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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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nick1816
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If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   04 Jun 2019, 19:27
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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
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nick1816
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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   10 Jun 2019, 11:40
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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.
Area of Non-shaded part=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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chetan2u
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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   04 Jun 2019, 19:59
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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 \(ab+bc+cd=1*31+31*30+30*1=31+930+30=991\)

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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   26 Mar 2024, 11:47
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ab+bc+cd=(a+c)(b+d)−ad

To maximize the objective, set ad=1 and make (a+c)×(b+d) as close to square as possible.

32×31−1=991

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If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   09 Jun 2019, 18:29
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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.
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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   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?
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Sreeragc
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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   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.
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rabjeet
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If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   27 Dec 2022, 03:53
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?


One hint could be to see that b and c are repeated twice in ab + bc + cd, whereas a and d are repeated only once.
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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink] New post   26 Mar 2024, 01:12
­bad question - pls remove
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Re: If a,b,c and d are positive integers and their sum is 63, then find th [#permalink]
26 Mar 2024, 01:12
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