Finding the Principal Using Interest Differences

This problem involves finding the initial amount of money (the principal) invested based on the difference between the compound interest earned and the simple interest earned over a period of three years.

Understanding the Concepts

• Simple Interest (SI): Interest calculated only on the initial principal amount.
• Compound Interest (CI): Interest calculated on the initial principal amount plus the accumulated interest from previous periods.
• Difference between CI and SI: For longer periods, compound interest typically yields more than simple interest. The difference arises because interest is earned on interest in CI.

Key Formula for 3 Years

The specific formula relating the difference between Compound Interest (CI) and Simple Interest (SI) for 3 years is:

Difference = $P \times (\frac{R}{100})^2 \times (3 + \frac{R}{100})$

Where:

• $P$ = Principal amount
• $R$ = Annual interest rate

Applying the Formula Step-by-Step

We are given:

• Difference (CI - SI) = ₹217
• Rate ($R$) = 10% per annum
• Time ($n$) = 3 years

Substitute the known values into the formula:

₹217 = $P \times (\frac{10}{100})^2 \times (3 + \frac{10}{100})$

Calculation Breakdown:

1. Simplify the rate fraction: $\frac{10}{100} = \frac{1}{10}$
2. Substitute the simplified rate back into the equation:

   ₹217 = $P \times (\frac{1}{10})^2 \times (3 + \frac{1}{10})$

3. Calculate the terms:

   ₹217 = $P \times (\frac{1}{100}) \times (\frac{30}{10} + \frac{1}{10})$

   ₹217 = $P \times (\frac{1}{100}) \times (\frac{31}{10})$

4. Multiply the fractions:

   ₹217 = $P \times \frac{31}{1000}$

5. Solve for Principal ($P$):

   $P = ₹217 \times \frac{1000}{31}$

6. Perform the division: $217 \div 31 = 7$

   $P = ₹7 \times 1000$

7. Final Principal Calculation:

   $P = ₹7,000$

Conclusion

Therefore, the principal amount is ₹7,000. This amount, when subjected to a 10% annual interest rate for three years, results in a difference of ₹217 between the compound interest and simple interest earned.