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Question 1206870: There were 5/7 as many red marbles as blue marbles in a jar. Dave took some blue marbles out of the jar and replaced them with the same number of red marbles. The number of red marbles became 5/9 of all the marbles in the jar. Which of the following is a possible number of blue marbles that were replaced?
Found 3 solutions by josgarithmetic, Edwin McCravy, greenestamps:
Answer by josgarithmetic(39616)   (Show Source):
Answer by Edwin McCravy(20054)  (Show Source):
You can put this solution on YOUR website!
Let R = the number of red marbles at the beginning.
Let B = the number of blue marbles at the beginning,
Let x = the the number of blue marbles exchanged for red marbles. <--the answer
There are R+B marbles at the beginning and at the end.
The given information
R, B, and x are all positive integers.
Since  B is a multiple of 7, say B=7p, where p is a positive
integer.
Then 
or R=5p
So p must be a multiple of 3, say p=3n, where n is a positive integer.
So every solution in integers is of the form.
You did not list the possible choices, but all you have to do is look for x
being a multiple of 5.
Let's check the smallest possible solution where n=1. B=21, R=15, x=5 There were 5/7 as many red marbles as blue marbles in a jar.That checks because 15=(5/7)(21). Now there are R+B=21+15=36 marbles total, before
and after the swapping is done. Dave took some blue marbles out of the jar and replaced them with
the same number of red marbles. The number of red marbles became 5/9
of all the marbles in the jar.When 5 blues are swapped for reds there are 21-5=16 blues and 15+5=20 reds,
and indeed, 20 = (5/9)(36). So that checks also.
Edwin
Answer by greenestamps(13198)  (Show Source):
You can put this solution on YOUR website!
Note that it is hard to know how to answer the question, since you didn't provide the answer choices.
But we can do some analysis to determine what kind of answers are possible.
Initially the number of red marbles was 5/7 the number of blue marbles. That means that the red marbles were 5/12 of the total and the blue marbles were 7/12 of the total.
Let x be the total number of marbles. Then (5/12)x is the original number of red marbles.
Let y be the number of blue marbles that were removed and replaced with red marbles.
After the change, the number of red marbles was 5/9 of the total number:
x and y are whole numbers. Since y=(5/36)x, the total number of marbles must be a multiple of 36, and the number of blue marbles that were removed and replaced with red marbles must be a multiple of 5.
There is not enough information to find a single possible answer for the number of blue marbles that were removed and replaced with red marbles. But we do know that...
ANSWER: any number that is a multiple of 5
Some possible sets of numbers for the problem....
36 total marbles; originally 15 red and 21 blue; 5 marbles changed to give 20 red and 16 blue; 20/(20+16) = 20/36 = 5/9
72 total; originally 30 and 42; 10 change to give 40 and 32; 40/(40+32) = 40/72 = 5/9
etc....
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