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A jar is filled with red, white, and blue tokens that are eq

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A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 11 Nov 2013, 19:23
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A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?

(A) 9
(B) 12
(C) 15
(D) 18
(E) 21
[Reveal] Spoiler: OA
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Re: An jar is filled with red, white, and blue tokens that are e [#permalink]

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New post 11 Nov 2013, 19:37
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AccipiterQ wrote:
An jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?

(A) 9
(B) 12
(C) 15
(D) 18
(E) 21



This is from the Manhattan Advanced Quant Guide. Here's how I solved it:

Since we know that the probability of drawing R then W = probability of drawing b we have an equation

\(\frac{r}{(b+w+r)} * \frac{w}{(b+w+r)}=\frac{b}{(b+w+r)}\)

\(\frac{rw}{(b+w+r)^2}=\frac{b}{(b+w+r)}\)

rw*(b+w+r)=b*(b+w+r)^2

rw=b(b+w+r)

now at this point I paid attention to the fact that b, w, and r have to be multiples of 3. Also, there has to be fewer blue chips than red or white. So (A) was out, since the only way to break 9 into 3's is 3-3-3. So now looking at it again I realized the answer had to be able to be broken down into factors of 3, which could add back up to the answer (since b+w+r=total chips). So (B) wouldn't work, since the only way to break it into 3 factors of 3 that add up to 12 is 3,3,6, and both R AND W have to be larger than B. Next I looked at (C), the only way for that to work is 3, 6, 6, with b=3, r=6, w=6. So I plugged that into the equation

6*6=3(3+6+6)
36=45

So C was out. Next I looked at (D). With D the only way to break down 18 into 3 factors of 3 where at least 2 of them were larger than the third is 3,6, and 9. 12-3-3 wouldn't work, because again you need both r&w to be larger than b. 6-6-6 is also out for the same reason. So I checked it out:

6*9=3(3+6+9)

54=54

YAAAY! The answer is D.

This is probably the hardest problem I've solved on my own, so I was quite happy
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Re: An jar is filled with red, white, and blue tokens that are e [#permalink]

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New post 07 Apr 2014, 00:31
Oh!took me 10 minutes to figure out!!
And that too by options.

All of these options are are multiples of \(3\).So dividing them equally will not help
because in that case we will never have equal probability of selecting two tokens on the one hand
and selecting just one on the other.
By same logic,we can never have two same no. of differently colored tokens.
To fulfill the condition,we need to have all different nos. of differently colored tokens.

Then I checked the first three multiples of \(3:3,6\) & \(9\) added up to \(18\).

With all mental capacity exhausted,clicked \(18\)!
(Now I realize the question asks for smallest no. of tokens.I know this is shoddy logic :twisted: :twisted:
but still got it right!)
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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 07 Apr 2014, 02:42
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AccipiterQ wrote:
A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?

(A) 9
(B) 12
(C) 15
(D) 18
(E) 21


Let us say that there are r, w and b tokens.

The question says that (r + w)/(r + w + b) x 2! = (b)/(r + w + b)
Hence 2 r+ 2 w = b

Number of tokens is a multiple of 3 hence the minimum number will be r, b, w = 3, 3, 2(3) + 2(3) = 3, 3, 12 = 18

(The key is 2! - which is the number of ways r and w can arrange itself)
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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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AccipiterQ wrote:
A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?

(A) 9
(B) 12
(C) 15
(D) 18
(E) 21


Say, no of red balls is R, white balls is W and blue balls is B and total number of balls is T (= R + W + B)

(R/T) * (W/T) = B/T (Select a red token, replacing and then white token = Select a blue token)
RW = BT
3a*3b = 3c*(3a + 3b + 3c) (Since R, W and B must be multiples of 3, a, b and c must be positive integers)
ab/c = a+b+c
Since a, b and c must be 1 at least, a+b+c must be at least 3. But that will make the left hand side 1. Let's try to keep c as 1 since it reduces complexity and keeps the right hand side low. Now, which two numbers when multiplied give 1 more than when added. Think of small numbers like 1, 2 and 3 since the total is small as is apparent from the options. The values that satisfy this are 2 and 3.
2*3 = 6 which is 1 more than 2+3 = 5.

