For three numbers, M is the median and A is the average. is

Question

For three numbers, M is the median and A is the average. is M<A?
1) M is less than the average of the greatest number and the least number.
2) The greatest number is 250 more than A. The least number is 150 less than A.

Vithal · Answer

I like your smart explanations sparky - please explain!

Is the strategy to pick numbers? (if so, don't bother to type in your explanation)

sparky · Answer

picking numbers here will lead you nowhere. draw a line, put x1 and x3 on the ends, now divide the line in the middle. if x2 is in the middle then M=A. now move x2 towards x1 (the smallest number), average will move towards x1 too but at a slower rate than the one at which you move x2 because it's slowed down (weighed up) by x3 (the largest number). This means that MA) 3*x2 > (x1+x2+x3) and M < (x1+x3)/2 or 2*x2<(x1+x3) 3*x2 > (x1+x2+x3) => 2*x2 > x1+ x3 2*x2<(x1+x3) a contradiction, therefore MA if x2 is closer to x3, by the same reasoning.

HongHu · Answer

A=(x+y+z)/3
M=y

1)
M<(x+z)/2 => x+z>2M
A=(x+y+z)/3>(2M+M)/3=M
A>M
Sufficient

2)
z=A+250
x=A-150
x+z=2A+100
A=(2A+100+M)/3
A=M+100
A>M
Sufficient

D

Vithal · Answer

That is very clear and concise Hong! - It just shows a very high degree of structured thinking! Thank you! :thanks

FN · Answer

D it is..



all we need to know is that max and numbers are not zero!

Statment 1 says that! sufficient!

statement 2 is obvious...sufficient!

whatever statmenet we have, we always will know that M>A!

AJB77 · Answer

D for me too.

HongHu and Sparky -> Both of you rule! You remind me of a couple of smart and helpful classmates I had back in freshman year of college.

rthothad · Answer

Honghu, Shouldn't M<(x+z)/2 equate to 2M < x+z and not 2M > x+z. Can you kindly explain.

AJB77 · Answer

Let the numbers be X,M,Y (M is the median). Since A is the average, we have: 3A = X+M+Y --------------(*) S1 => 2M < X+Y Plug this in (*) and we get: 3A = M+(X+Y) < M+2M =>3A<3M or A < M => Sufficient S2 => X=A-150;Y=A+250=>Plug those in (*) and get: 3A = M+A+250+A-150 = M+2A+100 =>A = M+100 => A > M => sufficient D is the answer.

Darth_McDaddy · Answer

Wow, you guys rock. I didn't substitute numbers. I thought it was fairly straight forward

1. M is less than the average of the greatest number and the least number. As the Median of the three numbers is the middle number, Statement 1 implies that the median will be closer to the least number than the greatest number, and thus less than the average. Therefore M < A

2. The greatest number is 250 more than A. The least number is 150 less than A. Again, this clearly implies that the median lies closer to the least number, and hence will be less than the average. 

But Kudos to Honghu and AJB... for their detailed explanation

For three numbers, M is the median and A is the average. is

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