If the Average of Four Distinct Positive Integers is 60, How Many Integers of These Four are Less than 50? GMAT Data Sufficiency

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

1. The median of the three largest integers is 51 and the sum of two largest integers is 190.
2. The median of the four integers is 50.

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are NOT sufficient.

“If the average of four distinct positive integers is 60, how many integers of these four are less than 50?”– is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken f0rom the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Answer

Approach Solution 1:

There is only one approach solution to this problem.

It’s almost always better to express the average in terms of the sum: the average of four distinct positive integers is 60, means that the sum of four distinct positive integers is 4 * 60 = 240
Say four integers are a, b, c and d so that 0 < a < b < c < d.

So, we have that a + b + c + d = 240

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190
The median of {b,c,d} is 51 means that c = 51. Now, if b = 50, then only a, will be less than 50, but if b < 50, then both a and b, will be less than 50.
But we are also given that c + d = 190.
Substitute this value in the above equation: a + b + 190 = 240, which boils down to a + b = 50
Now, given that all integers are positive then both a and b must be less than 50.
Hence this statement is sufficient.

(2) The median of the four integers is 50. The median of a set with even number of terms is the average of two middle terms, so median = \(\frac{b+c}{2}\)= 50
Since b < c then b < 50 < c, so both a and b are less than 50.
Hence this statement is sufficient.

Correct Answer: D

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