When the interest is compounded quarterly, the amount (A) is calculated by :
(P = Principal, r = rate of interest, n = time period)

Options

A. `A = P(1+ r/100)^n`
B. `A = P(1+ (r//4)/400)^(1//4n)`
C. `A = P(1+ r/400)^(4n)`
D. `A = P(1+ 4r/100)^(n/4)`

Answer

To calculate the amount (A) when the interest is compounded quarterly, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( P \) be the principal amount (the initial amount of money).
- Let \( r \) be the annual rate of interest (in percentage).
- Let \( n \) be the time period in years.

2. **Understand Quarterly Compounding**:
- Compounding quarterly means that the interest is calculated and added to the principal four times a year.
- Therefore, for each quarter, the interest rate will be divided by 4.

3. **Adjust the Rate of Interest**:
- The quarterly interest rate will be \( \frac{r}{4} \).

4. **Adjust the Time Period**:
- Since interest is compounded quarterly, the total number of compounding periods in \( n \) years will be \( 4n \) (because there are 4 quarters in a year).

5. **Write the Formula for Compound Interest**:
- The formula for the amount \( A \) when compounded quarterly is:
\[ A = P \left(1 + \frac{r}{400}\right)^{4n} \]
- Here, \( \frac{r}{400} \) is used because we convert the percentage into a decimal by dividing by 100, and since we are compounding quarterly, we divide by 4 again.

### Final Formula:
Thus, the final formula for the amount when interest is compounded quarterly is:
\[ A = P \left(1 + \frac{r}{400}\right)^{4n} \]