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16 Sep 2014, 00:49
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The average (arithmetic mean) of four distinct positive integers is 10. If the average of the smaller two of these four integers is 8, which of the following represents the maximum possible value of the largest integer?
A. 12
B. 14
C. 15
D. 16
E. 17
A. 12
B. 14
C. 15
D. 16
E. 17
16 Sep 2014, 00:49
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Official Solution:
The average (arithmetic mean) of four distinct positive integers is 10. If the average of the smaller two of these four integers is 8, which of the following represents the maximum possible value of the largest integer?
A. 12
B. 14
C. 15
D. 16
E. 17
Assume the four distinct positive integers are \(a, b, c\), and \(d\), such that \(0 < a < b < c < d \).
Given that the average of these integers is 10, we know that \(a+b+c+d=4*10=40\);
Furthermore, we're told that the average of the smallest two of these integers is 8, which means that \(a+b=2*8=16\). So, \(16+c+d=40\), and thus \(c+d=24\).
To find the maximum possible value for \(d\), we need to minimize \(c\). The smallest value of \(c\) can be 10, for \(a=7\) and \(b=9\) (this is because \(a + b = 16\) and since \(a\) and \(b\) must be distinct integers, it's impossible for both to be 8. Thus, the smallest possible value for \(b\) is 9). So, with the smallest possible value for \(c\) as 10, we have \(10+d=24\) which gives us \(d=14\).
Answer: B
The average (arithmetic mean) of four distinct positive integers is 10. If the average of the smaller two of these four integers is 8, which of the following represents the maximum possible value of the largest integer?
A. 12
B. 14
C. 15
D. 16
E. 17
Assume the four distinct positive integers are \(a, b, c\), and \(d\), such that \(0 < a < b < c < d \).
Given that the average of these integers is 10, we know that \(a+b+c+d=4*10=40\);
Furthermore, we're told that the average of the smallest two of these integers is 8, which means that \(a+b=2*8=16\). So, \(16+c+d=40\), and thus \(c+d=24\).
To find the maximum possible value for \(d\), we need to minimize \(c\). The smallest value of \(c\) can be 10, for \(a=7\) and \(b=9\) (this is because \(a + b = 16\) and since \(a\) and \(b\) must be distinct integers, it's impossible for both to be 8. Thus, the smallest possible value for \(b\) is 9). So, with the smallest possible value for \(c\) as 10, we have \(10+d=24\) which gives us \(d=14\).
Answer: B
General Discussion
15 Jun 2016, 00:07
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Bunuel : Why we minimize C=10???
