Adults at a concert before a bus arrives

At a concert, exactly $\tfrac{5}{12}$ of the audience are adults. A bus then drops off $50$ more people, after which exactly $\tfrac{11}{25}$ of the audience are adults. What is the smallest possible number of adults present before the bus came?

$125$. The initial crowd $x$ must be a multiple of $12$, and $x+50$ a multiple of $25$, so $x$ is a multiple of $25$ too; the least is $x=300$, giving $\tfrac5{12}\cdot300=125$ adults. This is feasible: afterwards there are $154$ adults out of $350$, so $29$ adults rode the bus.