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The average (arithmetic mean) of 4 different integers is 75.

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The average (arithmetic mean) of 4 different integers is 75. [#permalink]  26 Aug 2016, 07:40
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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)1
B)19
C)29
D)30
E)33
[Reveal] Spoiler: OA
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Re: The average (arithmetic mean) of 4 [#permalink]  26 Aug 2016, 09:21
Suppose four different integers are : a,b,c & d.The average of these four different integers is :

$$Avg$$ = (a+b+c+d)/4 = 75

=>

Sum of these four different integers is : sum = a+b+c+d = 300

The difference between the largest integer and the average is : 90-75 = 15

if largest integer is 90 (suppose d=90) then a+b+c = 300-90 = 210
=> Average of a,b & c is : (a+b+c)/3 = 210/3 = 70

How have you calculated that the least integer should be 33 (option E)?I have tried to calculate but finding it difficult;could you please share the detailed solution?
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Re: The average (arithmetic mean) of 4 [#permalink]  26 Aug 2016, 10:24
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Hi phoenixio,

Well the mean is given to be 75. Let the integers be x,y,z, 90.

So we know that:

$$\frac{(x+y+z+90)}{4}$$= 75

$$x+y+z= 210$$

Lets say z is the smallest. So in order for z to be smallest possible x and y should be largest values possible.

Now we also know that $$x,y,z < 90$$ (since 90 is the largest integer) and x,y,z, 90 are "different".

So x = 89 and y= 88 satisfy this criteria:

Hence $$z = 210 - 89 -88 =33$$.

Cheers

PS: Source of the Que ?
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Re: The average (arithmetic mean) of 4 [#permalink]  11 Nov 2017, 10:42
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a + b + c + d = 75*4
let a < b < c < d, so d = 90

a + b + c = 210
b+c = 210 - a => 1
we know that b < 90 and c < 90

b+c < 180

210 - a < 180 substitute from 1

a > 30
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Re: The average (arithmetic mean) of 4 [#permalink]  11 Nov 2017, 11:42
Expert's post
181charan wrote:
a + b + c + d = 75*4
let a < b < c < d, so d = 90

a + b + c = 210
b+c = 210 - a => 1
we know that b < 90 and c < 90

b+c < 180

210 - a < 180 substitute from 1

a > 30

Nice!!
Well done!
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Re: The average (arithmetic mean) of 4   [#permalink] 11 Nov 2017, 11:42
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The average (arithmetic mean) of 4 different integers is 75.

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