Compound Interest Rate Calculation

The first step is to determine the annual interest rate (R) using the given information for yearly compounding.

• Principal (P): ₹ 14000
• Amount after 2 years (A1): ₹ 18515
• Time (t): 2 years
• Compounding: Yearly

We use the compound interest formula: $A = P \times (1 + \frac{R}{100})^t$.

$18515 = 14000 \times (1 + \frac{R}{100})^2$

$(1 + \frac{R}{100})^2 = \frac{18515}{14000}$

$(1 + \frac{R}{100})^2 = 1.3225$

Taking the square root on both sides:

$1 + \frac{R}{100} = \sqrt{1.3225}$

$1 + \frac{R}{100} = 1.15$

$\frac{R}{100} = 1.15 - 1$

$\frac{R}{100} = 0.15$

$R = 15\%$ per annum.

8-Monthly Compounding Calculation

Next, we calculate the compound interest for the same principal (P = ₹ 14000) and rate (R = 15% p.a.) over 2 years, but with 8-monthly compounding.

• Principal (P): ₹ 14000
• Annual Rate (R): 15%
• Time (t): 2 years
• Compounding Period: 8 months.

Calculate the number of compounding periods in 2 years:

Number of periods = Time in years $\times$ (12 months / Period in months)

Number of periods (n) = $2 \times (\frac{12}{8}) = 2 \times 1.5 = 3$.

Calculate the interest rate per period:

Rate per period = $\frac{\text{Annual Rate}}{\text{Number of periods per year}} = \frac{15\%}{1.5} = 10\%$.

Now, calculate the amount (A2) using the compound interest formula with these adjusted values:

$A2 = P \times (1 + \frac{\text{Rate per period}}{100})^n$

$A2 = 14000 \times (1 + \frac{10}{100})^3$

$A2 = 14000 \times (1.1)^3$

$A2 = 14000 \times 1.331$

$A2 = 18634$

Final Compound Interest Calculation

The compound interest (CI) is the difference between the final amount (A2) and the principal (P).

$CI = A2 - P$

$CI = 18634 - 14000$

$CI = ₹ 4634$