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Mean and Product

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by dtweah » Mon May 04, 2009 12:54 pm
The Number K is the arithmetic mean of a^12 and b^14 and L is the product of (a^6 ) and ( b^7), where a and b are positive integers greater than 1. Which of the following CANNOT be False?

A. L>K
B. L> or equal to K
C. K>L
D. K> or equal to L
E. K=L
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by criszerriny » Mon May 04, 2009 3:55 pm
breaking it down...

((a^6)^2+(a^7)^2)/2=K
A6*A7=L

It seems like K will always be greater than A based on the above. Any other take?
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by Ian Stewart » Mon May 04, 2009 6:18 pm
criszerriny wrote:breaking it down...

((a^6)^2+(a^7)^2)/2=K
A6*A7=L

It seems like K will always be greater than A based on the above. Any other take?
Not quite always. If you look at (a^6 - b^7)^2, this must of course be greater than or equal to zero, since it's the square of an integer. Now,

(a^6 - b^7)^2 > 0
a^12 - 2(a^6)(b^7) + b^14 > 0
a^12 + b^14 > 2(a^6)(b^7)
(a^12 + b^14)/2 > (a^6)(b^7)

So in general, the average of a^12 and b^14 will be equal to or greater than (a^6)(b^7). They'll only be equal when a^6 - b^7 = 0, or when a^6 = b^7, which can happen for, say, a = 2^7 and b = 2^6, among many other possibilities.
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Re: Mean and Product

by dtweah » Tue May 05, 2009 5:48 am
dtweah wrote:The Number K is the arithmetic mean of a^12 and b^14 and L is the product of (a^6 ) and ( b^7), where a and b are positive integers greater than 1. Which of the following CANNOT be False?

A. L>K
B. L> or equal to K
C. K>L
D. K> or equal to L
E. K=L
The arithmetic mean of a given set of n integers is greater than or equal to the geometric mean of that set of n integers.

(a^12+b^12)/2 is greater than or equal to (a^12 x b14)^.5= (a^6)(b^7)= geometric mean.

D cannot be false.
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Re: Mean and Product

by Stuart@KaplanGMAT » Tue May 05, 2009 11:12 am
dtweah wrote:The Number K is the arithmetic mean of a^12 and b^14 and L is the product of (a^6 ) and ( b^7), where a and b are positive integers greater than 1. Which of the following CANNOT be False?

A. L>K
B. L> or equal to K
C. K>L
D. K> or equal to L
E. K=L
First, let's change "CANNOT be False" to "MUST be True". So, we're looking for the choice that's always correct.

Seemed very complicated conceptually, so I jumped in and picked some easy numbers.

Nowhere does it say that a and b are distinct, so I just picked a=b=2.

K = (2^12 + 2^14)/2 = 2^11 + 2^13

L = 2^6*2^7 = 2^13

Well, 2^13 + 2^11 is certainly more than 2^13, so we have K > L.

If it's possible that K>L, then (A), (B) and (E) are right out as "MUST BE TRUE".

Next, let's apply some logic (often underused on the GMAT). "K is greater than L" is a subset of "K is greater than or equal to L".

So, if it MUST BE TRUE that K>L, then it also MUST BE TRUE that "K>=L". Since they can't both be the right answer, we can eliminate (C) and choose (D).

In fact, merely by applying that logic we could have eliminated (A), (C) and (E) without doing any math at all. ((A) and (E) are subsets of (B); (C) and (E) are subsets of (D).)
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by avenus » Wed May 06, 2009 3:59 am
((a^6)^2+(a^7)^2)/2=K
A6*A7=L
First, let's change "CANNOT be False" to "MUST be True". So, we're looking for the choice that's always correct
good starting point(s).

(a^6 - b^7)^2 = a^12 + b^14 - 2a^6b^7 = 2(K-L)

since the left side of the equation must be >=0, it follows that K-L >= 0, i.e., K>=L

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Re: Mean and Product

by sureshbala » Wed May 06, 2009 4:37 am
dtweah wrote:The Number K is the arithmetic mean of a^12 and b^14 and L is the product of (a^6 ) and ( b^7), where a and b are positive integers greater than 1. Which of the following CANNOT be False?

A. L>K
B. L> or equal to K
C. K>L
D. K> or equal to L
E. K=L
Folks, this is simple if you know that AM >=GM>=HM

Since AM>=GM

we have (a^12 + b^12)/2 >= (a^12 x b^12)^1/2

i.e. (a^12 + b^12)/2 >= (a^6 x b^6)

i.e K >=L. (the equality holds if a =b)

Hence D cannot be false
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