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# If the average of four distinct positive integers is 60, how many inte

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If the average of four distinct positive integers is 60, how many inte  [#permalink]

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06 Feb 2019, 05:45
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If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.

M08-28

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If the average of four distinct positive integers is 60, how many inte  [#permalink]

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Updated on: 06 Feb 2019, 08:11
Let the four numbers be a,b,c and d in increasing order.
$$\frac{a+b+c+d}{4}$$=60
a+b+c+d = 240

1. Median of 3 largest integers is 51.
therefore c=51
c+d=190 ---(i)

we know that a+b+c+d = 240 ---(ii)
(ii) - (i)
a+b = 50
There are 2 numbers less than 50 which are a & b. SUFFICIENT

2. Median of 4 integers is 50.
$$\frac{b+c}{2}$$=50
b+c=100
Values can be b =49 and c=51 or b=48 and C= 52 etc
Hence there are 2 numbers less than 50 which are a & b. SUFFICIENT

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Originally posted by ak31 on 06 Feb 2019, 06:46.
Last edited by ak31 on 06 Feb 2019, 08:11, edited 1 time in total.
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If the average of four distinct positive integers is 60, how many inte  [#permalink]

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Updated on: 06 Feb 2019, 08:53
Bunuel wrote:
If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.

M08-28

given
a+b+c+d= 240

#1
median of three largest no 51
and two largest integer sum = 190
so
b+c+d /3 = 51= b+c+c = 153
and c+d= 190
b = 190-153 ; 37

a+b+37+190 = a+b +227 = 240
a+b = 13
so a,b & c are <50
sufficient

#2
The median of the four integers is 50.
median = 50 ; b+c/2
two no are less than 50 and two over 50
IMO D
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Originally posted by Archit3110 on 06 Feb 2019, 07:06.
Last edited by Archit3110 on 06 Feb 2019, 08:53, edited 1 time in total.
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Re: If the average of four distinct positive integers is 60, how many inte  [#permalink]

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06 Feb 2019, 08:04
If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

Let the numbers be a, b, c and d in increasing order

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.
So, median of b, c, d is 51 and hence c is 51. Now c+d=190, so a+b=240-190=50.
Thus each of a and b have to be less than 50 as a and b are positive integers.
Sufficient

(2) The median of the four integers is 50.
Note that Median is of even number of elements
So median is the middle of b and c, that is $$\frac{b+c}{2}=50$$..
As both b and c are distinct, b<50 and c>50..
Thus two numbers are less than 50.
Sufficient

D
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Re: If the average of four distinct positive integers is 60, how many inte  [#permalink]

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06 Feb 2019, 08:24
Bunuel wrote:
If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.

M08-28

Key word: four distinct positive integers

From 1
When median of 3 largest numbers is 51, this means that the series is
x y 51 139

The sum of all 4 numbers will be 240, Now when we subtract 190, will give value as 50

so we get 2 numbers less than 50

From 2, lets look at our key word

since 50 is the median this means, center term, when n is even, median = (2nd term + 3rd term)/ 2

so if median has to be 50, which we can get in many ways such as, 15+85, 45+55

So in all those cases we will get two values, Sufficient

D
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Re: If the average of four distinct positive integers is 60, how many inte  [#permalink]

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04 Jul 2019, 02:42
Bunuel wrote:
If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190.

(2) The median of the four integers is 50.

M08-28

If the average of four distinct positive integers is 60, how many integers of these four are less than 50?

It's almost always better to express the average in terms of the sum: the average of four distinct positive integers is 60, means that the sum of four distinct positive integers is $$4*60=240$$.

Say four integers are $$a$$, $$b$$, $$c$$ and $$d$$ so that $$0 \lt a \lt b \lt c \lt d$$. So, we have that $$a+b+c+d=240$$.

(1) The median of the three largest integers is 51 and the sum of two largest integers is 190. The mdian of $$\{b,c,d\}$$ is 51 means that $$c=51$$. Now, if $$b=50$$, then only $$a$$, will be less than 50, but if $$b \lt 50$$, then both $$a$$ and $$b$$, will be less than 50. But we are also given that $$c+d=190$$. Substitute this value in the above equation: $$a+b+190=240$$, which boils down to $$a+b=50$$. Now, since given that all integers are positive then both $$a$$ and $$b$$ must be less than 50. Sufficient.

(2) The median of the four integers is 50. The median of a set with even number of terms is the average of two middle terms, so $$\text{median}=\frac{b+c}{2}=50$$. Since given that $$b \lt c$$ then $$b \lt 50 \lt c$$, so both $$a$$ and $$b$$ are less than 50. Sufficient.

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Re: If the average of four distinct positive integers is 60, how many inte   [#permalink] 04 Jul 2019, 02:42
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