M08-28

Official Solution:


If the average of four distinct positive integers is 60, how many of these four integers are less than 50?

It is generally more helpful to express the average in terms of the sum. Since the average of four distinct positive integers is 60, this means that the sum of these four integers is \(4 * 60 = 240\).

Let the four integers be \(a\), \(b\), \(c\), and \(d\), such that \(0 < a < b < c < d\). Therefore, we have \(a + b + c + d = 240\).

(1) The median of the three largest integers is 51, and the sum of the two largest integers is 190.

The median of \(\{b, c, d\}\) is 51, which means that \(c = 51\). If \(b = 50\), then only \(a\) will be less than 50. However, if \(b < 50\), then both \(a\) and \(b\) will be less than 50. We are also given that \(c + d = 190\). Substituting this value into the equation above, we get \(a + b + 190 = 240\), which simplifies to \(a + b = 50\). Since all integers are positive, both \(a\) and \(b\) must be less than 50. Sufficient.

(2) The median of the four integers is 50.

The median of a set with an even number of terms is the average of the two middle terms, so \(\text{median} = \frac{b + c}{2} = 50\). Since \(b < c\), we have \(b < 50 < c\), which means both \(a\) and \(b\) are less than 50. Sufficient.


Answer: D
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I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

Bunuel

What if the numbers are

x1,x2,x3,x4 where x2=50 and x3=51?

Then clearly (2) would be insufficient.

Could you please clarify.


Thanks,

ucb2k7

(2) says that the median is 50. If x2=50 and x3=51, the median is 50.5 NOT 50.
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Why can't any number be zero?

If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

0 is neither positive nor negative.
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I think this is a high-quality question and I agree with explanation.

I think this is a high-quality question and I agree with explanation.

Bunuel
Do we really need to consider the first part of statement 1 here "The median of the three largest integers is 51"
I think 2nd part of Statement suffices the need to prove that the 2 integers will be less than 50.
Am I interpreting something incorrect here ?

If you just consider second part of 1st statement then the possibility of the numbers = { 20,30,40, 150} or {10,40, 94,96} which does not give one answer.

Thank you for creating such a question, what an incredible eye-opener.

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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In statement a) why can’t A be 0?

Posted from my mobile device

That's why:

If the average of four distinct positive integers is 60, how many of these four integers are less than 50?
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I think this is a high-quality question and I agree with explanation. a high-quality question