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Difficulty:
95%
(hard)
Question Stats:
54% (02:19) correct
46%
(02:32)
wrong
based on 224
sessions
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Not Attempted Yet
a, b, and c are distinct non-zero integers, and the standard deviation of {a, b, c} is greater than the standard deviation of {|a|, |b|, |c|}. If the range of {a, b, c} is 4, which of the following could be true?
I. The median of {a, b, c} is -2
II. The product of a, b, and c is a prime number
III. The mode of {|a|, |b|, |c|} is 1
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
M46-13
I. The median of {a, b, c} is -2
II. The product of a, b, and c is a prime number
III. The mode of {|a|, |b|, |c|} is 1
A. I only
B. II only
C. III only
D. I and III only
E. I, II, and III
M46-13
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GMAT Club Official Explanation:
Since the standard deviation of {a, b, c} is greater than that of {|a|, |b|, |c|}, {a, b, c} must contain both positive and negative integers. This is because if all numbers were positive or all negative, the standard deviations of {a, b, c} and {|a|, |b|, |c|} would be the same.
This fact, together with the range being 4, implies that the least and greatest integers in the set can be (-3, 1), (-2, 2), or (-1, 3). So we can have the following sets:
{-3, -2, 1}
{-3, -1, 1}
{-2, -1, 2}
{-2, 1, 2}
{-1, 1, 3}
{-1, 2, 3}
Let's evaluate the options:
I. The median of {a, b, c} is -2
This could be true if we have the set {-3, -2, 1}.
II. The product of a, b, and c is a prime number
This could be true if we have the set {-3, -1, 1}.
III. The mode of {|a|, |b|, |c|} is 1
This could be true if we have the set {-3, -1, 1} or {-1, 1, 3}.
Thus, all three options could be true.
Answer: E.
General Discussion
Let speeds be v1 and v2, lengths be l1 and l2.
Passing the man:
l1/v1 = 40
=> l1=40v1
l2/v2 = 25
=> l2=25v2
passing each other:
(l1+l2 ) / (v1+v2) = 30
Substitute:
(40v1+25v2)/(v1+v2) = 30
40v1 + 25v2 = 30v1+30v2
10v1 = 5v2
v1/v2 = 1:2
Answer: C
Passing the man:
l1/v1 = 40
=> l1=40v1
l2/v2 = 25
=> l2=25v2
passing each other:
(l1+l2 ) / (v1+v2) = 30
Substitute:
(40v1+25v2)/(v1+v2) = 30
40v1 + 25v2 = 30v1+30v2
10v1 = 5v2
v1/v2 = 1:2
Answer: C
