Two persons A and B lent certain amount at the same rate of interest for two and three years respectively under compound interest. If their final amounts are in the ratio of 3:5, B’s amount at the end of the first year being Rs.8,000 and A earned an interest of Rs.480 for the first year, then find the ratio of their principal amounts.

Let the rate of interest be R% p.a. Let the principal for A be P_a and for B be P_b. For the first year, CI = SI. Interest earned by A in 1st year = P_a × R/100 = 480 ...(i) B's amount after 1st year = P_b (1 + R/100) = 8000 ...(ii) Final Amounts: Amount for A (after 2 years) = P_a (1 + R/100)² Amount for B (after 3 years) = P_b (1 + R/100)³ Their ratio is given as 3:5: [P_a (1 + R/100)²] / [P_b (1 + R/100)³] = 3/5 => P_a / [P_b (1 + R/100)] = 3/5 ...(iii) From (ii), P_b (1 + R/100) = 8000. Substitute in (iii): P_a / 8000 = 3/5 => P_a = (3/5)×8000 = 4800 From (i): 4800 × R/100 = 480 => R = (480 × 100)/4800 = 10% From (ii): P_b (1 + 10/100) = 8000 => P_b × 1.1 = 8000 => P_b = 8000/1.1 = 7272.73 (approx.) The ratio P_a : P_b = 4800 : 7272.73 ≈ 4800 : (8000/1.1) = 4800 * 1.1 : 8000 = 5280 : 8000 = 528 : 800 = 66 : 100 = 33 : 50. Thus, the ratio is 33:50.