# Marble Puzzle

Infront of you is an urn containing an unknown mixture of red, white, and blue marbles. The marbles are thoroughly mixed.

- You reach into the urn and pull out marble, which turns out to be red. Your friend then looks inside the urn and states,
*"One ninth of the marbles remaining in the urn are red"*. - You replace the red marble, mix them up again, and pull out two new marbles (without replacement); both of these marbles are blue. Your friend looks inside the urn again and this time states,
*"Now, one seventh of the marbles remaining in the urn are red"*. - He then tells you,
*"Oh, and there are ten white marbles in the urn"*. - How many blue marbles are there in total?

*How many blue marbles are there in total?*

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## Solution

Let

**R**,**W**,**B**be the number of marbles in the appropriate colour. We'll also use**M**to represent the total number of marbles**(R+W+B)**.From the first pull, we known that

**(R - 1)/(M - 1) = 1/9**.Re-arranging this, we determine

**M = 9R - 8**From the second pull, we know that

**R/(M - 2) = 1/7**.Substituting

**M**from a above we get**7R = 9R - 8 - 2**.Solving this, we find that

**R = 5**. There are five red marbles.Plugging this back in we can determine that

**M = 37**. There are 37 marbles in total.Because we know

**W = 10**, then we can determine that**B = 22**.There are 22 blue marbles in the urn.

*There are 22 blue marbles in the urn.*