Compound Interest
Understand the Problem
The image features the phrase 'Compound Interest'. The context suggests a focus on finance and mathematical calculations related to interest rates over time.
Answer
The total amount after 10 years is approximately $1628.89.
Answer for screen readers
The total amount after 10 years is approximately $1628.89.
Steps to Solve
- Define the formula for compound interest
The formula for calculating compound interest is
$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$
where:
- ( A ) = the amount of money accumulated after n years, including interest.
- ( P ) = the principal amount (the initial money).
- ( r ) = annual interest rate (decimal).
- ( n ) = number of times that interest is compounded per year.
- ( t ) = number of years the money is invested or borrowed.
- Substitute values into the formula
Identify the values to substitute into the formula from the problem context. For example, if the principal amount is $1000, the interest rate is 5% (0.05 in decimal), compounded annually (n = 1), for 10 years, then:
- ( P = 1000 )
- ( r = 0.05 )
- ( n = 1 )
- ( t = 10 )
- Calculate the amount ( A )
Substituting the values into the formula gives:
$$ A = 1000 \left(1 + \frac{0.05}{1}\right)^{1 \cdot 10} $$ $$ A = 1000 \left(1 + 0.05\right)^{10} $$ $$ A = 1000 \left(1.05\right)^{10} $$
Now calculate ( \left(1.05\right)^{10} ).
- Perform the calculations
Calculate the power:
$$ (1.05)^{10} \approx 1.62889 $$
Now multiply by the principal:
$$ A \approx 1000 \times 1.62889 \approx 1628.89 $$
So, the total amount after 10 years is approximately $1628.89.
The total amount after 10 years is approximately $1628.89.
More Information
This calculation illustrates how money can grow through compound interest, which allows interest to be earned on previously accumulated interest. Understanding this concept is crucial in finance for investing and saving.
Tips
- Forgetting to convert the interest rate from a percentage to a decimal before calculations.
- Not adjusting the number of compounding periods (n) based on the frequency of compounding (e.g., monthly vs. annually).
- Miscalculating power when raising to ( nt ).