The compound interest on a sum of money at 5% per annum for 3 years is Rs. 2522. What would be the simple interest on this sum at the same rate and for the same period ?

Options

A. Rs. 2500
B. Rs. 2400
C. Rs. 2450
D. Rs. 2350

Answer

To solve the problem, we need to find the simple interest on a sum of money given that the compound interest for the same sum, rate, and time is Rs. 2522 at a rate of 5% per annum for 3 years.

### Step-by-Step Solution:

**Step 1: Understand the relationship between Compound Interest (CI) and Simple Interest (SI)**
The formula for Compound Interest is given by:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
Where:
- \( CI \) = Compound Interest
- \( P \) = Principal amount (the initial sum of money)
- \( r \) = Rate of interest per annum
- \( n \) = Number of years

The formula for Simple Interest is:
\[ SI = \frac{P \times r \times n}{100} \]

**Step 2: Calculate the Principal Amount (P)**
We know the Compound Interest (CI) is Rs. 2522, the rate (r) is 5%, and the time (n) is 3 years. We can rearrange the CI formula to find P:
\[ 2522 = P \left(1 + \frac{5}{100}\right)^3 - P \]
\[ 2522 = P \left(1.05^3 - 1\right) \]

Calculating \( 1.05^3 \):
\[ 1.05^3 = 1.157625 \]
Thus,
\[ 2522 = P (1.157625 - 1) \]
\[ 2522 = P (0.157625) \]
Now, solving for P:
\[ P = \frac{2522}{0.157625} \]
Calculating P:
\[ P \approx 16000 \]

**Step 3: Calculate Simple Interest (SI)**
Now that we have the principal amount (P), we can calculate the Simple Interest using the formula:
\[ SI = \frac{P \times r \times n}{100} \]
Substituting the values:
\[ SI = \frac{16000 \times 5 \times 3}{100} \]
Calculating SI:
\[ SI = \frac{240000}{100} \]
\[ SI = 2400 \]

### Final Answer:
The Simple Interest on this sum at the same rate and for the same period is Rs. 2400.

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