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(a) 33 years

(b) 40 years

(c) 45 years

(d) 50 years

Answer

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Let P be the principal amount which compounds at a rate of interest R.

It is given that it takes 15 years for the money to get doubled.

Thus, after 15 years, future value = 2(Principal amount)

$ \Rightarrow $ FV = 2P

So, we will substitute FV = 2P and n = 15 in the formula for future value.

$ \Rightarrow $ 2P = $ {{\left( 1+\dfrac{R}{100} \right)}^{15}}\times $ P

$ \Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{15}} $ = 2……(1)

Now, we need to find the number of years required for the future value to be 8 times the principal amount.

Let the number of years be n.

We will substitute future value = 8(principal amount)

Thus, FV = 8(P)

$ \Rightarrow $ 8P = $ {{\left( 1+\dfrac{R}{100} \right)}^{n}}\times $ P

$ \Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{n}} $ = 8

$ \Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{n}}={{2}^{3}} $

From (1), we know that $ {{\left( 1+\dfrac{R}{100} \right)}^{15}} $ = 2

$ \begin{align}

& \Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{n}}={{\left[ {{\left( 1+\dfrac{R}{100} \right)}^{15}} \right]}^{3}} \\

& \Rightarrow {{\left( 1+\dfrac{R}{100} \right)}^{n}}={{\left( 1+\dfrac{R}{100} \right)}^{45}} \\

\end{align} $

$ \Rightarrow $ n = 45

Therefore, it takes 45 years for the principal amount to become 8 times.