HMMT-Nov 2019 thm, Problem 1

For breakfast, Mihir always eats a bowl of Lucky Charms cereal, which consists of oat pieces and marshmallow pieces. He defines the luckiness of a bowl of cereal to be the ratio of the number of marshmallow pieces to the total number of pieces. One day, Mihir notices that his breakfast cereal has exactly 90 oat pieces and 9 marshmallow pieces, and exclaims, "This is such an unlucky bowl!" How many marshmallow pieces does Mihir need to add to his bowl to double its luckiness?

Solution: Let $x$ be the number of marshmallows to add. We are given that \[ 2 \cdot \frac{9}{99}=\frac{9+x}{99+x} \] Rearanging this gives \[ 2(99+x)=11(9+x) \] Thus $9 x=99$ and $x=11$. $\fbox{11}$.

Answer: 11