Secret Numbers

Algebra Level 2

I have a secret set of 3 integers. If I add every possible pairing of them, I get the numbers $0,\ 4,\ 10.$ What is the absolute difference between the largest two of my secret numbers?

The answer is 4.

3 solutions

Jason Dyer Staff
Oct 24, 2016

Two of the numbers must be $a$ and $-a$ to add up to 0.

It must be true that

$-a + b = 4$

$a + b = 10$

Adding these obtains $2b = 14$ so that $b = 7 .$ Using the original equations, $a = 3$ and $-a = -3 .$ So the absolute difference between the larger numbers is $7 - 3 = 4 .$

Kai Ott
Nov 2, 2016

Let the numbers be $a, b, c$ respectively with $c \geq b \geq a$ . Then

$a+b=0 (I)$

$a+c=4 (II)$

$b+c=10 (III)$ $(II)-(I): c + a - (b + a) = c - b = 4$

Nice identification of the variables for the equation!

Calvin Lin Staff - 4 years, 7 months ago

Wait,I have a question,if a is smaller than b and c,why c-b could be the biggest difference?

Gary Jiang - 4 years, 5 months ago

Note that the question asks for the "absolute difference between the largest two of my secret numbers".

It does not ask for the "largest absolute difference between two of my secret numbers".

Calvin Lin Staff - 4 years, 5 months ago

Roy Bertoldo
Nov 29, 2016

Solve the following simultaneous equations:

x+y=0;

x+z=4;

y+z=10.

Soln: x=-3, y=3, z=7

Largest pair in (x,y,z): 7, 3

Difference betwn elements in largest pair: 7-3=4

Nice and way

Yash Pagare - 4 years, 5 months ago

Secret Numbers

The answer is 4.

3 solutions

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