Question

find two numbers whose sum is 25 and the sum of whose reciprocals is one sixth

Answer 1

Solve this pair of equations:
x + y = 25
1/x + 1/y = 1/6

Rewrite as
1/x + 1/(25-x) = 1/6
[(25-x) + x]/[x(25-x)] = 1/6
150 = x(25-x)
x^2 -25x + 150 = 0
(x-30)(x-5) = 0
One more step

Answer 2

I tried to finish drwls solution and verify the answers, but they would not check out.

After my second cup of coffee, I realized that my good collegue made what we used to call "wishful thinking solutions" error.

the factors would be (x-15)(x-10)
and we get symmetric solutions, that is, when x=15, y=10 and when x=10 y = 15

Answer 3

Correct, my mistake.

Answer 4

x+25x+150=0