rxs0005 wrote:
If the average (arithmetic mean) of four positive numbers is 40, how many of the numbers are less than 40?
(1) The two smallest numbers are identical.
(2) The average (arithmetic mean) of the two largest numbers is 50.
Let's say these 4 positive numbers
in ascending order are \(a\), \(b\), \(c\), and \(d\). Given: \(a+b+c+d=160\). Question: how many are less than 40? Obviously less than 40 can be only 0 (in case all numbers are 40), 1, 2, or 3 numbers (all 4 can not be less than 40 as in this case their sum won't be to 4*40=160).
(1) The two smallest numbers are identical --> \(a=b\) --> \(2a+c+d=160\) --> 0 is out as all numbers are not identical and 1 is also out as 2 smallest are equal and if 1 is less than 40 then another is also, but still two answers are possible: 2 or 3 numbers are less than 40. For example: {20, 20, 50, 50} or {20, 20, 30, 70}. Not sufficient.
(2) The average (arithmetic mean) of the two largest numbers is 50 --> \(c+d=100\) --> \(a+b=60\). Again 1, 2, or 3 numbers can be less than 40. For example: {10, 50, 50, 50} or {30, 30, 50, 50} or {30, 30, 30, 70}. Not sufficient.
(1)+(2) As from (1) \(a=b\) and from (2) \(a+b=60\) then \(a=b=30\), so two smallest numbers are 30 and 30: {30, 30, c, d}. But still 2 or 3 numbers can be less than 40: {30, 30, 50, 50} or {30, 30, 35, 65}. Not sufficient.
Answer: E.
Well, I am little bit confused with the answer. Rather C seems more tempting to me. If we combine both statements then we come to know there two numbers greater than 40 and two numbers less than 40. We can come up with the answer using both the statements. I have interpreted smallest number as less than 40 whereas largest as more than 40. So, {30, 30, 35, 65} cant be present. Kindly rectify me where I went wrong.