Mathematics for FinanceMTP June 24 Series IIQuestion 1414 of 512
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A man invests an amount of 15,860\displaystyle 15,860 in the names of his three sons A, B and C in such a way that they get the same interest after 2,3\displaystyle 2,3 and 4\displaystyle 4 years respectively. If the rate of interest is 5%\displaystyle 5\%, then the ratio of amount invested in the name of A, B and C is.

Options

A6:4:3\displaystyle 6:4:3
B3:4:6\displaystyle 3:4:6
C30:12:5\displaystyle 30:12:5
DNone of these
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Correct Answer

✅ Option a — 6:4:3\displaystyle 6:4:3

All Options:

  • A6:4:3\displaystyle 6:4:3
  • B3:4:6\displaystyle 3:4:6
  • C30:12:5\displaystyle 30:12:5
  • DNone of these

Detailed Solution & Explanation

Let the amounts invested in the names of A, B, and C be PA\displaystyle P_A, PB\displaystyle P_B, and PC\displaystyle P_C respectively. Given parameters: * Total investment = Rs. 15,860\displaystyle \text{Rs. }15,860 * Rate of interest (r\displaystyle r) = 5%\displaystyle 5\% p.a. Simple Interest * Times (tA,tB,tC\displaystyle t_A, t_B, t_C) = 2,3,\displaystyle 2, 3, and 4\displaystyle 4 years respectively The simple interests obtained from the three investments are: IA=PA×5×2100=0.10PAI_A = \frac{P_A \times 5 \times 2}{100} = 0.10 P_A IB=PB×5×3100=0.15PBI_B = \frac{P_B \times 5 \times 3}{100} = 0.15 P_B IC=PC×5×4100=0.20PCI_C = \frac{P_C \times 5 \times 4}{100} = 0.20 P_C Since the interests earned are equal: 0.10PA=0.15PB=0.20PC0.10 P_A = 0.15 P_B = 0.20 P_C Dividing the entire relation by 0.05\displaystyle 0.05 (or multiplying by 20\displaystyle 20): 2PA=3PB=4PC2 P_A = 3 P_B = 4 P_C Let 2PA=3PB=4PC=k\displaystyle 2 P_A = 3 P_B = 4 P_C = k. Then: PA=k2,PB=k3,PC=k4P_A = \frac{k}{2}, \quad P_B = \frac{k}{3}, \quad P_C = \frac{k}{4} The ratio of the investments (PA:PB:PC\displaystyle P_A : P_B : P_C) is: PA:PB:PC=k2:k3:k4P_A : P_B : P_C = \frac{k}{2} : \frac{k}{3} : \frac{k}{4} PA:PB:PC=12:13:14P_A : P_B : P_C = \frac{1}{2} : \frac{1}{3} : \frac{1}{4} Multiplying each term by the LCM of 2,3,\displaystyle 2, 3, and 4\displaystyle 4 (which is 12\displaystyle 12): PA:PB:PC=6:4:3P_A : P_B : P_C = 6 : 4 : 3 Thus, the ratio of the amounts invested is 6:4:3\displaystyle 6:4:3. Hence, **Option A** is the correct answer.

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