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# M13-16

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Math Expert
Joined: 02 Sep 2009
Posts: 58372

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16 Sep 2014, 00:49
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65% (hard)

Question Stats:

49% (01:15) correct 51% (00:52) wrong based on 99 sessions

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The average (arithmetic mean) of four distinct positive integers is 10. If the average of the smaller two of these four integers is 8, which of the following represents the maximum possible value of the largest integer?

A. 12
B. 14
C. 15
D. 16
E. 17

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Posts: 58372

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16 Sep 2014, 00:49
Official Solution:

The average (arithmetic mean) of four distinct positive integers is 10. If the average of the smaller two of these four integers is 8, which of the following represents the maximum possible value of the largest integer?

A. 12
B. 14
C. 15
D. 16
E. 17

Given: $$0 \lt a \lt b \lt c \lt d$$. Notice that $$a, b, c$$ and $$d$$ are distinct positive integers.

The average of four distinct positive integers is 10: $$a+b+c+d=4*10=40$$;

The average of the smaller two of these four integers is 8: $$a+b=2*8=16$$. So, $$16+c+d=40$$, which leads to $$c+d=24$$.

We want to maximize $$d$$, so we should minimize $$c$$. The minimum value of $$c$$ is 10, for $$a=7$$ and $$b=9$$. So, $$10+d=24$$ and $$d=14$$.

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15 Jun 2016, 00:07
Bunuel : Why we minimize C=10???
Math Expert
Joined: 02 Sep 2009
Posts: 58372

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15 Jun 2016, 02:41
Channarong22 wrote:
Bunuel : Why we minimize C=10???

We have that a + b = 16. Since a and b are distinct integers, then a = b = 8 is not possible, so the least value of b is 9 (7 + 9 = 16). Hence the least value of c is 10.
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Joined: 02 Feb 2010
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10 Oct 2016, 07:09
I'm sorry. Maybe I'm not following the whole logic. But wouldn't the set 6,10,11,13 do the trick? I didn't understand why you assume the values of a and b in that way.

Thanks
Math Expert
Joined: 02 Aug 2009
Posts: 7958

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10 Oct 2016, 07:23
chambreau wrote:
I'm sorry. Maybe I'm not following the whole logic. But wouldn't the set 6,10,11,13 do the trick? I didn't understand why you assume the values of a and b in that way.

Thanks

Hi i
It is done to ensure that the largest number has maximum possible value
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17 Mar 2018, 08:14
Channarong22 wrote:
Bunuel : Why we minimize C=10???

It is given a,b are smaller of the 4 integers ..so definitely C and D have to be greater than that..
Now since we know a and b are 7 and 9 from above explanation.C has to be greater than 9 and so it is 10.
Re: M13-16   [#permalink] 17 Mar 2018, 08:14
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# M13-16

Moderators: chetan2u, Bunuel