1. **Stating the problem:** Simplify the expression $$\frac{R \times \frac{R}{3}}{R + \frac{R}{3}}$$. 2. **Formula and rules:** When dividing fractions or expressions, remember that dividing by a fraction is the same as multiplying by its reciprocal. Also, combine like terms in the denominator. 3. **Simplify numerator:** $$R \times \frac{R}{3} = \frac{R^2}{3}$$ 4. **Simplify denominator:** $$R + \frac{R}{3} = \frac{3R}{3} + \frac{R}{3} = \frac{4R}{3}$$ 5. **Rewrite the expression:** $$\frac{\frac{R^2}{3}}{\frac{4R}{3}}$$ 6. **Divide by a fraction by multiplying by its reciprocal:** $$\frac{R^2}{3} \times \frac{3}{4R}$$ 7. **Simplify:** $$\frac{R^2 \times 3}{3 \times 4R} = \frac{R^2}{4R}$$ 8. **Cancel common terms:** $$\frac{R^2}{4R} = \frac{R}{4}$$ **Final answer:** $$\frac{R}{4}$$