If the Average (Arithmetic Mean) of Four Different Numbers is 30, How GMAT Data Sufficiency

Question: If the average (arithmetic mean) of four different numbers is 30, how many of the numbers are greater than 30?

1. None of the four numbers is greater than 60
2. Two of the four numbers are 9 and 10, respectively

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.

Answer:
Approach Solution (1):
a + b + c + d = 4 * 30 = 120
S1: None of the four numbers is greater than 60. Many combinations are possible. For example: numbers can be: 20-25-30-45 (only one number is greater than 30) or 15-20-40-45 (two numbers are greater than 30)
Not sufficient
S2: Two of the four numbers are 9 and 10 respectively. Also, not sufficient, consider 0-9-10-101 or 9-10-35-66
Not sufficient
(1) + (2) As two of the four numbers are 9 and 10 then the sum of other two must be 120 – (9 + 10) = 101
Now as the greatest number can be at most 60, then the least value of othe other one is 41, so in any case two numbers will be more than 30
Sufficient
Correct option: C

Approach Solution (2):
Let x1, x2, x3, x4 be the #s
Given average = 30 i.e. sum = 120
S1: All #s < 60
Can come up with two sets {10,10,50,50} or {35,35,35,15} which has different answers (2 and 3)
Not sufficient
S2: x1 = 9, x2 = 10
x3 + x4 = 120 – 19 = 101
Possible values:
x3 = 100, x4 = 1
or
x3 = 50, x4 = 51
Two different answers
Not sufficient
S1 and S2:
x3 + x4 = 101
Maximum possible value for {x3, x4} = 59
As sum has to be 101, other value has to be 41
Therefore enough to answer the question: 2
Sufficient
Correct option: C

Approach Solution (3):
Sum of the 4 numbers = (quantity)(average) = 4 * 30 = 120
S1:
Case 1: The four numbers are 32, 31, 30, 27
In this case, two of the numbers are greater than 30
Case 2: The four numbers are 33, 32, 31, 24
In this case, three of the numbers are greater than 30
Insufficient
S2:
Case 1: The four numbers are 9, 10, 31, 70
In this case, two of the numbers are greater than 30
Case 2: The four numbers are 9, 10, 30, 71
In this case, one of the numbers is greater than 30
Insufficient
Statements combined:
Since the sum of the four numbers = 120, and two of the numbers are 9, 10, the sum of the other two numbers = 120 – 9 – 10 = 101
Since neither of the two remaining numbers may exceed 60, a sum of 101 is possible only if each of the two remaining is greater than 30
Thus, the two of the numbers must be greater than 30
Sufficient
Correct option: C

“If the average (arithmetic mean) of four different numbers is 30, how many of the numbers are greater than 30?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from 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.

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