Let the numbers be x and \( \Large \frac{1}{x} \)

Then, \( \Large x + \frac{1}{x} = \frac{10}{3} \)

=> \( \Large \frac{x^{2} + 1}{x} = \frac{10}{3} \)

=> \( \Large 3x^{2} - 10x + 3 = 0 \)

=> \( \Large 3x^{2} - 9x - x + 3 = 0 \)

=> \( \Large 3x \left(x - 3\right) - 1 \left(x-3\right) = 0 \)

=> \( \Large \left(3x - 1\right) \left(x - 3\right) = 0 \)

\( \Large x = \frac{1}{3}, x = 3 \)