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.