# The mean of five numbers is -5. The sum of the positive numbers in the set is 37 greater than the sum of the negative numbers in the set. What could the numbers be?

One possible set of numbers is $- 20 , - 10 , - 1 , 2 , 4$. See below for restrictions on making further lists:

#### Explanation:

When we look at mean, we're taking the sum of the values and dividing by the count:

$\text{mean"="sum of values"/"count of values}$

We're told that the mean of 5 numbers is $- 5$:

$- 5 = \text{sum of values"/5=>"sum} = - 25$

Of the values, we're told the sum of the positive numbers is 37 greater than the sum of the negative numbers:

$\text{positive numbers"="negative numbers} + 37$

and remember that:

$\text{positive numbers "+" negative numbers} = - 25$

I'll use P for the positives and N for the negatives, then substitute in our first expression into the second:

$\left(N + 37\right) + N = - 25$

$2 N + 37 = - 25$

$2 N = - 62 \implies N = - 31$

which means:

$P - 31 = - 25 \implies P = 6$

And now we have all the restrictions we have to work within. We can have any number of positive numbers and negative numbers, so long as the total number of numbers is 5 and the values of the negatives equals $- 31$ while the values of the positives equals $6$.

One possible set of numbers is:

$- 20 , - 10 , - 1 , 2 , 4$