Fraction Simplification D1592C

1. **State the problem:** Simplify the expression $$\frac{3.2 - \frac{3}{5}}{1 \frac{3}{10} + 5}$$. 2. **Convert mixed numbers and decimals to improper fractions:** - Convert $3.2$ to fraction: $3.2 = \frac{32}{10} = \frac{16}{5}$. - Convert $1 \frac{3}{10}$ to improper fraction: $1 \frac{3}{10} = \frac{10}{10} + \frac{3}{10} = \frac{13}{10}$. 3. **Rewrite the expression:** $$\frac{\frac{16}{5} - \frac{3}{5}}{\frac{13}{10} + 5}$$ 4. **Subtract the fractions in the numerator:** $$\frac{16}{5} - \frac{3}{5} = \frac{16 - 3}{5} = \frac{13}{5}$$ 5. **Add the fractions in the denominator:** Convert $5$ to fraction with denominator $10$: $5 = \frac{50}{10}$. $$\frac{13}{10} + \frac{50}{10} = \frac{63}{10}$$ 6. **Rewrite the expression:** $$\frac{\frac{13}{5}}{\frac{63}{10}}$$ 7. **Divide the fractions by multiplying numerator by reciprocal of denominator:** $$\frac{13}{5} \times \frac{10}{63}$$ 8. **Simplify before multiplying:** $$\frac{13}{\cancel{5}} \times \frac{10}{63} = \frac{13}{\cancel{5}} \times \frac{2 \times 5}{63}$$ Cancel $5$: $$\frac{13}{1} \times \frac{2}{63} = \frac{26}{63}$$ 9. **Final answer:** $$\frac{26}{63}$$