To solve this problem, we need to determine which son gets the maximum amount when two amounts are distributed among the sons P, Q, R, and S based on given ratios.

Step 1: Understanding the ratios

• The first amount is distributed in the ratio \(5 : 4 : 3 : 2\).
• The second amount is distributed in the ratio \(6 : 7 : 8 : 9\).
• The second amount is three times the first amount.

Step 2: Let the first amount be \(x\)

• The second amount is therefore \(3x\).

Step 3: Calculating the distribution for the first amount

• Total parts in the first ratio: \(5 + 4 + 3 + 2 = 14\).
• The shares from the first amount are calculated as follows:
  • Share of P from first amount = \(\frac{5}{14}x\)
  • Share of Q from first amount = \(\frac{4}{14}x\)
  • Share of R from first amount = \(\frac{3}{14}x\)
  • Share of S from first amount = \(\frac{2}{14}x\)

Step 4: Calculating the distribution for the second amount

• Total parts in the second ratio: \(6 + 7 + 8 + 9 = 30\).
• The shares from the second amount are calculated as follows:
  • Share of P from second amount = \(\frac{6}{30} \times 3x = \frac{18}{30}x\)
  • Share of Q from second amount = \(\frac{7}{30} \times 3x = \frac{21}{30}x\)
  • Share of R from second amount = \(\frac{8}{30} \times 3x = \frac{24}{30}x\)
  • Share of S from second amount = \(\frac{9}{30} \times 3x = \frac{27}{30}x\)

Step 5: Total shares for each son

• Total for P = \(\frac{5}{14}x + \frac{18}{30}x\)
• Total for Q = \(\frac{4}{14}x + \frac{21}{30}x\)
• Total for R = \(\frac{3}{14}x + \frac{24}{30}x\)
• Total for S = \(\frac{2}{14}x + \frac{27}{30}x\)

Step 6: Simplifying and comparing the totals

• To compare, calculate each total and it becomes evident:
• Total for S is \(\frac{2}{14}x + \frac{27}{30}x\)
• Total for S calculation yields a greater number compared to others when simplified, hence S gets the maximum amount.

Conclusion:

Based on the calculated totals, son S receives the maximum amount. Therefore, the correct answer is S.