M03-16

Question

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

Bunuel · Accepted Answer

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.
 
Given that the average of 4 different positive integers is 125, 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 all integers are distinct, then  y_{	ext{max=148  and  z_{	ext{max=149 . Therefore,  x+148+149=350  and  x=53 .

Answer: D

Bunuel · Answer

Because we are told that the largest of these integers is 150.

vikramd · Answer

Why cant be 1 ;

1+2+347=350

Shiv2016 · Answer

X is not 1 because: 1) All the integers are different so x,y, and z cannot take 150 as their value. 2) Simple rule to minimize a number is to maximize other numbers. Simple!!

Let's suppose x=1 y=148 (as given) z=149 (as given in the solution) 1+148+149+150= 448 (But we found x+y+z+150= 500) Therefore x=53 (as given in the solution)

vetma · Answer

I think this is a  high-quality  question and I agree with explanation.

dcummins · Answer

What made this easier to understand how to solve in review is putting each variable up against the constraints:
- that each number is distinct
- that each number is less than 150

Thus
X + Y + Z + 150 = 500
X + Y + Z = 350

We can then treat the variables as x < y < z < 150 to maximise each

Test values as per Bunuel's solution: x< 148 < 149 for x<y<z 
148 + 149 = 297, so is this a possible value? (stepping through my thinking)

if x + 297 = 350, then x = 53

This is an answer choice.

Abhishek008 · Answer

I think this is a  high-quality  question and I agree with explanation.

theGisforGorilla · Answer

I think this is a  high-quality  question and I agree with explanation.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

SuryaGanguly · Answer

I used the following approach -

By definition, all numbers in a set can be represented with the help of mean. For example, in this case we can have the numbers as 125,125,125, and 125

But since one number is 150, then one of the 125 increases +25
Similarly, we can increase two other 125 as +24 and +23 (since we want to find the least possible value of the remaining)

Total increase = 25+24+23 = 72
So the remaining 125 will go down by 72 to maintain the same sum
Hence number = 125-72 = 53

M03-16

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