1. **Problem:** Given that $(4a + 2b) : a = 19 : 4$ and $b : c = 7 : 3$, find the ratio $a : b : c$. 2. **Step 1: Express the first ratio as an equation.** $\frac{4a + 2b}{a} = \frac{19}{4}$ 3. **Step 2: Simplify the left side.** $\frac{4a + 2b}{a} = \frac{4a}{a} + \frac{2b}{a} = 4 + 2\frac{b}{a}$ 4. **Step 3: Set up the equation and solve for $\frac{b}{a}$.** $$4 + 2\frac{b}{a} = \frac{19}{4}$$ $$2\frac{b}{a} = \frac{19}{4} - 4 = \frac{19}{4} - \frac{16}{4} = \frac{3}{4}$$ $$\frac{b}{a} = \frac{3}{8}$$ 5. **Step 4: Express $b$ in terms of $a$.** $$b = \frac{3}{8}a$$ 6. **Step 5: Use the second ratio $b : c = 7 : 3$ to express $c$ in terms of $b$.** $$\frac{b}{c} = \frac{7}{3} \Rightarrow c = \frac{3}{7}b$$ 7. **Step 6: Substitute $b = \frac{3}{8}a$ into $c = \frac{3}{7}b$.** $$c = \frac{3}{7} \times \frac{3}{8}a = \frac{9}{56}a$$ 8. **Step 7: Write the ratio $a : b : c$ using a common denominator.** $$a : b : c = a : \frac{3}{8}a : \frac{9}{56}a$$ 9. **Step 8: Multiply all terms by 56 to clear denominators.** $$56a : 56 \times \frac{3}{8}a : 56 \times \frac{9}{56}a = 56a : 21a : 9a$$ 10. **Step 9: Simplify by dividing all terms by $a$.** $$56 : 21 : 9$$ **Final answer:** $$a : b : c = 56 : 21 : 9$$