Question 48155
Let n be the required number.

{{{n+(1/n) = 10/3}}} Simplift this.
{{{(n^2+1)/n = 10/3}}} Multiply both sides by n.
{{{n^2+1 = 10n/3}}} Multiply both sides by 3.
{{{3n^2+3 = 10n}}} Subtract 10n from both sides.
{{{3n^2-10n+3 = 0}}} Solve this quadratic equation by factoring.
{{{3n^2-10n+3 = (3n-1)(n-3)}}} Apply the zero products principle.
{{{3n-1 = 0}}} and/or {{{n-3 = 0}}}
If {{{3n-1 = 0}}} then {{{3n = 1}}} and {{{n = 1/3}}}
If {{{n-3 = 0}}} then {{{n = 3}}}

The answer is there are two numbers that meet the given criteria:

{{{1/3}}} and {{{3}}}