Question - Solving Compound Interest with Quarterly Compounding
Solution:
The formula in the image is a compound interest formula where interest is compounded quarterly. The standard compound interest formula is:\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]Where:- $$ A $$ is the amount of money accumulated after n years, including interest.- $$ P $$ is the principal amount (the initial amount of money).- $$ r $$ is the annual interest rate (decimal).- $$ n $$ is the number of times that interest is compounded per year.- $$ t $$ is the time the money is invested for, in years.But in the formula provided in the image, it is shown as:\[ \text{Total amount} = 20,000 \left(1 + \frac{1}{1000}\right)^{20} \]Firstly, we need to relate this to the standard formula to find $$ r $$ and $$ n $$.Given that the interest is compounded quarterly, $$ n $$ should be 4 (because there are four quarters in a year).From the standard formula, we can adjust $$ \frac{r}{n} $$ to look like $$ \frac{1}{1000} $$, which implies:\[ \frac{r}{n} = \frac{1}{1000} \]\[ \frac{r}{4} = \frac{1}{1000} \]\[ r = \frac{1}{1000} \times 4 \]\[ r = \frac{4}{1000} \]So the annual interest rate $$ r $$ as a decimal is:\[ r = 0.004 \]To convert this to a percentage, we multiply by 100:\[ r \times 100 = 0.004 \times 100 = 0.4\% \]Now, we need to identify $$ t $$ from the formula provided in the image:The $$ 20 $$ in the exponent should represent $$ nt $$, which implies:\[ nt = 20 \]Since we already determined $$ n = 4 $$, we can solve for $$ t $$:\[ 4t = 20 \]\[ t = \frac{20}{4} \]\[ t = 5 \]Therefore, the annual interest rate $$ r $$ is 0.4% per year and the time $$ t $$ is 5 years.
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