1. **State the problem:** We have a square divided into four triangles P, Q, R, and S. The area of P is $\frac{1}{9}$ of the square's area. The area of Q is $\frac{1}{6}$ of the square's area. The total area of P and Q is 10 cm$^2$. We need to find: (a) The length of each side of the square. (b) The fraction of the square that triangle S occupies. 2. **Set up the known information:** Let the area of the square be $A$. Then: $$\text{Area}(P) = \frac{1}{9}A$$ $$\text{Area}(Q) = \frac{1}{6}A$$ $$\text{Area}(P) + \text{Area}(Q) = 10$$ 3. **Write the equation for total area of P and Q:** $$\frac{1}{9}A + \frac{1}{6}A = 10$$ 4. **Find a common denominator and combine:** The common denominator of 9 and 6 is 18. $$\frac{2}{18}A + \frac{3}{18}A = 10$$ $$\frac{5}{18}A = 10$$ 5. **Solve for $A$:** Multiply both sides by the reciprocal of $\frac{5}{18}$: $$A = 10 \times \frac{18}{5}$$ Show cancellation: $$A = 10 \times \frac{\cancel{18}}{\cancel{5}} \times \frac{\cancel{1}}{\cancel{1}} = 10 \times \frac{18}{5}$$ Simplify: $$A = 10 \times 3.6 = 36$$ 6. **Find the side length of the square:** Since the square's area is $A = s^2$, where $s$ is the side length, $$s = \sqrt{A} = \sqrt{36} = 6$$ 7. **Find the fraction of the square that S occupies:** The total area of the square is $A = 36$. The areas of P and Q combined are $\frac{5}{18}A$. The remaining area for R and S is: $$1 - \frac{5}{18} = \frac{18}{18} - \frac{5}{18} = \frac{13}{18}$$ Since the problem does not give the area of R, we cannot find S directly from the total. But the problem states the square is divided into four triangles P, Q, R, and S. Assuming the four triangles cover the entire square without overlap, $$\text{Area}(P) + \text{Area}(Q) + \text{Area}(R) + \text{Area}(S) = 1 \times A$$ Given the problem's diagram and typical division, the fraction of S is: $$\text{Area}(S) = 1 - \text{Area}(P) - \text{Area}(Q) - \text{Area}(R)$$ Without $\text{Area}(R)$, we cannot find $\text{Area}(S)$ exactly. However, if the problem implies that $R$ and $S$ together fill the rest of the square, and no further info is given, the fraction of S cannot be determined uniquely. **Final answers:** (a) The side length of the square is $6$ cm. (b) The fraction of the square that S occupies cannot be determined with the given information.