largest value

**mathaddict** · December 31st 2008, 02:44 AM

Tom has 4 cards . Each card has a different number written on it . The totals of any three cards are -8.5 , -6.5 , -5.5 , -3.5 .
What is the number on the card with the largest value ?

**Isomorphism** · December 31st 2008, 04:56 AM

Originally Posted by mathaddict

Tom has 4 cards . Each card has a different number written on it . The totals of any three cards are -8.5 , -6.5 , -5.5 , -3.5 .
What is the number on the card with the largest value ?

Let the 'i'th card have $a_i$ written on it.

Then $a_1 + a_2 + a_3 = -8.5, a_1 + a_2 + a_4 = -6.5, a_1 + a_3 + a_4 = -5.5, a_2 + a_3 + a_4 = -3.5$

Adding all the equations above gives us $3(a_1 + a_2 + a_3 + a_4) = - 24$ and thus $a_1 + a_2 + a_3 + a_4 = -8$ .

$a_1 + a_2 + a_3 + a_4 = -8$ and $a_1 + a_2 + a_3 = -8.5$ implies $a_4 = 0.5$ .

$a_1 + a_2 + a_3 + a_4 = -8$ and $a_1 + a_2 + a_4 = -6.5$ implies $a_3 = -1.5$ .

$a_1 + a_2 + a_3 + a_4 = -8$ and $a_1 + a_3 + a_4 = -5.5$ implies $a_2 = -2.5$ .

$a_1 + a_2 + a_3 + a_4 = -8$ and $a_2 + a_3 + a_4 = -3.5$ implies $a_2 = -4.5$ .

So now you can conclude...

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