M03-16

Why cant be 1 ;

1+2+347=350

Yes, but lets take the example wherein the other numbers are 118, 116,115 and 1? In this case, we have the sum of the other four numbers to be 350 and the least value of the smallest integer to be 1?

What am I missing here?

Official Solution:

The average (arithmetic mean) of 4 different positive integers is 125 and the largest of these integers is 150, what is the least possible value of the smallest of the 4 integers?

A. 1
B. 2
C. 12
D. 53
E. 100


Suppose the 4 integers in ascending order are \(x\), \(y\), \(z\) and 150.

Since the average of 4 different positive integers is 125 then their sum is \(125*4=500\). So, \(x+y+z+150=500\), which simplifies to \(x+y+z=350\).

We want to minimize \(x\), so we should maximize \(y\) and \(z\), since given that all integers are distinct then \(y_{\text{max}}=148\) and \(z_{\text{max}}=149\). Therefore \(x+148+149=350\) and \(x=53\).


Answer: D