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# If a,b,c and d are positive integers and their sum is 63, then find th

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Director
Joined: 19 Oct 2018
Posts: 705
Location: India
If a,b,c and d are positive integers and their sum is 63, then find th  [#permalink]

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04 Jun 2019, 19:27
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95% (hard)

Question Stats:

45% (02:22) correct 55% (02:17) wrong based on 76 sessions

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If a,b,c and d are positive integers and their sum is 63, then find the maximum value of $$ab+bc+cd$$?

A. 552
B. 652
C. 752
D. 991
E. 1041
Math Expert
Joined: 02 Aug 2009
Posts: 7764
Re: If a,b,c and d are positive integers and their sum is 63, then find th  [#permalink]

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04 Jun 2019, 19:59
nick1816 wrote:
If a,b,c and d are positive integers and their sum is 63, then find the maximum value of $$ab+bc+cd$$?

A. 552
B. 652
C. 752
D. 991
E. 1041

We should get two of them the highest, so let us take other two as 1 each..
a=d=1, as a and d are used only once in the calculations.

max values of b and c will be 31 and 30 in any order.

so $$ab+bc+cd=1*31+31*30+30*1=31+930+30=991$$

D
_________________
Intern
Joined: 04 Apr 2019
Posts: 10
Location: Spain
Re: If a,b,c and d are positive integers and their sum is 63, then find th  [#permalink]

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07 Jun 2019, 14:23
chetan2u wrote:
nick1816 wrote:
If a,b,c and d are positive integers and their sum is 63, then find the maximum value of $$ab+bc+cd$$?

A. 552
B. 652
C. 752
D. 991
E. 1041

We should get two of them the highest, so let us take other two as 1 each..
a=d=1, as a and d are used only once in the calculations.

max values of b and c will be 31 and 30 in any order.

so $$ab+bc+cd=1*31+31*30+30*1=31+930+30=991$$

D

How do you come to the conclusion that we have to take 2 of them the highest?
Intern
Joined: 04 Sep 2018
Posts: 20
If a,b,c and d are positive integers and their sum is 63, then find th  [#permalink]

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09 Jun 2019, 18:29
1
manpaslop wrote:
chetan2u wrote:
nick1816 wrote:
If a,b,c and d are positive integers and their sum is 63, then find the maximum value of $$ab+bc+cd$$?

A. 552
B. 652
C. 752
D. 991
E. 1041

We should get two of them the highest, so let us take other two as 1 each..
a=d=1, as a and d are used only once in the calculations.

max values of b and c will be 31 and 30 in any order.

so $$ab+bc+cd=1*31+31*30+30*1=31+930+30=991$$

D

How do you come to the conclusion that we have to take 2 of them the highest?

Exactly how do you get to know that we have to maximize two values out of four? I took 15,15,15 and 18 to solve it.
Director
Joined: 19 Oct 2018
Posts: 705
Location: India
Re: If a,b,c and d are positive integers and their sum is 63, then find th  [#permalink]

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10 Jun 2019, 11:40
We need to maximize the shaded portion area that is ab+bc+cd, hence we have to minimize the area of non-shaded part that is ad.
To minimize the ad, we have to assign minimum positive numbers to both a and d.

That's why he maximizes only 2 numbers (b and c).
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Intern
Joined: 03 Jun 2019
Posts: 18
Re: If a,b,c and d are positive integers and their sum is 63, then find th  [#permalink]

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05 Jul 2019, 03:11
chetan2u wrote:
nick1816 wrote:
If a,b,c and d are positive integers and their sum is 63, then find the maximum value of $$ab+bc+cd$$?

A. 552
B. 652
C. 752
D. 991
E. 1041

We should get two of them the highest, so let us take other two as 1 each..
a=d=1, as a and d are used only once in the calculations.

max values of b and c will be 31 and 30 in any order.

so $$ab+bc+cd=1*31+31*30+30*1=31+930+30=991$$

D

chetan2u

i though that if we have a sum of three number for example a+b+c=25, then the maximum product of ab+bc will be when 25 is equally divided between three number plus the extra one to the most occurring number. So a=8,b=9,c=8 and will give ab+bc as 144 and if we take 1,23,1 it will just be 46.

Just put this example to show an approach, not sure its right or not.

So thinking this in mind, i took this as a=15,b=17,c=16,d=15 but you proved that is wrong.

Could you please help in providing a view to approach such problems in 3 variables and 4 variables.
Re: If a,b,c and d are positive integers and their sum is 63, then find th   [#permalink] 05 Jul 2019, 03:11
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