Is the average (arithmetic mean) height of Porthos, Athos, and Aramis

Question

Is the average (arithmetic mean) height of Porthos, Athos, and Aramis greater than the median height of Porthos, Athos, and Aramis?

(1) Porthos is 10 centimetres taller than Athos, and 9 centimeters taller than Aramis.
(2) Porthos height is 190 centimetres

M36-121

Bunuel · Accepted Answer

Official Solution:

Is the average (arithmetic mean) height of Porthos, Athos, and Aramis greater than the median height of Porthos, Athos, and Aramis? 
 
(1) Porthos is 10 centimetres taller than Athos, and 9 centimeters taller than Aramis.
 
The heights of the three musketeers in ascending order are  x ,  x+1 , and  x+10 . The median is  x+1  and the mean is  \frac{x+(x+1)+(x+10)}{3}=x+2.something . Obviously the mean ( x+2.something ) is greater than the median ( x+1 ). Sufficient.
 
(2) Porthos height is 190 centimetres.
 
Clearly insuficient.

Answer: A

LeoN88 · Answer

We have to find the relation between mean and median of theses three heights.

Statement 2, gives us only th height of one person clearly insufficient.

Statement 1

Athos, Aramis, Porthos
x, (x+1), (10+x) , X>0 as it's height.

Median is= x+1 and mean is clearly greater than median height. So A.

Archit3110 · Answer

#1 Porthos is 10 centimetres taller than Athos, and 9 centimeters taller than Aramis.
let P = x
so Athos = x-10
Aramis = x-9
so avg ; x+x-9+x-10/3 ; 3x-19/3
median ; x-9 
value of x is not know ; insufficient
#2
P = 190 
but relation and value with other 2 are not given; insufficient
from 1 & 2
x=190
so 
we get
551>543
sufficient 
IMO C

LeoN88 · Answer

Archit3110  The median x-9 has to be a positive number because it's height of a person. Also for any value of x (here x>9) we can compare mean and median of the list. Please correct me if I'm wrong.

exc4libur · Answer

Porthos = p, Athos = t, Aramis = r;

(1) Porthos is 10 centimetres taller than Athos, and 9 centimeters taller than Aramis.   sufic.

p=t+10…p=r+9…average=sum/num=\frac{p+t+r}{3}=\frac{p+p-10+p-9}{3}…median=p-9 
 Question:\frac{p+p-10+p-9}{3}>p-9…3p-19>3p-27…19<27?…Answer:yes

(2) Porthos height is 190 centimetres   insufic.

Answer (A)

PVOG · Answer

Using Statement 1, your working out was spot on, the only thing is you walked past the answer, you actually had enough info in your solution to determine that A was sufficient alone.

As you correctly mentioned, x-9 is the median, in addition to this, the average height can be expressed as (3x-19)/3 (as you all correctly noted). We can rewrite this average height as x - 19/3

But the thing you walked past was that these two values are indeed comparable (also bear in mind that heights can never be negative).

We can compare the median of x-9 to the average of x-19/3. 19/3 is approx 6.33

X-9 is clearly going to be LESS than x-6.33, therefore, the average is indeed greater than the median and (1) is sufficient alone. The key point is, it does NOT matter what the value of 'x' is, x will always be positive, and the other heights can never be negative either, so x must be 9 at the very least.

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Is the average (arithmetic mean) height of Porthos, Athos, and Aramis

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