# In what time will the sum of Rs. 1600 at 5% p.a. CI amounts to Rs. 1764?

A. 1

B. 1.5

C. 2

D. 3

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**Hint:**

We know that the interest on the given principal is being compounded annually. So let use the formula of compound interest which is given below:

\[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\], where P is the principal amount, R is the rate of interest and T is the time taken.

**Complete step by step answer:**

It is given in the problem that the principal amount is Rs.1600, the rate of interest is 5% compounded annually and the total amount becomes Rs.1764.

We have to find the time taken in which the principal rises to the amount Rs.1774

According to the question, we know that the sum invested is Rs. 1600 at a rate of 5% compounded annually. Assume the principal amount as P and the rate of interest

$P = Rs.1600$and$R = 5\% p.a.$

We need to calculate the time in which the principal rises to the amount Rs. 1774

\[A{\text{ }} = {\text{ }}Rs.{\text{ }}1774\]

We have the formula of the amount is:

\[A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}\]

Substituting the values of P, R, and A we get,

$1764 = 1600{\left( {1 + \dfrac{5}{{100}}} \right)^T}$

Simplifying the above equation:

$ \Rightarrow \dfrac{{1764}}{{1600}} = {\left( {\dfrac{{21}}{{20}}} \right)^T}$

$ \Rightarrow \dfrac{{441}}{{400}} = {\left( {\dfrac{{21}}{{20}}} \right)^T}$

$ \Rightarrow {\left( {\dfrac{{21}}{{20}}} \right)^2} = {\left( {\dfrac{{21}}{{20}}} \right)^T}$

By the law of exponents, we know that when the bases are the same across the equal too, the powers are equal. Thus we have,

$T = 2$

Hence, the principal will take 2 years to reach the amount to Rs.1764.

**Therefore, option (C) is correct.**

**Note:**The simple interest is cheaper than the compound interest because the simple interest applies to the whole amount for the whole time but in the case of compound interest, we have to pay the interest on the interest.