Question 24958
Let n = the required number. You can write:
{{{n + 1/n = 25/12}}} Simplify and solve for n.
{{{(n^2 + 1)/n = 25/12}}} Multiply both sides of the equation by n.
{{{n^2 + 1 = 25n/12}}} Multiply both sides by 12 and simplify.
{{{12n^2 + 12 = 25n}}} Subtract 25n from both sides.
{{{12n^2 - 25n + 12 = 0}}} Solve for n by factoring.
{{{(3n - 4)(4n - 3) = 0}}} Apply the zero products principle.
{{{3n - 4 = 0}}} and/or {{{4n - 3 = 0}}}
If {{{3n - 4 = 0}}} then {{{3n = 4}}} and {{{n = 4/3}}}
If {{{4n - 3 = 0}}} then {{{4n = 3}}} and {{{n = 3/4}}}

The answer is: There are two numbers that fit this situation.
{{{4/3}}} and {{{3/4}}}  Check:

{{{4/3 + 3/4 = (16 + 9)/12}}} = {{{25/12}}}
{{{3/4 + 4/3 = (9 + 16)/12}}} = {{{25/12}}}