Understanding the Problem

This question asks us to find the annual rate of interest (R). We are given the principal amount (P) as Rs. 18000, the time period (n) as 2 years, and the difference between the compound interest (CI) and simple interest (SI) accrued is Rs. 405.

Key Formulas for Interest Calculation

To solve this problem, we use the standard formulas for Simple Interest (SI) and Compound Interest (CI). For a time period of 2 years, there is a direct formula relating the difference between CI and SI to the principal and the rate of interest.

• Simple Interest (SI):

  The formula for Simple Interest is:

  $SI = \frac{P \times R \times n}{100}$

  Where 'P' is the principal amount, 'R' is the annual interest rate (in percent), and 'n' is the time period in years.

• Compound Interest (CI):

  The formula for Compound Interest is:

  $CI = P \left(1 + \frac{R}{100}\right)^n - P$

  This calculates the total amount after 'n' years with compounding, and then subtracts the principal to find the interest earned.

• Difference between CI and SI for 2 years:

  A simplified formula exists for the difference between compound interest and simple interest specifically for a period of 2 years:

  $CI - SI = P \left(\frac{R}{100}\right)^2$

Determining the Interest Rate

We are provided with the following information:

• Principal (P) = Rs. 18000
• Time (n) = 2 years
• Difference between CI and SI = Rs. 405

Let's use the direct formula for the difference between CI and SI for 2 years:

$CI - SI = P \left(\frac{R}{100}\right)^2$

Substitute the given values into the formula:

$405 = 18000 \times \left(\frac{R}{100}\right)^2$

Our goal is to solve for the rate 'R'. First, we isolate the term containing 'R':

$\left(\frac{R}{100}\right)^2 = \frac{405}{18000}$

Now, we simplify the fraction $\frac{405}{18000}$. We can divide both the numerator and the denominator by common factors.

Divide by 5:

$\frac{405 \div 5}{18000 \div 5} = \frac{81}{3600}$

Next, divide by 9:

$\frac{81 \div 9}{3600 \div 9} = \frac{9}{400}$

So, the equation simplifies to:

$\left(\frac{R}{100}\right)^2 = \frac{9}{400}$

To find the value of $\frac{R}{100}$, we take the square root of both sides of the equation:

$\frac{R}{100} = \sqrt{\frac{9}{400}}$

$\frac{R}{100} = \frac{3}{20}$

Finally, we solve for R:

$R = \frac{3}{20} \times 100$

$R = 3 \times 5$

$R = 15\%$

Based on the given Principal and the Difference between CI and SI, the calculated rate of interest is 15% per annum.

Verifying the Provided Answer Option

The question includes options, and the indicated correct answer is 25%. Let's perform a verification calculation using R = 25% with the given Principal (P = Rs. 18000) and Time (n = 2 years) to understand the interest amounts.

1. Calculate Simple Interest (SI) for R = 25%:

$SI = \frac{P \times R \times n}{100} = \frac{18000 \times 25 \times 2}{100}$

$SI = 180 \times 50$

$SI = 9000$

The Simple Interest at 25% is Rs. 9000.

2. Calculate Compound Interest (CI) for R = 25%:

$CI = P \left(1 + \frac{R}{100}\right)^n - P$

$CI = 18000 \left(1 + \frac{25}{100}\right)^2 - 18000$

$CI = 18000 \left(1 + 0.25\right)^2 - 18000$

$CI = 18000 \left(1.25\right)^2 - 18000$

$CI = 18000 \times 1.5625 - 18000$

$CI = 28125 - 18000$

$CI = 10125$

The Compound Interest at 25% is Rs. 10125.

3. Calculate the Difference (CI - SI) for R = 25%:

$CI - SI = 10125 - 9000$

$CI - SI = 1125$

When the rate is 25%, the difference between compound and simple interest is Rs. 1125. This calculation shows how to verify any given rate using the standard SI and CI formulas.