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**Hint:**Here, we will first calculate the rate percent using the given condition. Then, we will use that value of rate to find the sum that will amount to Rs. 450 in two years at the same rate. We will use the formula of simple interest which is given as $SI=\dfrac{P\times R\times T}{100}$.

**Complete step by step answer:**

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.

$SI=\dfrac{P\times R\times T}{100}.........\left( 1 \right)$

Here, P is principal, R is interest rate and T is time.

When one makes a payment on a simple interest loan, the payment first goes towards that month’s interest and the remainder goes towards the principal.

Here, let us assume that the principal in the first case is P and the rate of interest is R.

Since, it is given that the interest is one-fourth of the principal and time is given as 4 years.

Therefore, we can write from equation (1) that:

\[\begin{align}

& \dfrac{P}{4}=\dfrac{P\times R\times 4}{100} \\

& \Rightarrow \dfrac{1}{4}=\dfrac{R\times 4}{100} \\

& \Rightarrow R=\dfrac{100}{4\times 4}=6.25\% \\

\end{align}\]

Now, in the second case the total amount is given as Rs. 450.

If the principal in this case is x, then we can write:

$\begin{align}

& \text{Principal + Interest = Toal amount} \\

& \Rightarrow \text{x + interest = Rs}\text{.450 } \\

& \Rightarrow \text{Interest = Rs}\text{.450 - x } \\

\end{align}$

Therefore, we can write:

$\begin{align}

& 450-x=\dfrac{x\times 6.25\times 2}{100} \\

& \Rightarrow \left( 450-x \right)\times 100=12.5\times x \\

& \Rightarrow 45000-100x=12.5x \\

& \Rightarrow 112.5x=45000 \\

& \Rightarrow x=\dfrac{45000}{112.5}=400 \\

\end{align}$

The value of x comes out to be 400.

Therefore, the principal in this case is Rs. 400.

**So, the correct answer is “Option a”.**

**Note:**Students should note here that the total amount at the end is given as the interest added to the principal amount. So, we can write the equation $x+\text{interest=450}$.The formula for calculating simple interest must be remembered by the students.

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