System Of Equations Related Question

[dagr8est]

a<b<c<d represent four numbers that can be added in pairs in 6 different ways. If all of the sums are unique and the four smallest sums are 1, 2, 3, and 4, what are all possible values of d?

Here's what I did.

smallest
a+b = 1
a+c = 2
a+d = 3
b+c = 4
b+d = y
c+d = z
largest

(a+d)+(b+c) = 3+4
a+b+c+d = 7

(a+b)+(a+c)+(b+c) = 1+2+4
2a+2b+2c = 7
a+b+c = 7/2

Substitute into previous equation:
(9/2)+d = 7
d = 7-(7/2)
d = (14/2)-(7/2)
d = 7/2

The question asks for all possible values which implies that there is more than one answer. This is the only one answer I could come up with though. Anyone see another possible value for d?

 

[Guest]

I also would agree with your single answer approach.

 

[dagr8est]

Hmm, so 7/2 is the only possible value for d then? I got a quick peak at the answer key and I'm pretty sure I saw two values, but I didn't get to write them down because the contest guy had to leave. :roll:

 

[Guest]

ok when I think more on it.....

a + b = 1
a + c = 2
b + c = 3
a + d = 4
b + d =y
c + d = x

then
a + b +a + c = 1 + 2
a + (a + b + c) = 3

and
a + b + a + c + b + c = 1 + 2+ 3
2(a + b + c) =6
a + b + c = 3

sub this back into the first eqn.

a + 3 = 3
then a= 0

then b= 1,
c= 2
d= 4

how does this look

 

[dagr8est]

I didn't think about a+d > b+c but it works. I remember that the second answer was an integer so that looks correct. Thanks.

 

[Guest]

glad to help and it did make me think a little longer - thanks for the interesting problem.