1. **State the problem:** We need to find the width of a rectangle given its area and length. 2. **Given:** - Area $= 1 \frac{5}{7}$ cm$^2$ - Length $= 3 \frac{3}{4}$ cm 3. **Formula:** The area of a rectangle is given by: $$\text{Area} = \text{Length} \times \text{Width}$$ 4. **Convert mixed numbers to improper fractions:** - $1 \frac{5}{7} = \frac{12}{7}$ - $3 \frac{3}{4} = \frac{15}{4}$ 5. **Set up the equation:** $$\frac{12}{7} = \frac{15}{4} \times \text{Width}$$ 6. **Solve for Width:** $$\text{Width} = \frac{\frac{12}{7}}{\frac{15}{4}} = \frac{12}{7} \times \frac{4}{15}$$ 7. **Multiply fractions:** $$\text{Width} = \frac{12 \times 4}{7 \times 15} = \frac{48}{105}$$ 8. **Simplify the fraction:** The greatest common divisor of 48 and 105 is 3. $$\frac{48}{105} = \frac{48 \div 3}{105 \div 3} = \frac{16}{35}$$ 9. **Final answer:** The width of the rectangle is $$\boxed{\frac{16}{35} \text{ cm}}$$