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04 Dec 2023, 09:49
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If y is the average (arithmetic mean) of 15 consecutive positive integers, which of the following must be true?
I. y is an integer.
II. y > 7
III. y < 100
A. I only
B. II only
C. III only
D. I and II
E. II and III
2023-12-04_20-37-35.png [ 29.05 KiB | Viewed 4161 times ]
I. y is an integer.
II. y > 7
III. y < 100
A. I only
B. II only
C. III only
D. I and II
E. II and III
Attachment:
2023-12-04_20-37-35.png [ 29.05 KiB | Viewed 4161 times ]
04 Dec 2023, 09:53
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guddo
If y is the average (arithmetic mean) of 15 consecutive positive integers, which of the following must be true?
I. y is an integer.
II. y > 7
III. y < 100
A. I only
B. II only
C. III only
D. I and II
E. II and III
I. y is an integer.
II. y > 7
III. y < 100
A. I only
B. II only
C. III only
D. I and II
E. II and III
Attachment:
2023-12-04_20-37-35.png
Firstly, the average of any evenly spaced set is equal to its median, and the median of an odd number of integers is the middle number when arranged in order.
Therefore, the average/median will always be an integer, making I always true.
The lowest set of 15 consecutive positive integers is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. The average/median in this case is 8, so it is greater than 7. Hence, all other sets will also have averages greater than 7, making II also true.
As for III, it's not always true. We can make the set as large as we want, so the upper limit of the average is not restricted at all.
Answer: D.
General Discussion
09 Dec 2023, 06:30
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guddo
Asked: If y is the average (arithmetic mean) of 15 consecutive positive integers, which of the following must be true?
I. y is an integer. For odd number of integers (15), the average is middle value (8th integer) MUST BE TRUE
II. y > 7: If starting integer = 1(lowest); y=8>7. For all other starting integers > 1; y > 7; MUST BE TRUE
III. y < 100; If starting integer =1 ; No; If starting integer = 100; Yes. COULD BE TRUE
A. I only
B. II only
C. III only
D. I and II
E. II and III
IMO D
Asked: If y is the average (arithmetic mean) of 15 consecutive positive integers, which of the following must be true?
I. y is an integer. For odd number of integers (15), the average is middle value (8th integer) MUST BE TRUE
II. y > 7: If starting integer = 1(lowest); y=8>7. For all other starting integers > 1; y > 7; MUST BE TRUE
III. y < 100; If starting integer =1 ; No; If starting integer = 100; Yes. COULD BE TRUE
A. I only
B. II only
C. III only
D. I and II
E. II and III
IMO D
