1. The problem is to subtract the mixed numbers $2\frac{2}{25}$ and $1\frac{1}{3}$.\n\n2. First, convert the mixed numbers to improper fractions.\n\nFor $2\frac{2}{25}$: $$2\frac{2}{25} = \frac{2 \times 25 + 2}{25} = \frac{50 + 2}{25} = \frac{52}{25}.$$\n\nFor $1\frac{1}{3}$: $$1\frac{1}{3} = \frac{1 \times 3 + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3}.$$\n\n3. Now subtract the improper fractions: $$\frac{52}{25} - \frac{4}{3}.$$\n\n4. Find the least common denominator (LCD) of 25 and 3, which is $75$.\n\n5. Convert both fractions to have denominator 75: $$\frac{52}{25} = \frac{52 \times 3}{25 \times 3} = \frac{156}{75},$$ $$\frac{4}{3} = \frac{4 \times 25}{3 \times 25} = \frac{100}{75}.$$\n\n6. Subtract the numerators: $$\frac{156}{75} - \frac{100}{75} = \frac{156 - 100}{75} = \frac{56}{75}.$$\n\n7. The fraction $\frac{56}{75}$ is already in simplest form because 56 and 75 have no common factors other than 1.\n\n8. Therefore, the final answer is $\boxed{\frac{56}{75}}$.