The area of a square is $$𝑑.$$ What is the area of the circle which has the diagonal of the square as its diameter?

“A _________ investigation can sometimes yield new facts, but typically organized ones are more successful.”

The word that best fills the blank in the above sentence is

What would be the smallest natural number which when divided either by $$20$$ or by $$42$$ or by $$76$$ leaves a remainder of $$7$$ in each case?

In appreciation of the social improvements completed in a town, a wealthy philanthropist decided to gift Rs $$750$$ to each male senior citizen in the town and Rs $$1000$$ to each female senior citizen. Altogether, there were $$300$$ senior citizens eligible for this gift. However, only $$8/9$$^th of the eligible men and $$2/3$$^rd of the eligible women claimed the gift. How much money (in Rupees) did the philanthropist give away in total?