# largest value

• Dec 31st 2008, 02:44 AM
largest value
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 ?
• Dec 31st 2008, 04:56 AM
Isomorphism
Quote:

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... :D