The ratio of the sum of two numbers to the difference of the two numbers is 9. the product of the two numbers is 80. find the numbers.

There are two equations here:

(a+b)/(a-b) = 9 and ab=80

the first one simplifies to 8a = 10b or a = (4/5)b.

You can plug this last part in for 'a' in the second equation:

(4/5)b*b = (4/5)b^2 = 80 which means b^2 = 64 or b = +8 or -8.

I guess the question asks for the positive solution, so then you will find that a = 10 if you plug the value of b in for one of the original equations. Then your final answer would be a = 10 and b = 8. But, there is another solution, I believe, that is a = -10 and b = -8, since ab would still equal 80 and (a+b)/(a-b) would be -18/-2 which still equals 9. Notice that either a and b are both positive or both negative, because this is the only time that the product of both numbers would be also positive.