Question

What are the values? (full question in Details)

The sum of five numbers is -1/4. The numbers include two pairs of opposites. The quotient of two values is 2. The quotient of to different values is -3/4. What are the values?

Answer 1

If you get this one, what do you win?
MULTIPLE SOLUTIONS:
`1/2, -1/2, 3/16, -3/16, -1/4`
or
`1/8, -1/8, 1/3, -1/3, -1/4`
(there are still more...)

Explanation:

...I had to look up "opposite numbers", which is embarrassing.

A number's opposite is the same distance from zero on the number-line, but in the other direction. 7's opposite is -7, for example.

So, if I understand it right, we have:

`a + (-a) + b + (-b) + c = -1/4`

We know the 2 pairs of opposites cancel each other out, so we can say that:
`c = -1/4`

Now for the quotients. We know that the quotient of a number divided by its opposite is -1, so to analyze the 2 quotients (2 and -3/4), we have to divide c/a or c/-a (or vice versa), and c/b or c/-b (or vice versa.

Let's say `a/c = 2` - this would make `a = 2 * (-1/4)`, so `a = -1/2 and -a = 1/2`

Okay, then. Let's say `b/c = -3/4`, so `b = -3/4 * (-1/4) = -3/16`, and then `-b = 3/16`

So `3/16, -3/16, 8, -8, and -1/4` meet the criteria and are a solution.

NOT THE ONLY SOLUTION.

Lets say `c/a = 2`, so `c/2 = a`, so `-1/(4*2) = -1/8 = a`.

Or, `c/b = -3/4`, so `c = -3/4b`, so `c(-4/3) = b`, so `-1/4(-4/3) = 4/12 = 1/3 = b`