Question: 2 - \frac{3}{5} \n----------\n1 \frac{3}{10} + 5
1. **State the problem:** Simplify the expression $$\frac{2 - \frac{3}{5}}{1 \frac{3}{10} + 5}$$.
2. **Convert mixed numbers to improper fractions:**
- Convert $$1 \frac{3}{10}$$ to an improper fraction:
$$1 \frac{3}{10} = \frac{10}{10} + \frac{3}{10} = \frac{13}{10}$$.
3. **Simplify the numerator:**
- Calculate $$2 - \frac{3}{5}$$.
- Express 2 as $$\frac{10}{5}$$ to have a common denominator:
$$2 = \frac{10}{5}$$.
- Subtract:
$$\frac{10}{5} - \frac{3}{5} = \frac{10 - 3}{5} = \frac{7}{5}$$.
4. **Simplify the denominator:**
- Add $$\frac{13}{10} + 5$$.
- Express 5 as $$\frac{50}{10}$$:
$$5 = \frac{50}{10}$$.
- Add:
$$\frac{13}{10} + \frac{50}{10} = \frac{63}{10}$$.
5. **Rewrite the entire expression:**
$$\frac{\frac{7}{5}}{\frac{63}{10}}$$.
6. **Divide the fractions:**
$$\frac{7}{5} \div \frac{63}{10} = \frac{7}{5} \times \frac{10}{63}$$.
7. **Multiply numerators and denominators:**
$$\frac{7 \times 10}{5 \times 63} = \frac{70}{315}$$.
8. **Simplify the fraction:**
- Find the greatest common divisor (GCD) of 70 and 315, which is 35.
- Cancel common factors:
$$\frac{\cancel{35} \times 2}{\cancel{35} \times 9} = \frac{2}{9}$$.
**Final answer:** $$\frac{2}{9}$$.