# 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?

Oct 26, 2017

If you get this one, what do you win?
MULTIPLE SOLUTIONS:
$\frac{1}{2} , - \frac{1}{2} , \frac{3}{16} , - \frac{3}{16} , - \frac{1}{4}$
or
$\frac{1}{8} , - \frac{1}{8} , \frac{1}{3} , - \frac{1}{3} , - \frac{1}{4}$
(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:

$a + \left(- a\right) + b + \left(- b\right) + c = - \frac{1}{4}$

We know the 2 pairs of opposites cancel each other out, so we can say that:
$c = - \frac{1}{4}$

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 $\frac{a}{c} = 2$ - this would make $a = 2 \cdot \left(- \frac{1}{4}\right)$, so $a = - \frac{1}{2} \mathmr{and} - a = \frac{1}{2}$

Okay, then. Let's say $\frac{b}{c} = - \frac{3}{4}$, so $b = - \frac{3}{4} \cdot \left(- \frac{1}{4}\right) = - \frac{3}{16}$, and then $- b = \frac{3}{16}$

So $\frac{3}{16} , - \frac{3}{16} , 8 , - 8 , \mathmr{and} - \frac{1}{4}$ meet the criteria and are a solution.

NOT THE ONLY SOLUTION.

Lets say $\frac{c}{a} = 2$, so $\frac{c}{2} = a$, so $- \frac{1}{4 \cdot 2} = - \frac{1}{8} = a$.

Or, $\frac{c}{b} = - \frac{3}{4}$, so $c = - \frac{3}{4} b$, so $c \left(- \frac{4}{3}\right) = b$, so $- \frac{1}{4} \left(- \frac{4}{3}\right) = \frac{4}{12} = \frac{1}{3} = b$