Question

The average (arithmetic mean) of 4 different integers is 75. If the largest integer is 90, what is the least possible value of the smallest integer?
A. 19
B. 29
C. 30
D. 33

Answer 1

this is not a nice question. So before I spout into something, what level math is this for?

Answer 2

Set up some equations describing this. That tends to make word problems a little more transparent.

Answer 3

I'm in Algebra 2...I'm not sure really, this is a extra credit question my teacher came up with but I don't know how to start at all.

Answer 4

ohhhk. Well then

Answer 5

If the biggest number is 90, how big can the next two largest numbers be?

Answer 6

That kicks my number theory approach out of here.

Answer 7

I would say, let's set up a few things we know as Kainui recommended

Answer 8

so how do we find the mean of 4 numbers?

Answer 9

Well first off let's just give the variables names:

\[\Large a,b,c,d\]

What do we know about them? We know their average is equal to 75 right? \[\Large \frac{a+b+c+d}{4}=75\] What else do we know? We know they're different integers right? So that means we can just write them as \[\Large a<b<c<d\] What else do we know? We know the largest one is 90, so that must be d. \[\Large d=90\] What can we do from here?

Answer 10

We can now say that we're looking for "a" since that will be the smallest right? So how can we do that? Let's maximize b and c. So b and c equal 88 and 89 since that's the largest they can be. Plug them all in and we can solve for a! Done!

Answer 11

Nice! but we also must take into account that it needs to be divisible by 4

Answer 12

which we will if we solve for a :)

Answer 13

Do you follow @EclipsedStar ?

Answer 14

(I might be late posting this, but I'll post it anyways) I don't think we do, do we? For the more general case we are looking at: \[\Large \frac{a+b+c+d}{4}=t \\ \Large a+b+c+d = 4t \\ \Large a= 4t-b-c-d\] So we don't need a to be divisible by 4.

Answer 15

I meant that t needs to be divisible by a, so it can't be even. (so we could get rid of an answer)

Answer 16

The answer choices also don't add up. ^

Answer 17

There's no need for any equations. It's as simple as:
What is the sum of the four numbers?
What is the biggest we can make the other three numbers?
What does that leave for the smallest number?

Answer 18

let's see: \[a+88+89+90=4(75)\]solve this @EclipsedStar

Answer 19

The answer is there

Answer 20

That's what I've been saying... #_#

Answer 21

I know

Answer 22

You have my best response for it Kainui

Answer 23

33. XD

Answer 24

D! :'D

Answer 25

Haha, I just mean I was trying to avoid just sort of giving the answer too much. Oh well

Answer 26

Thanks! I get it now. :)

Answer 27

True, I should have been less to the point

Answer 28

My bad, sorry @Kainui