If $10,000 is invested at 16% compounded quarterly, then what is the compound amount at the end of six years?
Understand the Problem
The question is asking to calculate the future value of an investment with compound interest. We need to use the compound interest formula considering the principal amount, interest rate, compounding frequency, and the investment period to determine the final amount.
Answer
$25,633.04
Answer for screen readers
$25,633.04
Steps to Solve
- Identify the given values
Principal amount $P = 10000$ Annual interest rate $r = 16% = 0.16$ Compounding frequency $n = 4$ (quarterly) Time in years $t = 6$
- Apply the compound interest formula
The formula for compound amount $A$ is given by: $$A = P(1 + \frac{r}{n})^{nt}$$
- Plug in the values
Substitute the given values into the formula: $$ A = 10000(1 + \frac{0.16}{4})^{(4 \times 6)} $$
- Simplify the expression
First, simplify the fraction inside the parentheses: $$ \frac{0.16}{4} = 0.04 $$ Now, simplify the expression inside the parentheses plus one: $$ 1 + 0.04 = 1.04 $$ Next, calculate the exponent: $$ 4 \times 6 = 24 $$ Now we have: $$ A = 10000(1.04)^{24} $$
- Calculate the final amount
Calculate $(1.04)^{24}$: $$ (1.04)^{24} \approx 2.563304 $$ Multiply this value by the principal amount: $$ A = 10000 \times 2.563304 \approx 25633.04 $$
$25,633.04
More Information
The investment grows to approximately $25,633.04 after six years with quarterly compounding. This shows the power of compound interest over time.
Tips
A common mistake is not dividing the annual interest rate by the number of compounding periods per year or not multiplying the number of years by the number of compounding periods per year. Also, rounding intermediate calculations too early can lead to inaccuracies in the final answer.
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