video explanation

text explanation

(in this explanation, we'll introduce variables to keep things organized. if you prefer to avoid variables, definitely watch the video explanation above.)

let's start by highlighting a very important restriction in the question: we are dealing with 4 different integers. this means that all 4 of our numbers must be different from each other and must be integers.

next, we are told that the average (arithmetic mean) of these 4 numbers is 75 and that the largest of these 4 numbers is 90.

finally, we are asked to find the least possible value of the smallest integer. let's label this unknown integer as .

we'll label the next two largest integers as and . this means our 4 integers are, in order from least to greatest,



now we can write our average formula as an equation:









our goal is to minimize the value of . to do this, we need to maximize the other unknown integers. therefore, we should make and  as large as possible.

since 90 is the largest integer in our set and we can have no repeats, the next largest integer possible is 89.





applying the same logic, the next largest integer after 90 and 89 is 88.





this shows that 33 is the least possible value for our smallest integer.

answer choice (e) is correct.


by the way, there is an alternative approach through which we can find the correct answer choice by eliminating all the incorrect answer choices.

if we were allowed repeated numbers in our set of 4 integers, then the set that has 3 90s would give us the least possible value of the smallest integer.





but, because we are not allowed repeats, our actual integers will be,



where represents a number less than 90.



the least possible value of the smallest integer must be larger than 30! this eliminates answer choices (a) through (d)! :-)

faq: after a little calculation, we get a sum for 3 other integers equal to 210. why can't the smallest integer be 1 for those 3 integers?

the trick here is in noting that the largest integer is 90. so not only do the other three numbers have to total 210, but none of them can be larger than 90. if the smallest were 1, then the last two would have to add up to 209. this is impossible.

faq: why are the numbers in consecutive order: 88, 89, and 90?

we are not choosing them because they are consecutive. this is the result of following the parameter set by the problem. no number is larger than 90. so, to find the least value of a number, we stack as much "weight" on the other numbers, and choose the next largest number possible after 90 (89). and since the numbers are all different, we have to again choose the next largest number (88). thus 88 + 89 + 90.