Let \(x,y\) be the numbers.
\(\frac{1}{x} + \frac{1}{y} = 7\), let \(\frac{1}{x}\) be the larger reciprocal.
\(\frac{1}{x} = \frac{1}{y} + \frac{7}{3}\)
\(\frac{1}{y} + \frac{7}{3} + \frac{1}{y} = 7\)
\(\frac{2}{y} = \frac{14}{3}\)
\(\frac{3}{7} = y\)
\(\frac{1}{x} =\frac{7}{3} + \frac{7}{3}\)
\(\frac{1}{x} = \frac{14}{3}\)
\(\frac{3}{14} = x\)
Thus the numbers are \(\frac{3}{7} and \frac{3}{14}\)