Question 57369
Let the two numbers be a and b.

a = 5b  One number is 5 times another.

{{{1/a + 1/b = 6/35}}} The sum of their reciprocals is {{{6/35}}} Simplifying this:
{{{(a+b)/ab = 6/35}}} Cross-multiply.
{{{35a + 35b = 6ab}}} Substitute a = 5b.
{{{175b + 35b = 30b^2}}} Simplify.
{{{210b = 30b^2}}}
{{{30b^2 - 210b = 0}}} Factor out a b.
{{{b(30b - 210) = 0}}} Apply the  zero product principle.
{{{b = 0}}} and/or {{{30b - 210 = 0}}}
{{{b = 0}}} Discard this solution as not meaningful.
{{{30b - 210 = 0}}} Add 210 to both sides.
{{{30b = 210}}} Divide both sides by 30.
{{{b = 7}}} 
{{{a = 5b}}}
{{{a = 35}}}

The two numbers are:
7 and 35

Check:

{{{1/7 + 1/35 = 5/35 + 1/35}}} = {{{6/35}}}
{{{35 = 5(7)}}}