Calculate $0 . \dot{2} \times 0 . \dot{5} \dot{4}$ Give your answer as a fraction in its simplest form.

Step 1 ：\(x = 0.\overline{2}\)

Step 2 ：\(10x = 2.\overline{2}\)

Step 3 ：\(9x = 2\)

Step 4 ：\(x = \frac{2}{9}\)

Step 5 ：\(y = 0.\overline{54}\)

Step 6 ：\(100y = 54.\overline{54}\)

Step 7 ：\(99y = 54\)

Step 8 ：\(y = \frac{54}{99}\)

Step 9 ：\(y = \frac{6}{11}\)

Step 10 ：\(\frac{2}{9} \times \frac{6}{11} = \frac{2 \times 6}{9 \times 11}\)

Step 11 ：\(\boxed{\frac{12}{99}}\)

Step 12 ：\(\boxed{\frac{4}{33}}\)