Mathematics for FinanceMTP Dec 23 Series IQuestion 3950 of 507
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A sum of money invested in compounded interest doubles itself in four years. In how many years it becomes 32 times of itself as the same rate of compound interest?

Options

A12 years
B16 years
C20 years
D24 years
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Correct Answer

Option c20 years

All Options:

  • A12 years
  • B16 years
  • C20 years
  • D24 years

Detailed Solution & Explanation

Let the principal amount be P\displaystyle P and the rate of interest per annum be i\displaystyle i. Under compound interest, the amount A\displaystyle A after t\displaystyle t years is: A=P(1+i)tA = P(1+i)^t Given that the sum of money doubles itself in 4\displaystyle 4 years: 2P=P(1+i)4    (1+i)4=22P = P(1+i)^4 \implies (1+i)^4 = 2 We want to find the number of years t\displaystyle t in which the sum will become 32\displaystyle 32 times of itself: 32P=P(1+i)t    (1+i)t=3232P = P(1+i)^t \implies (1+i)^t = 32 We can express 32\displaystyle 32 as a power of 2\displaystyle 2: 32=2532 = 2^5 Substituting (1+i)4=2\displaystyle (1+i)^4 = 2 into the equation: (1+i)t=(2)5=((1+i)4)5=(1+i)20(1+i)^t = (2)^5 = \left((1+i)^4\right)^5 = (1+i)^{20} Comparing the exponents: t=20 yearst = 20 \text{ years} Thus, the sum becomes 32\displaystyle 32 times of itself in 20\displaystyle 20 years. Hence, **Option C** is the correct answer.

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