To solve for compound interest, we use the formula:
[tex]\[CI = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right]\][/tex]
Here, [tex]\(CI\)[/tex] represents the compound interest, [tex]\(P\)[/tex] is the principal amount, [tex]\(R\)[/tex] is the annual rate of interest, and [tex]\(T\)[/tex] is the time in years.
Let's break down the steps to understand how this formula works:
1. Identify the Principal Amount (P): This is the initial sum of money that you have invested or loaned. For example, let’s assume [tex]\(P = 1000\)[/tex].
2. Determine the Rate of Interest (R): This is the percentage of interest you earn on the principal annually. Let's assume [tex]\(R = 5\% \)[/tex].
3. Specify the Time Period (T): This is the number of years for which the money is invested or borrowed. Let's assume [tex]\(T = 2\)[/tex] years.
4. Calculate the Compound Factor:
- First, we convert the interest rate from percentage to a decimal by dividing by 100:
[tex]\(\frac{R}{100} = \frac{5}{100} = 0.05\)[/tex]
- Then, add 1 to this decimal:
[tex]\(1 + 0.05 = 1.05\)[/tex]
5. Raise this compound factor to the power of time (T):
[tex]\((1.05)^2\)[/tex]
6. Subtract 1 from the result of step 5:
[tex]\((1.05)^2 - 1 = 1.1025 - 1 = 0.1025\)[/tex]
7. Multiply this result by the Principal (P):
[tex]\(1000 \times 0.1025 = 102.5\)[/tex]
Hence, the compound interest is given by:
[tex]\[CI = 102.5\][/tex]
So, the correct choice that represents the formula for calculating compound interest is:
[tex]\[c. \ P\left[\left(1+\frac{R}{100}\right)^T-1\right]\][/tex]
