1. **State the problem:** Simplify the expression given: $$20f \frac{2}{3} \div 1 \frac{2}{3} + 3 \quad \text{and}\quad \frac{1}{2} + 3 \frac{1}{2} \times 2 - 1 \frac{1}{2}$$ 2. **Convert mixed numbers to improper fractions:** - For $20f \frac{2}{3}$, since $20f$ is a notation, assume it means $20 \times f$, and $\frac{2}{3}$ is a fraction multiplying $f$: - Let's write $20f \frac{2}{3}$ as $20f \times \frac{2}{3} = \frac{40f}{3}$. - For $1 \frac{2}{3}$: $$1 \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3}$$ - For $3 \frac{1}{2}$: $$3 \frac{1}{2} = \frac{7}{2}$$ - For $1 \frac{1}{2}$: $$1 \frac{1}{2} = \frac{3}{2}$$ 3. **Rewrite the expression with improper fractions:** $$\frac{40f}{3} \div \frac{5}{3} + 3 \quad\text{and}\quad \frac{1}{2} + \frac{7}{2} \times 2 - \frac{3}{2}$$ 4. **Simplify the division:** Dividing by a fraction is the same as multiplying by its reciprocal: $$\frac{40f}{3} \times \frac{3}{5} = \frac{40f \times 3}{3 \times 5} = \frac{40f}{5} = 8f$$ So the first part simplifies to: $$8f + 3$$ 5. **Simplify the second part:** Calculate $\frac{7}{2} \times 2$: $$\frac{7}{2} \times 2 = 7$$ Now substitute back: $$\frac{1}{2} + 7 - \frac{3}{2}$$ Combine $\frac{1}{2} - \frac{3}{2} = -1$: $$-1 + 7 = 6$$ 6. **Therefore, the two simplified expressions are:** $$8f + 3$$ and $$6$$