This GMAT practice question tests your understanding of 3 topics. The core topic tested is Descriptive Statistics. Basic understanding of Prime Numbers and Progressions is needed to solve this question.
Question
The ages of three friends are prime numbers. The sum of the ages is less than 51. If the ages are in Arithmetic Progression (AP) and if at least one of the ages is greater than 10, what is the difference between the maximum possible median and minimum possible median of the ages of the three friends?
(A) 0
(B) 1
(C) 13
(D) 6
(E) 8
Correct Answer
Choice D. The difference is 6
Video Explanation
Explanatory Answer
The ages are prime numbers in arithmetic progression i.e., they have a common difference. Furthermore, at least one of them is greater than 10.
Let the ages be a, b, and c such that a < b < c.
a + b + c < 51
Step 1: Compute the maximum value of the median
Because a, b, and c are in AP, a + b + c = 3b
3b < 51 or b < 17
The largest prime number less than 17 is 13.
So, the maximum value of b is 13.
i.e., the maximum value of the median of the ages is 13.
The ages of the friends could be {3, 13, and 23. Common difference 10} OR {7, 13, and 19. Common difference 6}.
Step 2: Compute the minimum value of the median
The ages are prime numbers and in AP and their sum is less than 51. At least one number is more than 10.
Minimum value of b will be a prime number less than 13.
Let us start with b = 11: The possibilities satisfying all these conditions are {5, 11, 17} and {3, 11, 19}
If b = 7: The only possibility is {3, 7, 11}
If b = 5: The only possibility is {3, 5, 7}. This does not satisfy the condition that at least one number has to be greater than 10.
Therefore, the minimum value of b, the median = 7
Step 3: Compute the difference
The required difference is 13 – 7 = 6.
The correct answer is D