1. **State the problem:** Simplify the expression $$\frac{2}{1} \div \frac{1}{2} \times \left(\frac{2}{5} + \frac{1}{3}\right)$$ using BEDMAS (Brackets, Exponents, Division and Multiplication, Addition and Subtraction). 2. **Recall the rules:** - Division by a fraction is the same as multiplying by its reciprocal. - Addition of fractions requires a common denominator. - Multiplication and division are performed from left to right. 3. **Simplify inside the brackets first:** $$\frac{2}{5} + \frac{1}{3} = \frac{2 \times 3}{5 \times 3} + \frac{1 \times 5}{3 \times 5} = \frac{6}{15} + \frac{5}{15} = \frac{6 + 5}{15} = \frac{11}{15}$$ 4. **Rewrite the expression:** $$\frac{2}{1} \div \frac{1}{2} \times \frac{11}{15}$$ 5. **Convert division to multiplication by reciprocal:** $$\frac{2}{1} \times \frac{2}{1} \times \frac{11}{15}$$ 6. **Multiply the numerators and denominators:** $$\frac{2 \times 2 \times 11}{1 \times 1 \times 15} = \frac{44}{15}$$ 7. **Final answer:** $$\boxed{\frac{44}{15}}$$ This fraction is an improper fraction and can also be expressed as a mixed number: $$2 \frac{14}{15}$$.