1. The problem is to subtract the mixed numbers $6 \frac{6}{10}$ and $3 \frac{7}{10}$.\n\n2. First, convert the mixed numbers to improper fractions.\n\nFor $6 \frac{6}{10}$: $$6 \times 10 + 6 = 60 + 6 = 66$$ so it becomes $$\frac{66}{10}$$.\n\nFor $3 \frac{7}{10}$: $$3 \times 10 + 7 = 30 + 7 = 37$$ so it becomes $$\frac{37}{10}$$.\n\n3. Now subtract the improper fractions: $$\frac{66}{10} - \frac{37}{10} = \frac{66 - 37}{10} = \frac{29}{10}$$.\n\n4. Convert the result back to a mixed number. Divide 29 by 10: quotient is 2 and remainder is 9, so $$\frac{29}{10} = 2 \frac{9}{10}$$.\n\n5. Therefore, the answer is $2 \frac{9}{10}$.