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# you have 1 minute! If a, b, c and d are four positive

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CEO
Joined: 15 Aug 2003
Posts: 3454
you have 1 minute! If a, b, c and d are four positive [#permalink]

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17 Sep 2003, 00:49
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you have 1 minute!

If a, b, c and d are four positive real numbers such that
abcd = 1, what is the minimum value of (1 + a) (1 + b) (1 + c) (1 + d).
a) 4
b) 1
c) 16
d) 18

thanks
Senior Manager
Joined: 22 Aug 2003
Posts: 257
Location: Bangalore

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17 Sep 2003, 23:21
We use the property that:
Arithemetic Mean of two numbers is g.t.e.q Geometric mean.
g.t.e.q = greater than or equal to.

Thus (1+a)/2 >= sqrt(a)
(1+b)/2 >= sqrt(b)
--------
(1+d)/2 >= sqrt(d)

Multiplying above equations (we can mutilply them without chanding sign of inequality because it is given that a,b,c & d are postive real num's)

we get (1 + a) (1 + b) (1 + c) (1 + d)/2^4 >= sqrt (abcd)
Thus minimum is 16.
-Vicks
ps: let me know if u know a faster way to solve this.
Manager
Joined: 26 Aug 2003
Posts: 233
Location: United States

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18 Sep 2003, 00:51
Vicky wrote:
We use the property that:
Arithemetic Mean of two numbers is g.t.e.q Geometric mean.
g.t.e.q = greater than or equal to.

Thus (1+a)/2 >= sqrt(a)
(1+b)/2 >= sqrt(b)
--------
(1+d)/2 >= sqrt(d)

Multiplying above equations (we can mutilply them without chanding sign of inequality because it is given that a,b,c & d are postive real num's)

we get (1 + a) (1 + b) (1 + c) (1 + d)/2^4 >= sqrt (abcd)
Thus minimum is 16.
-Vicks
ps: let me know if u know a faster way to solve this.

I'm kinda confused as to why you're using Geometric Mean here provided you don't know if these numbers are in sequence or not.
CEO
Joined: 15 Aug 2003
Posts: 3454

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18 Sep 2003, 01:38
Vicky wrote:
We use the property that:
Arithemetic Mean of two numbers is g.t.e.q Geometric mean.
g.t.e.q = greater than or equal to.

Thus (1+a)/2 >= sqrt(a)
(1+b)/2 >= sqrt(b)
--------
(1+d)/2 >= sqrt(d)

Multiplying above equations (we can mutilply them without chanding sign of inequality because it is given that a,b,c & d are postive real num's)

we get (1 + a) (1 + b) (1 + c) (1 + d)/2^4 >= sqrt (abcd)
Thus minimum is 16.
-Vicks
ps: let me know if u know a faster way to solve this.

Vicky

Unless Akamai or Stolyar have a good idea,yours is really the best analytical approach...very convincing and leaves no room for error.

as for me, I did it this way.

Since the answer choices are integers..i find it very likely that a,b,c,d will be integers

As a check...For example let a=0.5 ,b=2 ,c=2 , d=0.5

a*b*c*d=0.5*2*2*0.5 =1 ...here we have integers and non integers..and we satsify the abcd =1

But (1+a)(1+b)(1+c)(1+d)= 1.5*3*3*1.5 =>> not an integer ...

Another one
3*1/3*2*1/2 =1
but 4*4/3*3*3/2 ==>>not an integer

So lets work with integers first

The answer cannot be 1 as the product is 1 + something ...and the something is positive

4= 1*1*2*2 ..compare with (1+a) (1+b)(1+c)(1+d)

We get a=0 and b =0 ..not possible..as abcd = 1....doesnt satisfy

18= 1*2*3*3 again comparison gives a=0...not possible

16= 2*2*2*2 ...comparison gives a=1,b=1,c=1,d=1...its the only one that holds true.

thanks
praetorian
CEO
Joined: 15 Aug 2003
Posts: 3454

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18 Sep 2003, 01:43
wonder_gmat wrote:
Vicky wrote:
We use the property that:
Arithemetic Mean of two numbers is g.t.e.q Geometric mean.
g.t.e.q = greater than or equal to.

Thus (1+a)/2 >= sqrt(a)
(1+b)/2 >= sqrt(b)
--------
(1+d)/2 >= sqrt(d)

Multiplying above equations (we can mutilply them without chanding sign of inequality because it is given that a,b,c & d are postive real num's)

we get (1 + a) (1 + b) (1 + c) (1 + d)/2^4 >= sqrt (abcd)
Thus minimum is 16.
-Vicks
ps: let me know if u know a faster way to solve this.

I'm kinda confused as to why you're using Geometric Mean here provided you don't know if these numbers are in sequence or not.

Wonder

I think Geometric Mean and Geometric Progression are two different things.
Senior Manager
Joined: 22 Aug 2003
Posts: 257
Location: Bangalore

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18 Sep 2003, 01:44
Arithemtic mean (average of two numbers) and Geometric means are properties (or rather expressions) for numbers. Numbers dont necesarily have to be in sequence to use them.
I guess u are confusing them with Arithmetic and geometric progression & for them, YES the numbers should be in sequence.
-Vicks
18 Sep 2003, 01:44
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