Compound Interest Calculation

To determine the compound rate of interest, we need to understand the concept of compound interest and the formula used to calculate it.

Compound interest is the interest calculated on both the initial principal amount and the accumulated interest from previous periods. It differs from simple interest, which is calculated only on the principal amount.

The formula for compound interest is:

A = P(1 + r/n)^(nt)

Where:
- A is the future value of the investment/loan, including interest
- P is the principal amount (the initial amount)
- r is the annual interest rate (in decimal form)
- n is the number of times that interest is compounded per year
- t is the number of years

In this case, we are given that the amount will be 9 times its principal in 2 years. Let's assume the principal amount is P.

Step 1: Formulate the equation
Based on the given information, we can set up the equation as follows:

9P = P(1 + r/n)^(nt)

Step 2: Simplify the equation
To simplify the equation, we can divide both sides by P:

9 = (1 + r/n)^(nt)

Step 3: Solve for the compound rate of interest
To find the compound rate of interest, we need to solve the equation for r.

Taking the natural logarithm (ln) of both sides:

ln(9) = ln[(1 + r/n)^(nt)]

Using the logarithmic property, we can bring down the exponent:

ln(9) = nt ln(1 + r/n)

Next, divide both sides by nt:

ln(9) / nt = ln(1 + r/n)

Now, we can isolate r by multiplying both sides by n:

n * ln(9) / t = ln(1 + r/n)

Finally, we can solve for r by multiplying both sides by n and rearranging the equation:

r = n * [exp(n * ln(9) / t) - 1]

This formula gives us the compound rate of interest required to make the amount 9 times its principal in 2 years.