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

[sambati]

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

[deloitte247]

Let the 4 different positive integers be a, b, c, and d
Largest integer = 150
Let the largest integer = d
d = 150


$$\frac{a+b+c+150}{4}=125$$
$$a+b+c+150=125\cdot4$$
$$a+b+c=500-150$$
$$a+b+c=350$$


To get the least possible value of the smallest of the 4 integers, we have to maximize the other 2 integers
Let least/smallest integer = a


Since largest integer = 150, the maximum possible value for other 2 integer= 150 - 1 and 150 - 2 respectively


Smallest integer => a + (150 - 2) + (150 - 1) = 350
=> a + 148 + 149 = 350
a + 297 = 350
a = 350 - 297
a = 53

Answer = D

[Scott@TargetTestPrep]

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

Solution:

The sum of the 4 integers is 125 x 4 = 500. Since the largest integer is 150, the sum of the remaining 3 integers is 500 - 150 = 350. Since we want the smallest integer to be as small as possible, we can choose the second and third largest integers to be as large as possible, so we choose 149 and 148, respectively. Therefore, the least possible value of the smallest integer is 350 - 149 - 148 = 53.

Answer: D