A jar contains x red marbles, y white marbles and z blue marbles wherE

Official Solution:


A jar contains \(x\) red marbles, \(y\) white marbles, \(z\) blue marbles, where \(x > y > z\), and no other marbles. How many red marbles are there in the jar?

(1) To ensure that at least one marble of each color is removed from the jar, minimum 55 marbles must be removed

How many marbles should we remove to ensure, to absolutely guarantee that we have at least one marble of each color? Three? No, because all three might be say red and in this case we won't have at least one marble of each color. Ten? No, because all ten might be say red and in this case we won't have at least one marble of each color. So, how many? To answer this question we should consider the worst case scenario, the scenario in which we for sure, for 100% are getting at least one marble of each color.

The worst case scenario will be if we pick all \(x\) red marbles first and all \(y\) white marbles next. At this stage we still won't have one marble of each color but the next marble will for sure be blue and thus we will have one marble of each color.

So, this statement means that \(x + y + 1 = 55\). Not sufficient.

(2) To ensure that a red marble is removed from the jar, the smallest number of marbles that must be removed is 51

The worst case scenario will be if we pick all \(y\) white marbles first and all \(z\) blue marbles next. At this stage we still won't have a red marble but the next marble will for sure be red.

So, this statement means that \(y + z + 1 = 51\). Not sufficient.

(1)+(2) We have \(x + y= 54\), \(y + z = 50\) and \(x > y > z\).

Since \(x + y= 54\) and \(x > y\), then \(y z\), then \(y>25\) (\(y\) must be more than half of 50).

\(y < 27\) and \(y > 25\) means that \(y=26\).

So, \(x=28\), \(y=26\), and \(z= 24\). Sufficient.


Answer: C
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The constraint that z < y < x tells us that x,y,z are distinct integers.
Per Statement 2, y+z=50. So y and z cant both be 25. The next logical case we can examine is if y=26 and z=24.
Per Statement 1, x+y=54. Since y=26, that makes x=28. This gives us the answer everyone's posting, (x,y,z) = (28,26,24).

Let's take the next case, y=27 and z=23. This fulfills Statement 2. To fulfill Statement 1, x has to equal 27. But this violates the constraint that x,y,z are distinct integers. And for all other increasing values of y, x will be less than y, which violates the constraint that y<x.

Another way to think about it is to subtract Statement 2 from Statement 1. What you get: x-z=4. This means that the range of the set is 4. We found that (x,y,z) = (28,26,24) satisfies both statements. If we try other values of y and z that satisfy Statement 2, there's no way the set will have a range of 4.

Hope that helps.

Statement 1: x+y+1 = 55

NOT SUFFICIENT

Statement 2: y+z+1 = 51

NOT SUFFICIENT

Combining the two statements

x+y = 54
y+z = 50

i.e. x - z = 4

Case 1: x =28, y =26, z = 24

SUFFICIENT

Answer: Option C

Check that there are total x red, y white, and z blue marbles.
As per statement 1, minimum 55 marbles must be removed to ensure at least 1 marble of each color is removed

Let's assume we are to ensure that we will pick 1 blue marble
Now, in the worst case scenario, we will pick all the red and white marbles first; and once there are no more red or white marbles, the next pick will be by default blue marble
This is exactly what is given in statement 1

It is also mentioned that x > y > z, that is why when we consider the worst case scenario, we form the equation as x + y + 1 = 55
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statement 1... X+Y+1=55.....X+Y=54
statement 2...Y+Z+1=51......Z+Y=50
NOW X>Y>Z
This gives y=26,x=28,z=24

from statement 1, how do we know if x+y=54, wouldn't y+z or x+z be possibilities to consider as well? Thanks.

This is an excellent question which not only tests you on your probability concepts but also on your ability to conjure up numbers to fit a particular situation. It’s only fair that this is classified as a 700 level question.

In case you are not able to figure out how to interpret the statements given, you can take smaller values for the number of marbles, establish a pattern and apply it to the values given in the statements.

For example, if we assume 5 red marbles, 4 white marbles and 3 blue marbles – remember, all three values have to be distinct since it’s clearly mentioned that x>y>z – let’s see what happens with statement I and statement II.

To ensure that at least one marble of each color is removed from the jar, we have to remove a minimum of (5+4+1) = 10 marbles.

But what if we did the other way round – (3+4+1) = 8 marbles?

