Question 55652
reciprocal of x = the number which multiplied by x gives 1.

Therefore, we can express the reciprocal of x as {{{1/x}}}.

So now, let's assign the variables to the values:

First number = 2x+1
Second number = 2x+3
Reciprocal of first = {{{1/(2x+1)}}}
Reciprocal of second = {{{1/(2x+3)}}}

The assugnment of the two numbers requires ome explanation. As the problem requires the two numbers to be odd, we use a little trick (that 2x is always even), to determine that 2x+1 and 2x+3 will be always odd.

{{{1/(2x+1) + 1/(2x+3) = 8/15}}}
Take the lcm of the denominators (it's {{{15(2x+1)(2x+3)}}}):
{{{15(2x+3) + 15(2x+1) = 8(2x+3)(2x+1)}}}
{{{30x + 45 + 30x + 15 = 32x^2 + 64x + 24}}}
{{{60x + 60 = 32x^2 + 64x + 24}}}
{{{32x^2 + 4x - 36 = 0}}}
*[invoke quadratic "x", 32, 4, -36 ]

Since 1.125 is not an integer, x = 1, first = 3 and second = 5.