid censorship. But it is useful to produce a concrete example. We have $100$ people, of whom $10$ have all of A, B, C, D. 
Of the remaining $90$ people, $15$ have A, B, C; $20$ have A, B, D; $25$ have A, C, D; and $30$ have B, C, D. It is not hard to check that the sums all come out right.
A: The least percentage is:
$$100\% - (100\%-70\%) - (100\%-75\%) - (100\% -80\%) - (100\%-85\%) = 100\% - 30\% - 25\% - 20\% - 15\% = 10\% $$
The logic behind that is the following. We'll just check how many percent of the people have $A$ and $B$. From the condition $75\%$ have $B$. We know that $30\%$ of the people don't have A so in worst case scenario every people who don't have $A$ have $B$ so the number who have both $A$ and $B$
$$100\% - (100\%-70\%) - (100\%-75\%) = 100\% - 30\% - 25\% = 55\%$$
So at least 55% percent have $A$ and $B$. Now just expand that on 4 objects.
