A person invested a certain amount at simple interest at the rate of 6 per cent per annum earning Rs. 900 as an interest at the end of three years. Had the interest been compounded every year. How much more interest would he have earned on the same amount with the same interest after three years ?

Options

A. Rs.38.13
B. Rs25.33
C. Rs.55.08
D. Rs.35.30

Answer

To solve the problem step by step, we will first find the principal amount using the simple interest formula, then calculate the compound interest, and finally find the difference between the compound interest and simple interest.

### Step 1: Calculate the Principal Amount

The formula for Simple Interest (SI) is given by:

\[ SI = \frac{P \times R \times T}{100} \]

Where:
- \(SI\) = Simple Interest
- \(P\) = Principal Amount
- \(R\) = Rate of Interest (per annum)
- \(T\) = Time (in years)

Given:
- \(SI = 900\)
- \(R = 6\%\)
- \(T = 3\) years

We can rearrange the formula to find the principal \(P\):

\[ P = \frac{SI \times 100}{R \times T} \]

Substituting the values:

\[ P = \frac{900 \times 100}{6 \times 3} = \frac{90000}{18} = 5000 \]

### Step 2: Calculate the Compound Interest

The formula for Compound Interest (CI) is:

\[ CI = P \left(1 + \frac{R}{100}\right)^T - P \]

We already found \(P = 5000\), \(R = 6\%\), and \(T = 3\) years. Now we can substitute the values into the CI formula:

\[ CI = 5000 \left(1 + \frac{6}{100}\right)^3 - 5000 \]

Calculating \(1 + \frac{6}{100} = 1.06\):

\[ CI = 5000 \times (1.06)^3 - 5000 \]

Calculating \((1.06)^3\):

\[ (1.06)^3 \approx 1.191016 \]

Now substituting this back into the CI formula:

\[ CI = 5000 \times 1.191016 - 5000 \]

Calculating:

\[ CI \approx 5955.08 - 5000 = 955.08 \]

### Step 3: Find the Difference Between CI and SI

Now we find the difference between the compound interest and the simple interest:

\[ \text{Difference} = CI - SI = 955.08 - 900 = 55.08 \]

Thus, the amount of more interest earned with compound interest compared to simple interest after three years is:

\[ \boxed{55.08} \]

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