Question 131457
Call the larger number {{{a}}}
Call the smaller number {{{b}}}
{{{a - b = 4}}}
{{{a = b + 4}}}
{{{(1/b) - (1/a) = 4/21}}}
Notice that the reciprocal of a smaller 
number will be larger than the reciprocal
of a larger number
Substituting:
{{{(1/b) - (1/(b+4)) = 4/21}}}
multiply both sides by {{{b*(b+4)*21}}}
{{{21*(b+4) - 21*b = 4*b*(b+4)}}}
{{{21b + 84 - 21b = 4*b^2 + 16b}}}
{{{4*b^2 + 16b - 84 = 0}}}
{{{b^2 + 4b - 21 = 0}}}
solve by completing the square
{{{b^2 + 4b + (4/2)^2 = 21 + (4/2)^2}}}
{{{b^2 + 4b + 4 = 25}}}
{{{(b + 2)^2 = 25}}}
{{{b + 2 = 5}}} (only the + square root makes sense here)
{{{b = 3}}}
{{{a - b = 4}}}
{{{a - 3 = 4}}}
{{{a = 7}}}
The numbers are 7 and 3
check:
{{{(1/b) - (1/a) = 4/21}}}
{{{(1/3) - (1/7) = 4/21}}}
{{{(7/21) - (3/21) = 4/21}}}
{{{7 - 3 = 4}}}
{{{4 = 4}}}
OK