Compound Interest

Steps to Solve

1. 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.

2. 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 )

3. 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} ).

4. 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.