1. Problem: The patricians and plutocrats are in the ratio 2 to 13, and the total number is 315. Find the number of plutocrats. 2. Formula: If two quantities are in ratio $a:b$ and their total is $T$, then the parts are $\frac{a}{a+b} \times T$ and $\frac{b}{a+b} \times T$. 3. Here, $a=2$, $b=13$, and $T=315$. 4. Calculate the total parts: $$2 + 13 = 15$$ 5. Calculate the number of plutocrats: $$\frac{13}{15} \times 315$$ 6. Simplify the multiplication: $$\frac{13}{\cancel{15}} \times \cancel{315} = 13 \times 21 = 273$$ 7. Therefore, the number of plutocrats is **273**.