Answer

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**Hint:**Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments.

**Complete step by step solution**:

According to the question,

Let the initial amount be P and the rate be R%.

Then from the formula of Simple Interest we can write:

\[S.I = P \times \dfrac{R}{{100}} \times Time\]

Therefore the amount will be: $ = P + S.I$

$

\therefore this \Rightarrow A = P + S.I \\

\Rightarrow A = P + P \times \dfrac{R}{{100}} \times T \\

\Rightarrow A = P(1 + \dfrac{{RT}}{{100}}) \\

$

Now according to the question the sum P is increased 1.5 times in 3 years

Therefore $A = 1.5P$

$

this \Rightarrow A = 1.5P \\

\Rightarrow 1.5P = P(1 + \dfrac{{3R}}{{100}}) \\

\Rightarrow 1.5 = 1 + \dfrac{{3R}}{{100}} \\

\Rightarrow 0.5 = \dfrac{{3R}}{{100}} \\

\Rightarrow R = \dfrac{{50}}{3} \\

$

Now, in the second part the amount becomes 3 times of the sum in T years.

Therefore

$

A = 3P \\

\Rightarrow 3P = P(1 + \dfrac{{RT}}{{100}}) \\

\Rightarrow 3 = 1 + \dfrac{{\dfrac{{50}}{3}T}}{{100}} \\

\Rightarrow 2 = \dfrac{{50}}{{300}}T \\

\Rightarrow 2 = \dfrac{1}{6}T \\

\Rightarrow 12 = T \\

\Rightarrow T = 12years \\

$

Hence the required time period is equal to 12 years.

**Note:**Sometimes to create confusion time is converted in other units like in one part it is given in years but in other it is in months , so be careful about this to convert everything in one unit and then proceed.

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