Luke has red marbles and blue marbles in a bag. If he chooses a marble at random, the probability that he will choose a blue marble is $\frac{2}{5}$. Luke adds 5 blue marbles to the bag and removes 5 red marbles. If he chooses a marble at random, the probability that he will choose a blue marble is now $\frac{3}{5}$. How many marbles are in the bag?
(A) 20
(B) 10
(C) 45
(D) 50
(E) 25

Part C: Each correct answer is worth 8.
Solution 1

Since the probability that Luke randomly chooses a blue marble is $\frac{2}{5}$, then for every 2 blue marbles in the bag there are 5 marbles in total.

Since there are only red and blue marbles in the bag, then for every 2 blue marbles, there are $5-2$ or 3 red marbles.

Thus, if there are $2 k$ blue marbles in the bag then there are $3 k$ red marbles in the bag (where $k$ is some positive integer).

If Luke adds 5 blue marbles to the bag, then the number of blue marbles in the bag is $2 k+5$. If in addition to adding 5 blue marbles to the bag Luke removes 5 red marbles from the bag, then the total number of marbles in the bag, $2 k+3 k$, remains the same.

The probability that Luke randomly chooses a blue marble from the bag is determined by dividing the number of blue marbles in the bag, $2 k+5$, by the total number of marbles in the bag, $5 k$.

Thus, $\frac{2 k+5}{5 k}=\frac{3}{5}$ or $5(2 k+5)=3(5 k)$.

Dividing both sides of this equation by 5 we have, $2 k+5=3 k$ and so $k=5$.

Therefore, the total number of marbles in the bag is $5 k=5(5)=25$.

## Solution 2

Let the initial number of marbles in the bag be $N$.

Since the probability that Luke randomly chooses a blue marble from the bag is $\frac{2}{5}$, then there are $\frac{2}{5} N$ blue marbles initially.

When 5 blue marbles are added to the bag and 5 red marbles are removed from the bag, the number of marbles in the bag is still $N$.

Since the probability of choosing a blue marble at the end is $\frac{3}{5}$, then there are $\frac{3}{5} N$ blue marbles in the bag at the end.

The difference between the number of blue marbles initially and the number of blue marbles at the end is 5 , since 5 blue marbles were added.

Thus, $\frac{3}{5} N-\frac{2}{5} N=5$ or $\frac{1}{5} N=5$ or $N=25$.

Therefore, the total number of marbles in the bag is 25 .

ANSWER: (E)