Hence total number of balls = 3 ( 1 + 2 + 3) = 18
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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 19 Jul 2015, 19:11
Hello from the GMAT Club BumpBot!

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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 19 Jul 2015, 21:53
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AccipiterQ wrote:
A jar is filled with red, white, and blue tokens that are equivalent except for their color. The chance of randomly selecting a red token, replacing it, then randomly selecting a white token is the same as the chance of randomly selecting a blue token. If the number of tokens of every color is a multiple of 3, what is the smallest possible total number of tokens in the jar?

(A) 9
(B) 12
(C) 15
(D) 18
(E) 21


(Red / Total Coins)*(White / Total Coins) = (Blue / Total Coins)

i.e. Red*White = Blue*Total Coins

Let, Red = 3a
White = 3b
Blue = 3c
Total Coins = 3(a+b+c)

i.e. 3a * 3b = 3c*3(a+b+c)
i.e. a*b = c*(a+b+c)
For smallest values of a, b and c
2*3 = 1*(1+2+3)

i.e. Minimum Total Coins = 3*(1+2+3) = 18

Answer: Option D
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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 07 Oct 2015, 11:09
So as stated above after some manipulations we have:
r*w=b*(r+w+b), If we consider a hint about probability, it says that there are less white balls than white or red ones.
Let's look at the answer choices:
(A) 9: Is out, because than we have r=w=b and we know that b is less
(B) 12: Is out, smallest multiple of 3 is 3=Blue, than we have B=3, W(>B)=6 and r=b
(C) 15: b=3, r=w=6 let's plug this values in the equation above -> 6*6=3*(6+6+3) is not true OUT
(D) 18: b=r=w=6 wrong combination, we need b=3, w=6, r=9 -> 6*9=3*(9+6+3) TRUE
(E) 21: b=3, w=r=9 -> 9*9=3*(9+9+3) is not true OUT
Answer (D)
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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 14 Nov 2016, 03:23
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: A jar is filled with red, white, and blue tokens that are eq [#permalink]

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New post 18 Feb 2017, 06:49
So, why is this question considered brutal? #1 it deals with probability, not a topic most GMAT students are super-familiar with, and #2 the question doesn’t give us any actual numbers, except telling us that the number of tokens of every color is a multiple of 3.

Let’s start with what we do know. There are “R” red tokens, “W” white tokens, and “B” blue tokens, and R + W + B = Total “T”. The probability of selecting a red token is R/T, and the probability of selecting a white token is W/T. To find the probability of two events occurring, we multiple the individual probabilities: (R/T)*(W/T) = RT / TT.

The question tells us this is equivalent to the probability of selecting a blue token: B/T = (RW)/(TT). This simplifies when we cross-multiply to (RW)/B = T.

The correct numbers of tokens in the jar will allow us to break down the number of red, blue, and white tokens such that they have the relationship (RW)/B = T. So let’s backsolve, and since the question asks for “smallest possible,” we’ll start with (A).

If T = 9, the only possible numbers of tokens are B = 3, R = 3, and W = 3. But since (3*3)/3 doesn’t equal 9, we know this isn’t a possible value for the total.

If T = 12, the only possible numbers of tokens are 3, 3, and 6. We don’t know which color has 6 tokens, but there’s still no way to make (RW)/B = T true for these values.

If T = 15, the numbers could be 3, 3, 9 or 3, 6, 6. It should still be somewhat clear that these numbers won’t work, but try out a couple the combinations to see if they work if you’re unclear: (3*9)/3 = 9, not our total of 15. Try the other set: (6*6)/3 = 12, not our total of 15.

If T = 18, the numbers could be 3, 3, 12, or 3, 6, 9, or 6, 6, 6. Try a few arrangements to see if you land on one that makes (RW)/B) = T true.

(6*9)/3 = 54/3 = 18, our total! Finally, we’ve got a set of numbers that makes the relationship between the tokens work!

Takeaway: Use the answer choices to your advantage as much as possible, but thoroughly analyze the relationships in the question stem first!
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Re: A jar is filled with red, white, and blue tokens that are eq   [#permalink] 18 Feb 2017, 06:49
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