If a, b, c and d are four different positive integers selected from 1

Question

If a, b, c and d are four different positive integers selected from 1 to 25, then the highest possible value of  \frac{(a+b)+(c+d)}{(a+b)+(c−d)}  would be:

A. 47
B. 49
C. 51
D. 96
E. 101

chetan2u · Accepted Answer

Great question  Bunuel 
For max value make the denominator the least.

a+b+c-d is to be the least.  Make d the maximum as it is getting added in numerator and remaining a+b+c =d+1 
So d=25 and a+b+c=25+1=26

\frac{(a+b)+(c+d)}{(a+b)+(c−d)}

\frac{(a+b+c)+d}{(a+b+c)−d}

\frac{26+25}{26-25}=\frac{51}{1}=51

C

gmatophobia · Answer

\frac{(a+b)+(c+d)}{(a+b)+(c−d)}

\frac{(a+b)+(c+d)}{(a+b)-(d-c)}

We need to make the denominator the least for the value of the fraction to be greatest.

Before we proceed with further analysis, there are few things to that we can observe.

All the answers choices are positive integers so the numerator has to be a multiple of the denominator. 
 The numerator will always be positive integer, so the denominator also has to be positive integer.  
 We are subtracting (d - c) from (a + b), we need to keep the net result positive, at the same time ensure that the denominator is least. At this point, we can conclude that the denominator is 1. Even if we cannot say for sure, its always worth a try to see if the denominator can be 1.

We need to maximize d - c, so that we subtract the maximum possible from (a+b).

Let's take d = 25 & c = 1. d - c = 24

Can we have a + b = 25 ? yes when a = 13 and b = 12

So the denominator = 1

Numerator = (13 + 12) + (25 + 1) = 26 + 25 = 51.

Option C

sumit99kr · Answer

(a+b+c+d)/(a+b+c-d) can be written as {(a+b+c-d)+2d}/(a+b+c-d) which can be further split into (a+b+c-d)/(a+b+c-d) + 2d/(a+b+c-d) 
Finally we get 1 + 2d/(a+b+c-d).
Now lets focus on {d/(a+b+c-d)}, for this expression to be highest "d" should be max, hence d=25 and the denominator sould be minimum i.e, 1, which can be when a+b+c = 26 
hence, 1 + 2d/(a+b+c-d) = 1 + 2*25/1 = 1+50= 51.

Hence C is the correct choice.

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If a, b, c and d are four different positive integers selected from 1

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If a, b, c and d are four different positive integers selected from 1

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