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 ?

Printable View

- Dec 31st 2008, 02:44 AMmathaddictlargest 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 AMIsomorphism
Let the 'i'th card have $\displaystyle a_i$ written on it.

Then $\displaystyle 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 $\displaystyle 3(a_1 + a_2 + a_3 + a_4) = - 24$ and thus $\displaystyle a_1 + a_2 + a_3 + a_4 = -8$.

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

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

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

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

So now you can conclude... :D