Question 1167077
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A curious problem -- I have to wonder if the information given is not correct.<br>
Defining N as the number of red gumballs added to the bag suggests that N should be a positive integer; but it turns out to be negative....<br>
Given the original ratio 2:3:5...<br>
let 2x = number of blue originally
let 3x = number of red originally
let 5x = number of purple originally<br>
let A be the number of blue to be added.  Then after adding more gumballs,<br>
2x+A = number of blue
3x+N = number of red
5x+2N = number of purple<br>
After more gumballs are added, the probability of drawing a blue is 1/6, the probability of drawing a red is 2/6, and the probability of drawing a purple is 3/6.  That means the number of red is 2 times the number of blue, and the number of purple is 3 times the number of blue.<br>
{{{3x+N = 2(2x+A)}}}
{{{3x+N = 4x+2A}}}
{{{x = N-2A}}}<br>
{{{5x+2N = 3(2x+A)}}}
{{{5x+2N = 6x+3A}}}
{{{x = 2N-3A}}}<br>
So now<br>
{{{N-2A = 2N-3A}}}
{{{N = A}}}<br>
So the answer to the problem as stated is answer A: the number of blue gumballs to be added is equal to N.<br>
But now let's put some actual numbers in the problem.<br>
The number of blue gumballs is now 2x+N, and the number of red gumballs is 3x+N.<br>
And since the probability of drawing a red is 2/6 while the probability of drawing a blue is 1/6, the number of red gumballs is twice the number of blue gumballs:<br>
{{{3x+N = 2(2x+N)}}}
{{{3x+N = 4x+2N}}}
{{{N = -x}}}<br>
Now what we have is this:<br>
original number of blue gumballs: 2x
original number of red gumballs: 3x
original number of purple gumballs: 5x<br>
number of blue gumballs "added": -x
number of red gumballs "added": -x
number of purple gumballs "added": -2x<br>
ending number of blue gumballs: 2x-x = x
ending number of red gumballs: 3x-x = 2x
ending number of purple gumballs: 5x-2x = 3x<br>
And we see from this that the final condition of the problem is satisfied:<br>
P(blue) = x/6x = 1/6
P(red) = 2x/6x = 2/6
P(purple) = 3x/6x = 3/6<br>