Question 7718
Let n be the unknown number, then:

{{{n + 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 - 1)(n - 3) = 0}}} Apply the zero product principle.

If {{{3n - 1 = 0}}}, then {{{3n = 1}}}, and {{{n = 1/3}}} or
If {{{n - 3 = 0}}}, then {{{n = 3}}}

The two numbers are: 3 and 1/3

Check:

3 + 1/3 = 9/3 + 1/3 = 10/3

1/3 + 3/1 = 1/3 + 9/3 = 10/3