Remember that you have to ensure that you have 1 marble of each color. You can take out 5 of red and 3 of blue and still have 8 marbles. But this way, you don’t have 1 marble of all colors. So, if we have to ensure 1 marble of each color, we have to exhaust the red and white because they are higher in number. Exhausting them will automatically ensure the next one is blue. Sounds right?

Similarly, to ensure that a red marble is removed from the jar, we have to remove a minimum of (3+4+1) = 8 marbles.

We can now adopt a similar approach to interpret the data given in the statements.

From statement I alone, we can say x + y + 1 = 55, which leads us to x + y = 54. Since x and y are different, x can be 28 and y can be 26; or x can be 29 and y can be 25 and so on. Clearly, this is not sufficient to find the number of red marbles in the jar uniquely. Options A and D can be eliminated. Possible answers are B, C or E.

From statement II alone, we can say 1 + y + z = 51, which gives us y + z = 50. Again, y and z can be different values. But, whatever they are, we have no information about x from this statement. Hence, this is insufficient. Option B can be eliminated.

Combining both statements I and II, we have x + y = 54 and y + z = 50. Solving these equations, we get x – z = 4. Also, x>y>z. The only set of values to satisfy all the above conditions, is {x=28, y=26, z= 24}.
Therefore, the correct answer option is C.

Getting this question right, definitely depends on whether you were able to get the equations right. But, more importantly, it also depends on how you used the inequality – x>y>z.
This is a question which can take about 2 minutes to solve, considering the difficulty level and also because you need to try values.

Hope this helps!
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(1) To ensure that at least one marble of each color is removed from the jar, minimum 55 marbles must be removed insufic.

worst case scenario: x+y+1=55…x+y=54…x>y: (x,y)=(28,26;40,14;34,20…)

(2) To ensure that a red marble is removed from the jar, the smallest number of marbles that must be removed is 51 insufic.

worst case scenario: y+z+1=51…y+z=50…x=?

(1 & 2) sufic.

x+y=54-(y+z=50)…x-z=4…x=z+4…(z+4)>y>z…y=(z+1,z+2,z+3)
y=z+1: z+1+z=50…2z=49≠integer
y=z+3: z+3+z=50…2z=47≠integer
y=z+2: z+2+z=50…2z=48…z=24…x=z+4=28

Answer (C)

Can somebody please explain how you got x+y+1=55 from the 1st statement. There is no mention anywhere that we are taking the blue marble out. 1 could signify x,y or z.

Thank you.

My understanding is that the reason why
Statement 1: x+y+1 = 55
Statement 2: y+z+1 = 51
is sufficient even though we have 3 variables and 2 equations is because of the additional constraint that 0 > z > y > x. Is there an easy way to confirm that y=26,x=28,z=24 is the ONLY answer? What's a good way to solve this question?

(1) To ensure that at least one marble of each color is removed from the jar, minimum 55 marbles must be removed

We know that the most amount of marbles of a given color is x, since x > y > z, and then next greatest is y. Therefore, in order to guarantee that you have one of each color, you assume the worse case scenario, which is that you first pick out all x of the red marbles, then all y of the white marbles, then FINALLY you pick 1 of the blue marbles. Hence, you get \(x + y + 1 = 55 \implies x + y = 54\)

Par of GMAT CLUB'S New Year's Quantitative Challenge Set

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Given; A jar contains x red marbles, y white marbles, z blue marbles, where x > y > z, and no other marbles.

Asked: How many red marbles are there in the jar?

(1) To ensure that at least one marble of each color is removed from the jar, minimum 55 marbles must be removed
x + y = 54
NOT SUFFICIENT

(2) To ensure that a red marble is removed from the jar, the smallest number of marbles that must be removed is 51
y + z = 50
NOT SUFFICIENT

(1) + (2)
(1) To ensure that at least one marble of each color is removed from the jar, minimum 55 marbles must be removed
(2) To ensure that a red marble is removed from the jar, the smallest number of marbles that must be removed is 51
x + y = 54; (x,y) = {(28,26),(29,25),,,,,(28+x,26-x),,,,(53,1)}
y + z = 50; (y,z) = {(26,24),(27,23),,,,,(26+y,24-y),,,(49,1)}
x>y>z>,
(x,y,z) = (28,26,24)
SUFFICIENT

IMO C

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