In a bag with 20 marbles, five of them are blue. How many blue marbles must be added to the bag so that the probability of selecting a blue marble at random is $\frac{1}{2}$?
I want to find the number of blue marbles that would make the proportion of blue marbles in the bag equal to $\frac{1}{2}$. Let x be the number of blue marbles to add. Then the total number of marbles in the bag would be 20 + x, and the number of blue marbles would be 5 + x. The proportion of blue marbles would be $\frac{5 + x}{20 + x}$. I want this proportion to equal $\frac{1}{2}$, so I can set up an equation: $\frac{5 + x}{20 + x} = \frac{1}{2}$. To solve for x, I can cross-multiply and simplify: $10 + 2x = 20 + x$, so $x = 10$. This means I need to add 10 blue marbles to the bag. 