Re: GMAT CLUB OLYMPICS: If x and y are positive integers less than 10, wha
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28 Aug 2021, 02:21
{x, y, 1, 2, 3}
If x and y are positive integers less than 10, what is the median of the list above?
Let S = {x, y, 1, 2, 3}
x is one of {1, 2, 3, 4, 5, 6, 7, 8, 9}
y is one of {1, 2, 3, 4, 5, 6, 7, 8, 9}
Example S can be {1, 1, 1, 2, 3} or {1, 2, 3, 3, 8} and so on.
(1) When one number is chosen at random from the list, the probability of selecting a multiple of 2 is less than the probability of selecting a non-prime number
Let a = {2, x, y} or {2, x} or {2, y} or {2}
where x is one of {2, 4, 6, 8} and y is one of {2, 4, 6, 8}
Let b = {1, x, y} or {1, x} or {1, y} or {1}
where x is one of {1, 4, 6, 8, 9} and y is one of {1, 4, 6, 8, 9}
Let |a| = Number of elements in set a (elements are multiples of 2)
Let |b| = Number of elements in set b (elements are non-prime numbers)
|a|/5 < |b|/5
|a| < |b|
0<|a|<3
1<|b|<4
0<|a|<|b|<4
a has fewer elements than b
a can have at least one element and at most two elements
b can have at least two elements and at most three elements
a = {2, x} or {2, y} or {2}
where x is one of {2, 4, 6, 8} and y is one of {2, 4, 6, 8}
b = {1, x, y} or {1, x} or {1, y}
where x is one of {1, 4, 6, 8, 9} and y is one of {1, 4, 6, 8, 9}
say x = 8 and y = 9
Verify: a = {2, 8}, b = {1, 8, 9}, |a| < |b|
S = {1, 2, 3, 8, 9}
Median of S = 3
say x = 1 and y = 1
Verify: a = {2}, b = {1, 1, 1}, |a| < |b|
S = {1, 1, 1, 2, 3}
Median of S = 1
We have no unique value of the median of S. Therefore, statement 1 is insufficient.
(2) When one number is chosen at random from the list, the probability of selecting a multiple of 3 is greater than the probability of selecting a prime number
Let c = {3, x, y} or {3, x} or {3, y} or {3}
where x is one of {3, 6, 9} and y is one of {3, 6, 9}
Let d = {2, 3, x, y} or {2, 3, x} or {2, 3, y} or {2, 3}
where x is one of {2, 3, 5, 7} and y is one of {2, 3, 5, 7}
Let |c| = Number of elements in set c (elements are multiples of 3)
Let |d| = Number of elements in set d (elements are prime numbers)
|c|/5 > |d|/5
|c| > |d|
1<|c|<4
1<|d|<5 --> 1<|d|<3 since max |c| = 3
1<|d|<|c|<4
c has more elements than d
c can have at least three element and at most three elements
d can have at least two elements and at most two elements
c = {3, x, y}
where x is one of {6, 9} and y is one of {6, 9}
x and y can not be 3 even though 3 is a multiple of 3 because 3 is a prime and we do not want to increase |d|.
If |d| increases above 2 elements then |c| > |d| will not hold.
d = {2, 3}
where x is one of {} and y is one of {}
say x = 6 and y = 6
Verify: c = {3, 6, 6}, d = {2, 3}, |c| > |d|
S = {1, 2, 3, 6, 6}
Median of S = 3
say x = 6 and y = 9
Verify: c = {3, 6, 9}, d = {2, 3}, |c| > |d|
S = {1, 2, 3, 6, 9}
Median of S = 3
say x = 9 and y = 9
Verify: c = {3, 9, 9}, d = {2, 3}, |c| > |d|
S = {1, 2, 3, 9, 9}
Median of S = 3
We have a unique value of the median of S. Therefore, statement 2 is sufficient.
Option B