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 find the highest possible value of  \frac{(a+b)+(c+d)}{(a+b)+(c−d)} .

A. 47
B. 49
C. 51
D. 96
E. 120

chetan2u · Accepted Answer

Maybe it would be correct to mention that  a+b+c-d
eq{0} ..

We will make the denominator the least so that the fraction becomes the maximum...

So, to make the denominator 1, that is a+b+c-d=1, let us take the max values of a+b+c.
  As the max value is 25, take d as 25, making a+b+c=26  
 \frac{(a+b)+(c+d)}{(a+b)+(c−d)}=\frac{26+25}{26-25}=51

C

EncounterGMAT · Answer

chetan2u  sir, your solution is crisp and very clear. However, I have a doubt. From the solution, I am convinced that 51 is the only highest possible value. But in such min/max fraction questions, how to find out appropriate numbers quickly? Is it that we need to calculate the fraction in such a way that the denominator is always 1? Could you please share any tricks or tips related to such questions, because I literally spent more than 2-3 mins by testing numbers but in vain. Thank you for your time and consideration.

1616 · Answer

Hello  Bunuel ,

Do you have any way how to select the numbers to solve this/these kinda questions?

Regor60 · Answer

Since the objective is to maximize the expression, one needs to maximize the numerator and minimize the denominator.

Get rid of the parentheses since they don't matter and are designed to constrain your thinking:

a+b+c+d
_______

a+b+c-d

Now, to maximize the overall expression it makes sense to maximize d because that increases the numerator while at the same time decreases the denominator.

The maximum value d can take is 25, so the expression becomes:

a+b+c+25
________
a+b+c-25

Now we seem to be in a pickle because we'd like to maximize a,b and c to maximize the numerator, but that also maximizes the denominator, defeating the overall purpose.

Well, the smallest we can make the denominator without turning the whole expression negative is to make it equal to 1. Anything greater than 1 just turns the overall expression into a fraction, yes ?

So, making a+b+c = 26 renders the denominator 1 and at the same time the numerator 51.

Posted from my mobile device

1616 · Answer

Thanks for the explanation

bumpbot · Answer

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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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