Understanding Compound Interest vs Simple Interest Difference

This problem involves finding the principal amount ($P$) given the difference between compound interest (CI) and simple interest (SI) over a period of 3 years at a specific interest rate. The interest is compounded annually.

Given Information:

• Time period ($n$): 3 years
• Interest Rate ($R$): 10% per annum
• Difference between CI and SI for 3 years: ₹186

Calculating the Principal Amount

We can solve this by calculating the SI and CI separately and then using their difference, or by using a direct formula for the difference.

Method 1: Using the Formula for Difference

The formula for the difference between Compound Interest and Simple Interest for 3 years is:

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

Substitute the given values:

• Difference = ₹186
• $R$ = 10%

Plugging these into the formula:

$$ 186 = P \left( \frac{10}{100} \right)^2 \left( 3 + \frac{10}{100} \right) $$ $$ 186 = P (0.1)^2 \left( 3 + 0.1 \right) $$ $$ 186 = P (0.01) (3.1) $$ $$ 186 = P (0.031) $$

Now, solve for $P$:

$$ P = \frac{186}{0.031} $$ $$ P = \frac{186 \times 1000}{31} $$ $$ P = 6 \times 1000 $$ $$ P = 6000 $$

Method 2: Calculating SI and CI Separately

1. Calculate Simple Interest (SI) for 3 years:

   The formula for SI is $ SI = \frac{P \times R \times n}{100} $.

   $$ SI = \frac{P \times 10 \times 3}{100} = \frac{30P}{100} = 0.30P $$
2. Calculate Compound Interest (CI) for 3 years:

   The formula for the Amount ($A$) with CI is $ A = P \left(1 + \frac{R}{100}\right)^n $.

   $$ A = P \left(1 + \frac{10}{100}\right)^3 $$ $$ A = P (1 + 0.1)^3 $$ $$ A = P (1.1)^3 $$

   Calculate $(1.1)^3$:

   $$ (1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.21 \times 1.1 = 1.331 $$

   So, the Amount is $ A = 1.331P $.

   The Compound Interest is $ CI = A - P $.

   $$ CI = 1.331P - P = 0.331P $$
3. Find the difference between CI and SI:

   We are given that the difference is ₹186.

   $$ CI - SI = 186 $$ $$ 0.331P - 0.30P = 186 $$ $$ 0.031P = 186 $$

   Solve for $P$:

   $$ P = \frac{186}{0.031} $$ $$ P = 6000 $$

Conclusion

Both methods confirm that the principal amount is ₹6,000.