Certainly! Let's discuss the differences between simple and compound interest in detail:
Simple Interest:
1. Definition: Simple interest is calculated solely on the principal amount of the loan or investment.
2. Formula: The formula for simple interest is [tex]\( \text{SI} = P \times r \times t \)[/tex], where:
- [tex]\( \text{SI} \)[/tex] is the simple interest.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the rate of interest per period.
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for.
3. Payment: Interest is paid only on the original principal throughout the entire period. There is no compounding effect.
Compound Interest:
1. Definition: Compound interest is calculated on the principal amount as well as on the interest that has been added to the principal. This process is repeated over multiple periods.
2. Formula: The formula for compound interest is [tex]\( A = P \left(1 + \frac{r}{n}\right)^{nt} \)[/tex], where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n periods, including interest.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the annual interest rate.
- [tex]\( n \)[/tex] is the number of times that interest is compounded per unit [tex]\( t \)[/tex].
- [tex]\( t \)[/tex] is the time the money is invested or borrowed for.
3. Payment: Interest is paid on both the initial principal and the interest that accumulates on that principal. Thus, interest is "compounded" over time.
Comparative Analysis:
- Interest on Principal: Simple interest is paid only on the original principal amount, while compound interest is paid on both the original principal and on previously earned interest.
- Effect Over Time: Compound interest generally results in higher total interest payments compared to simple interest, especially as the number of compounding periods increases.
- Variability: Simple interest remains constant across each period, whereas compound interest grows over time due to the effect of compounding.
Given these differences, the first option is the correct one:
Simple interest is paid on the principal only; compound interest is paid on both the principal and on any interest already earned.
