1. **Problem statement:** The ratio of Mary's beads to Jane's beads is 4 : 7. The ratio of Jane's beads to Pauline's beads is 5 : 2. Mary has 6 more beads than Pauline. We need to find how many beads Jane has. 2. **Step 1: Express the ratios with variables.** Let Mary's beads = $4x$, Jane's beads = $7x$ (from the first ratio). From the second ratio, Jane's beads = $5y$, Pauline's beads = $2y$. 3. **Step 2: Equate Jane's beads from both ratios.** Since Jane's beads are the same, set $7x = 5y$. 4. **Step 3: Express $y$ in terms of $x$.** $$y = \frac{7x}{5}$$ 5. **Step 4: Write Mary's and Pauline's beads in terms of $x$.** Mary's beads = $4x$ Pauline's beads = $2y = 2 \times \frac{7x}{5} = \frac{14x}{5}$ 6. **Step 5: Use the condition that Mary has 6 more beads than Pauline.** $$4x - \frac{14x}{5} = 6$$ 7. **Step 6: Solve for $x$.** $$\frac{20x}{5} - \frac{14x}{5} = 6$$ $$\frac{6x}{5} = 6$$ $$6x = 30$$ $$x = 5$$ 8. **Step 7: Find Jane's beads.** Jane's beads = $7x = 7 \times 5 = 35$ **Final answer:** Jane has 35 beads.